Theory of Topological Spaces, 2nd ed..pdf

2000 , , R. Arens (1919—2000), A. Borel (1923—2003), J. J. Charatonik (1934—2004), B. Fitzpatrick (1931—1999), E. Hewitt (1920—1999), J. Isbell (1931—2005), F. B. Jones (1910—1999), M. Katětov (1918—1995), J. L. Kelley (1917—1999), K. Morita (1915— 1995), J. Nagata (1925—2007), R. H. Sorgenfrey (1915—1996), A. H. Stone (1916— 2000), M. H. Stone (1903—1998), J. W. Tukey (1915—2000), L. Vietoris (1891— 2002), A. Weil (1906—1998) . 20 F. Hausdorﬀ (1868—1942) , , 20 , , C. E. Aull R. Lowen [33, 34, 35] Handbook of the History of General Topology . . , [267, 268] . Encyclopedia of General Topology [180] , Open Problems in Topology [292, 333] , [332] , . 6 . . A. V. Arhangel’skiı̌ [259] : “ , , , .” , Arhangel’skiı̌ “ ” (1910—1988) (1919—2003) 20 70 . . . . . : . : . : . : . (). : ( ). : (). : ( ). : , 1994, 199−205. , 1998, 287−297. , 2000, 560−577. , 2002, 569−582. · iv · 1919 , , 20 , , , 20 70 , P. S. Alexandroﬀ (1896—1982) [5] A. V. Arhangel’skiı̌ [21] “ ” , , 30 . , F. Hausdorﬀ Grundzüge der Mengenlehre[181] , N. Bourbaki Topologie Générale[57] , K. Kuratowski Topologie[239] , J. L. Kelley General Topology[232] , [172] , J. Dugundji Topology[112] , S. Willard General Topology[411] , [233] , J. Nagata Modern General Topology[319] , R. Engelking General Topology [114] , [211] . , , . , 20 60 , , “ [142] . “”“ ” ” [24, 69, 166, 309] , , “Moore ” (R. L. Moore, 1882—1974) , , , “” , , . , . , . , , , . , , . , , . 90 , : , “” “” , “”, “” . . , , . “ ·v· ” ( 10571151) , . , . 2008 1 : 352100 E-mail: linshou@public.ndptt.fj.cn . 1979 , . , . . “” “ ” , . 60 80 , “” “ ” , , ! , 70 , , , . . ", , . “” . Arhangel’skiı̌ “” . , # . Σ , # . . , "!. " , $. . , , , ( #), , & , . %. ( ). , . . , , . ' , , . · viii · . , . 1999 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.......................................................................6 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 . . . . . . . . . . . . . . . . . . . . 27 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Tychonoﬀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 $# k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 # 3.6 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ·x· 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.1 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.1 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.4 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.5 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.6 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.6 Iso # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.1 Moore , Gδ % . . . . . . . . . . . . . . . . . . . . . . . . 199 7.2 w∆ M p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.3 σ Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.4 Mi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 . . . . . . . . . . . . . . . . 248 7.5 k , , 7.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 8 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 8.1 ℵ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 8.2 ℵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 8.3 cs cs-σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 8.4 σ · xi · k Lašnev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 8.5 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 0.1 (set) A, B (union) (intersection) (diﬀerence) A ∪ B = {x : x ∈ A x ∈ B}, A ∩ B = {x : x ∈ A x ∈ B}, / B}. A − B = {x : x ∈ A x ∈ “∈”“∈” / “”“”. (empty set) ∅ , A∩B = ∅ A B ; A − B = ∅ A ⊂ B, x ∈ A ⇒ x ∈ B. “⇒” “”. “⇔” “”. A ⊂ B A B (subset). A ⊂ B A = B A B (proper subset), A B. . , (family collection), , A , B . (index set), {Aγ }γ∈Γ , {Aγ : γ ∈ Γ }, Γ . {A1 , A2 , · · · , An , · · ·} , {An }n∈N , {An : n ∈ N}, N, {An } {An }∞ n=1 . γ∈Γ Aγ γ∈Γ Aγ ; ∞ n∈N An ( ∞ n=1 An ) n∈N An ( n=1 An ). , 0 . {Aγ }γ∈Γ X , B X , (i) B ∪ ( γ∈Γ Aγ ) = γ∈Γ (B ∪ Aγ ), B ∩ ( γ∈Γ Aγ ) = γ∈Γ (B ∩ Aγ ); (ii) X − ( γ∈Γ Aγ ) = γ∈Γ (X − Aγ ), X − ( γ∈Γ Aγ ) = γ∈Γ (X − Aγ ). (i) (distributive law), (ii) de Morgan (de Morgan formula). , X , B ⊂ X, X − B B X (complement). de Morgan = , = . X Y , X a Y b (ordinal pair) (a, b) X Y (product), ·2· X × Y = {(a, b) : a ∈ X, b ∈ Y }. X × Y R (relation), (a, b) ∈ R, aRb. f ⊂ X × Y X Y (mapping), x ∈ X, y ∈ Y , (x, y) ∈ f , y x , (x, y) ∈ f (x, y ) ∈ f ⇒ y = y . f : X → Y , x y f (x). f : x → f (x), x ∈ X, f (x) ∈ Y . A ⊂ X f (image) f (A) = {y : y = f (x), x ∈ A}. B ⊂ Y f (inverse image) (preimage) f −1 (B) = {x : f (x) ∈ B}. X, f (X) f . f : X → Y (injective mapping), f (x) = f (x ) ⇒ x = x ; (surjective mapping), f (X) = Y , f X Y (onto) . , (bijective mapping), (inverse mapping) f −1 : Y → X f −1 (y) = x ⇔ f (x) = y. f f (f −1 (B)) = B ∩ f (X) ⊂ B, f −1 (f (A)) ⊃ A. f , f (f −1 (B)) = B; f , f −1 (f (A)) = A. {Aγ }γ∈Γ , Γ γ∈Γ Aγ f (γ) ∈ Aγ (γ ∈ Γ ) f γ∈Γ Aγ , {Aγ }γ∈Γ (product of families). f ∈ γ∈Γ Aγ , f (γ) ∈ Aγ f γ (coordinate), f (γ) = xγ , γ∈Γ Aγ f xγ (γ ∈ Γ ) , {xγ }, {xγ }γ∈Γ ( ). γ∈Γ Aγ Aγ pγ {xγ } ∈ γ∈Γ Aγ , pγ ({xγ }) = xγ , γ∈Γ Aγ Aγ (projection). R ⊂ X × X X . X (equivalence relation), (i) x ∈ X, xRx (); (ii) xRy, yRx (); (iii) xRy yRz, xRz ( ). X = γ∈Γ Aγ , γ = γ ⇒ Aγ ∩ Aγ = ∅, {Aγ }γ∈Γ X (decomposition). X R X x, y Aγ , xRy; , X X RxRy γ ∈ Γ , x, y ∈ Aγ . 0.1 ·3· X < X (linear order) (total order), (i) x = y, x < y y < x ; (ii) x < y, y < x (); (iii) x < y, y < z, x < z. () < X (linearly ordered set) (totally ordered set) (chain), (X, <). x0 ∈ X X (minimum element)( (maximum element)), x ∈ X − {x0 }, x0 < x (x0 > x). X (well-ordered set), X . (X, <) (Y, < ) (order preserving), x, x ∈ X, x < x ⇒ f (x) < f (x ). X X (order) (partial order), (i) x ∈ X, x x; (ii) x y y x, x = y; (iii) x y y z, x z. (order set) (partially ordered set), (X, ). X <, x, y ∈ X xy⇔x α. < . ∅ 0, . , (ﬁnite ordinal number); (inﬁnite ordinal number). N , o(N) = ω, (minimal inﬁnite ordinal number). α, α + 1 α (successor), α α + 1 (predecessor). 0 (isolated ordinal), (limit ordinal). , X, Y X, Y , o(X) = o(Y ) ⇒ |X| = |Y |. α α , |α|. |α| ℵ0 , α (countable ordinal number); (uncountable ordinal number). ω () , |ω| = ℵ0 ; (minimal uncountable ordinal number) ω1 , |ω1 | = ℵ1 . αi < ω1 (i = 1, 2, · · ·), sup {αi : i = 1, 2, · · ·} < ω1 , αi < α i = 1, 2, · · · . α < ω1 , α, β , X, Y , o(X) = α, o(Y ) = β X ∩ Y = ∅. X ∪ Y x, y ∈ X ∪ Y , x, y ∈ X x, y ∈ Y , x, y ; x ∈ X, y ∈ Y , x < y. α + β = o(X ∪ Y ). α > 0, [0, α) α ( 0.7). α [0, α) , α, β > 0, α × β [0, α) [0, β) , α × β = {(x, y) : x ∈ [0, α), y ∈ [0, β)}. ·6· 0.3 (transﬁnite induction). α, P (α). , P (α) α (1) α = 0, P (0) ; (2) α < α0 α, P (α) , P (α0 ) . . (axiom of [114]. choice), (i) Zermelo . {Xγ }γ∈Γ , Γ γ∈Γ Xγ f γ ∈ Γ , f (γ) ∈ Xγ . (ii) Zermelo (). . (iii) Zorn . () X , X . A (ﬁnite character) , A A A A . (iv) Tukey . ( ⊂ ), A0 ∈ A , A ∈ A , A0 ⊂ A ⇒ A0 = A. 0.1 0 f : X → Y , A, B ⊂ X, C, D ⊂ Y . , “=” “⊂” ? ? “=” ? (i) f (A ∪ B) = f (A) ∪ f (B); (ii) f (A ∩ B) = f (A) ∩ f (B); (iii) f (A − B) = f (A) − f (B); (iv) f −1 (C ∪ D) = f −1 (C) ∪ f −1 (D); (v) f −1 (C ∩ D) = f −1 (C) ∩ f −1 (D); (vi) f −1 (C − D) = f −1 (C) − f −1 (D). 0.2 f : X → Y . A ⊂ X, C ⊂ Y , (i) f (A ∩ f −1 (B)) = f (A) ∩ B; (ii) f (A − f −1 (B)) = f (A) − B. 0.3 f : X → Y , A, C X, Y , f −1 (C) ⊂ A ⇔ C ⊂ Y − f (X − A). 0.4 A A α + n , α , , n 0 . 0.5 ω + 1 > ω, 1 + ω = ω, 0.6 0.7 0.8 {Xγ}γ∈Γ , Γ = ∅, . ω1 . α, [0, α) α. γ∈Γ Xγ =∅. 1 , (General Topology). . G. Cantor 1895 1897 , 20 F. Hausdorﬀ, M. Fréchet, C. Kuratowski , P. Urysohn, P. Alexandroﬀ A. [4] Tychonoﬀ . 1960 , P. Alexandroﬀ . C. E. Aull R. Lowen Handbook of the History of General Topology [33, 34, 35] . , . 1.1 1.1.1 , X, T X (O1) ∅ ∈ T , X ∈ T ; (O2) Ui ∈ T (i = 1, 2, · · · , n), n i=1 Ui ∈ T ; (O3) Uγ ∈ T (γ ∈ Γ ), ∪{Uγ : γ ∈ Γ } ∈ T , Γ , (X, T ) (topological space). T (topology), T (open set). X T , X . 1.1.2 T , T X , T ⊂ T , T T (coarse) T T (ﬁne). R ( ) R (usual topology). n (Euclidean space) Rn (y1 , y2 , · · · , yn ) n ρ(x, y) = (xi − yi )2 , i=1 , x = (x1 , x2 , · · · , xn ), y = ·8· 1 Sε (x) = {y : y ∈ Rn , ρ(x, y) < ε} ( x ε ), T = {U : U ⊂ Rn , x ∈ U ε > 0 Sε (x) ⊂ U } Rn , Rn (usual topology) (Euclidean topology). R Rn . X T1 ∅ X , T1 (indiscrete trivial topology), . T2 X , T2 (discrete topology), . (discrete space). 1.1.3 X , x ∈ X. X U , x, U x (neighborhood); U x ∈ U , U x (open neighborhood). R , . n Rn Sε (x) x (ε) , Sε (x) . Sε (x) Sε (x) . 1.1.1 U (x) x ∈ X , (N1) X ∈ U (x); (N2) U ∈ U (x), x ∈ U ; (N3) U ∈ U (x), V ⊃ U , V ∈ U (x); (N4) U, V ∈ U (x), U ∩ V ∈ U (x); x ∈ V ⊂ U x ∈ V, V ∈ U (x ). (N5) U ∈ U (x), V (N1)∼(N3) 1.1.3 , (N4) 1.1.3 (O2) . (N5). U ∈ U (x), ( 1.1.3), V x ∈ V ⊂ U , V x , V ∈ U (x ) (x ∈ V ). . 1.1.2 X U U . . , x ∈ U , 1.1.3, V (x), x ∈ V (x) ⊂ U , U = ∪{V (x) : x ∈ U }, U , 1.1.1 (O3) U . . 1.1.1 , . X x U (x) (N1)∼(N5), U (x) x , 1.1.2 , (N1), (N4) (N3) 1.2 ·9· 1.1.1 (O1), (O2) (O3), X . , . . 1.1.4 X F (closed set), X − F . 1.1.3 X F F (F1) ∅ ∈ F , X ∈ F ; (F2) Fi ∈ F (i = 1, 2, · · · , n), ni=1 Fi ∈ F ; (F3) Fγ ∈ F (γ ∈ Γ ), ∩{Fγ : γ ∈ Γ } ∈ F , Γ . (F3), . de Morgan X − ∩{Fγ : γ ∈ Γ } = ∪{X − Fγ : γ ∈ Γ }. Fγ , X − Fγ , 1.1.1, , . 1.1.4, ∩{Fγ : γ ∈ Γ } . . , (F1)∼(F3) X . . 1.1.4 F X (F1)∼(F3), F (X, T ) , T = {X − F : F ∈ F }. X A Gδ (Fσ ), A () () . 1.2 1.2.1 X U (, base), U ; V (, subbase), V . 1.2.1 U V x ∈ V , U ∈U, x∈U ⊂V. . R, (a, b) , a, b (a < b), a, b ; (a, +∞), (−∞, b) , a, b , . 1.2.2 U X , U (B1) U1 , U2 ∈ U x ∈ U1 ∩ U2 , U3 ∈ U , x ∈ U3 ⊂ U1 ∩ U2 ; · 10 · 1 (B2) ∪{U : U ∈ U } = X. , X , U X (B1) (B2), T = {∪V : V ⊂ U }, T (O1)∼(O3), X U . 1.2.1, , . ∅, (B2) (O1). T , (O3). T (O2), (B1) ( n − 1 ), . U1 , U2 , · · · , Un ∈ U x ∈ ni=1 Ui , U ∈ U , n x∈U ⊂ n Ui . (1.2.1) i=1 V1 , V2 , · · · , Vn ∈ T x ∈ i=1 Vi , x ∈ Vi (i = 1, 2, · · · , n). Vi U , Ui ∈ U x ∈ Ui ⊂ Vi . (1.2.1) , n n x ∈ U ⊂ i=1 Vi . i=1 Vi U , U ∈U, n (X, T ) . i=1 Vi ∈ T . T (O2). , U . V X , V (B2); , X , V X (B2), U V T , U (B1) (B2), V X , U , X V . 1.2.2 x X , x B(x) x (neighborhood base), x U , V ∈ B(x), x ∈ V ⊂ U. 1.2.3 B(x) X x , (NB1) B(x) = ∅; (NB2) U ∈ B(x), x ∈ U ; (NB3) U ∈ B(x) V ∈ B(x), W ∈ B(x), W ⊂U ∩V; (NB4) U ∈ B(x), V x ∈ V ⊂ U , x ∈ V , W ∈ B(x ) W ⊂ V . . , . 1.2.4 X , x ∈ X X B(x) (NB1)∼(NB4), U (x) = {U : U ⊃ V, V ∈ B(x)}, 1.2 · 11 · U (x) (N1)∼(N5). X U (x) , B(x) x . x . n Rn , x ε Sε (x) (ε > 0) x B(x), B(x) B (x) = {S1/n (x) : n = 1, 2, · · ·} x . ∪{B(x) : x ∈ Rn } ∪{B (x) : x ∈ Rn } Rn . , () (equivalent neighborhood base). 1.2.1 (Smirnov , B(x) = [372] ) R+ = [0, ∞) = {x : 0 x < +∞} {(x − ε, x + ε) ∩ R+ : ε > 0}, x = 0, {[0, ε) − {1, 1/2, 1/3, · · ·} : ε > 0}, x = 0, B(x) 1.2.3 (NB1)∼(NB4), R+ . R , Smirnov (Smirnov’s deleted sequence topology). 1.2.2 (Niemytzki [372] ) R R2 x , R = {(x, y) : y 0, −∞ < x < +∞}, B(x, y) = {Sε ((x, y)) ∩ R : ε > 0}, y = 0, {Sε ((x, ε)) ∪ {(x, 0)} : ε > 0}, y = 0, x . B((x, y)) (NB1)∼ (NB4), R . R2 , Niemytzki (Niemytzki’s half-plane topology) Niemytzki (Niemytzki’s tangent disc topology). . 1.2.3 ( [372] ) X , V = {(a, +∞) : a ∈ X} ∪ {(−∞, b) : b ∈ X} X ( 1.2.2 ), V X (linearly ordered topology), X (linearly ordered space). (a, b) = (a, +∞) ∩ (−∞, b) (a, b ∈ X) . X (i) X (ii) X (a, b), a, b ∈ X; a0 , [a0 , b), b ∈ X; · 12 · 1 (iii) X b0 , (a, b0 ], a ∈ X. R . (ordered topology), (ordered space). [0, ω1 ), α, β < ω1 , , (β, α] = (β, +∞) ∩ (−∞, α + 1). [0, ω1 ) , {{0}} ∪ {(β, α] : 0 β < α < ω1 } , B(α) = α ∈ [0, ω1 ) {{0}}, α = 0, {(β, α] : β < α}, α = 0 . T = {U ⊂ [0, ω1 ) : α ∈ U, α > 0 β < α, T [0, ω1 ) . (β, α] ⊂ U }, [0, ω1 ) . 1.3 1.3.1 A X , A A (closure), A , A, A− ClA. A A (contact point). 1.3.1 , X A , A = A; U , X − U = X − U . 1.3.1 (C1) ∅ = ∅; (C2) A ⊃ A; (C3) A ∪ B = A ∪ B; (C4) A = A. (C1)∼(C4) Kuratowski (Kuratowski’s closure axioms). (C1), (C2) 1.3.1 . (C3), 1.3.1, A ∪ B ⊃ A ⇒ A ∪ B ⊃ A. , A ∪ B ⊃ B, A ∪ B ⊃ A∪B. , A∪B A∪B , 1.3.1, A ∪ B A ∪ B , A ∪ B ⊂ A ∪ B. (C4), A A , A , . . 1.3.2 x ∈ A x A . x U A , U , X − U A , X − U ⊃ A, x ∈ / A. , x ∈ / A, X − A x A . . 1.3 · 13 · 1.3.3 U x x ∈ / X − U. U x , U ∩ (X − U ) = ∅, 1.3.2 x ∈ / X − U. , x ∈ / X − U, 1.3.2, x V V ∩ (X − U ) = ∅, V ⊂ U , U x . . X, X A A, A = A, , . X, A X, A = X (A = ∅), A = ∅ (A = ∅). 1.3.4 X , X A, A (C1)∼(C4), X U X − U = X − U, (1.3.1) (O1)∼(O3), X . (C2), X − ∅ = X = X = X − ∅, (1.3.1) , ∅ . (C1), X − X = ∅ = ∅ = X − X, (1.3.1) , X . Ui (i = 1, 2, · · · , n) , (1.3.1) , X − Ui = X − Ui (i = 1, 2, · · · , n), (C3), n n X− Ui = i=1 n (X − Ui ) = i=1 n n (X − Ui ) = X − = (1.3.1) , ni=1 Ui . Uγ (γ ∈ Γ ) , (1.3.1) , X − Ui i=1 i=1 Ui . i=1 X − Uγ = X − Uγ , γ ∈ Γ. , (C3) C ⊂ D ⇒ C ⊂ D. γ∈Γ (X − Uγ ) ⊂ X − Uγ (γ ∈ Γ ) (X − Uγ ) ⊂ γ∈Γ X − Uγ . γ∈Γ X− Uγ = γ∈Γ (X − Uγ ) ⊂ γ∈Γ X − Uγ γ∈Γ (X − Uγ ) = X − = γ∈Γ Uγ . γ∈Γ (O3). · 14 · 1 (C2) , X− Uγ = X − γ∈Γ (1.3.1) , γ∈Γ Uγ . . . 1.3.1 R R R∗ A, A A= Uγ . γ∈Γ x∗ R∗ , R∗ = R ∪ {x∗ }. (A − {x∗ } R ) ∪ {x∗ }, A , A, A . , A , 1.3.4, R∗ . , . 1.3.2 A X , A A (interior), A , A◦ IntA. 1.3.2 , X A , A◦ = A. A X , x A (inner point), A x , x U (x) ⊂ A; A (), A . . 1.3.5 A X , A◦ = X − X − A. X − A X − A , X − X − A X − (X − A) = A , A◦ , A◦ = X − X − A. . 1.3.1 1.3.5 de Morgan . . 1.3.6 (I1) X ◦ = X; (I2) A◦ ⊂ A; (I3) (A ∩ B)◦ = A◦ ∩ B ◦ ; (I4) (A◦ )◦ = A◦ . 1.3.5, , . 1.3.7 A X , A = X − (X − A)◦ . “ ” , 1.3.5 1.3.7 A◦ = A− A− = A◦ . 1.3.3 x A (accumulation point) (limit point), x ∈ A − {x}; A A (derived set), Ad ; A − Ad A (isolated point). 1.3 · 15 · , x X {x} . , X x ∈ X . [0, ω1 ) ( 1.2.3), α = β + 1, (β, α] = {α}, [0, ω1 ) , . x Ad 1.3.8 . x A x 1.3.3, x ∈ A − {x}. 1.3.2, x ∈ A − {x} x A − {x} , x A x . . 1.3.9 (D1) A = A ∪ Ad ; (D2) A ⊂ B, Ad ⊂ B d ; (D3) (A ∪ B)d = Ad ∪ B d ; (D4) γ∈Γ Adγ ⊂ ( γ∈Γ Aγ )d . . 1.3.4 A (dense) X, A = X; (, nowhere dense) X, (A)◦ = ∅; (, ﬁrst category), A ; (, second category), A . A V V ∩ A = ∅ ( 1.15). X U 1.3.2 R R, R. Q , , R ; I . . I , R = Q ∪ I . ∞ R= Ai , (1.3.2) i=1 Ai (i = 1, 2, · · ·) R. I1 = (0, 1), A1 , V1 ∩ A1 = ∅. R () V1 ⊂ I1 , , V1 , V1 ⊂ I1 , V1 , V2 , · · ·, V 1 ∩ A1 = ∅. · 16 · 1 V1 ⊃ V2 ⊃ · · · ; V i ∩ Ai = ∅ Vi (1.3.3) (i = 1, 2, · · ·); (1.3.4) 1/i. (1.3.5) ∞ (1.3.3) (1.3.5), {V i } , x ∈ i=1 V i , x ∈ R, (1.3.4), x ∈ / Ai (i = 1, 2, · · ·), (1.3.2) . . I 1.3.5 A X , A ∩ X − A A (boundary frontier), FrA ∂A; x ∈ A ∩ X − A A (boundary point frontier point). 1.3.5, FrA = A ∩ (X − A◦ ) = A − A◦ , x A x A , X − A . 1.3.10 A , (i) A◦ = A − FrA; (ii) A = A ∪ FrA; (iii) Fr(A ∪ B) ⊂ FrA ∪ FrB; (iv) Fr(A ∩ B) ⊂ FrA ∪ FrB; (v) A , FrA = A − A, A ∩ FrA = ∅; (vi) A , FrA = A − A◦ , FrA ⊂ A; (vii) A FrA = ∅. (i) (iii), . A − FrA = A − (A ∩ X − A) = (A − A) ∪ (A − X − A) = A − X − A = A ∩ A◦ = A◦ . Fr(A ∪ B) = A ∪ B ∩ X − (A ∪ B) = (A ∪ B) ∩ (X − A) ∩ (X − B) ⊂ (A ∪ B) ∩ (X − A ∩ X − B) ⊂ (A ∩ X − A) ∪ (B ∩ X − B) = FrA ∪ FrB. (i), (iii) . . 1.4 . 1.4 · 17 · , . ε- δ , f (x) x0 y0 = f (x0 ) f (Sδ (x0 )) ⊂ Sε (y0 ). ε Sε (y0 ), x0 δ Sδ (x0 ), , . ; . , . 1.4.1 F X , (Fl1) ∅ ∈ / F; (Fl2) A ∈ F B ⊃ A, B ∈ F ; (Fl3) A ∈ F B ∈ F , A ∩ B ∈ F , F (ﬁlter). F F , F (maximal ﬁlter) (ultraﬁlter). U (ﬁnite intersection property), U . 1.4.1 F , F F . Φ = {F : F F }, F1 ⊂ F2 Φ F1 , F2 , Φ . , Φ () . Zorn , Φ F , F F , A ∩ B = ∅, . F (Fl1). A, B ∈ F , F = {A ∩ B} ∪ F , F ∈ Φ; F Φ , A ∩ B ∈ F , (Fl3). (Fl2), B ⊃ A, A ∈ F , {B} ∪ F ∈ Φ, , B ∈ F . . 1.4.2 F F F . F , A F , F = {C : C ⊃ A ∩ B, B ∈ F }, F F , A . F , F = F , A ∈ F . , F 1.4.2 , F F , F ⊃ F . , . A ∈ F , (Fl3) (Fl1), A F , F , , A ∈ F , F ⊂ F. . 1.4.2 F X , x ∈ X. x F , F (converge to) x, F → x. · 18 · 1 1.4.1 (Fl2) F → x x U , A∈F A ⊂ U. 1.4.3 A ∈ F , x ∈ A, x F (cluster point of ﬁlter). 1.4.3 F x, x F ; , F , x F , F x. F → x A ∈ F , 1.4.2, x U ∈ F , (Fl3) (Fl1), A ∩ U = ∅, 1.3.2, x ∈ A, 1.4.3, x F . , x F , 1.4.3, x U F , 1.4.2, U ∈ F , 1.4.2, F → x. . 1.4.1 , F F F ∈ F , F ∈ F F ⊂ F , F F (ﬁlter base), F (FB1) ∅ ∈ / F ; (FB2) F1 , F2 ∈ F , F3 ∈ F , F3 ⊂ F1 ∩ F2 . , F (FB1) (FB2). F = {F : F ⊃ F , F ∈ F }, F F . F (generate) F . F x, x U , F ∈ F , F ⊂ U , F x, F → x. F F , F → x F → x. x U (x) (Fl1)∼(Fl3), , 1.4.2, U (x) x, . x F (x) 1.4.1, , , . F F (x) , F B x ∈ B, B {x} F , . B ∈ F (x), F = F (x), F (x) . (principle ultraﬁlter). , F , ∩F = ∅, (free ultraﬁlter). . Hi = (i, +∞) (i = 1, 2, · · ·). F = {Hi : i = 1, 2, · · ·}, F (Fl1) (Fl3), (Fl2). F = {A : A ⊃ Hi , i = 1, 2, · · ·}, F , F F , , F , F . x, y , F , x, y F , F . , 1.4 · 19 · 1.4.4 D , D > (i) a > b, b > c, a > c ; (ii) D a, b ∈ D, c ∈ D, c > a c > b, D (direct set). 1.4.5 ∆ , X , ∆ X ϕ(δ) (δ ∈ ∆), ∆ (net), , ϕ(∆; >), > ∆ . X {ϕ(δ) : δ ∈ ∆} ∆ . ϕ(δ) ϕ(∆; >) . 1.4.6 ϕ(∆; >) X , A ⊂ X, δ0 ∈ ∆ δ > δ0 ⇒ ϕ(δ) ∈ A, ϕ(∆; >) (eventually in) A; A ⊂ X, ϕ(∆; >) A X − A, ϕ(∆; >) (maximal net) (ultra net); δ0 ∈ ∆, δ ∈ ∆, δ > δ0 , ϕ(δ) ∈ A, ϕ(∆; >) (coﬁnal in) A. 1.4.7 ϕ(∆; >) x , x, ϕ(∆; >) → x; ϕ(∆; >) x , x (cluster point of net). F = {Aδ : δ ∈ ∆} , ∆ δ > δ Aδ ⊂ Aδ ( δ = δ ⇒ Aδ = Aδ ); ∆ ϕ(∆; >) δ ∈ ∆, ϕ(δ) ∈ Aδ , F (derived net). , ϕ(∆; >), F = {A : ϕ(∆; >) A}. F , F ϕ(∆; >) (derived ﬁlter). 1.4.4 F x F x. F → x, F = {Aδ : δ ∈ ∆}, ϕ(∆; >) F , ϕ(δ) ∈ Aδ , δ ∈ ∆. U x , 1.4.2, U ∈ F , U = Aδ0 (δ0 ∈ ∆), δ > δ0 , Aδ ⊂ Aδ0 = U , ϕ(δ) ∈ U , 1.4.7, ϕ(∆; >) → x. , F x, 1.4.2, x U ∈ / F , δ ∈ ∆, Aδ ⊂ U (, 1.4.1 (Fl2), U ∈ F ). ϕ(δ) ∈ Aδ −U, δ ∈ ∆, F ϕ(∆; >), x. . 1.4.5 ϕ(∆; >) x x. F ϕ(∆; >) , F = {A : ϕ(δ; >) A}. · 20 · 1 ϕ(∆; >) → x, x U, ϕ(∆; >) U , U ∈ F , F → x. , F → x, x U , U ∈ F , F , ϕ(∆; >) U , ϕ(∆; >) → x. . 1.4.1 ϕ(∆; >) . lim xn = a , ∆ N, > n→∞ . , ϕ(n) = xn (n ∈ N), lim ϕ(x) = b , ∆ = U (x0 ) − {x0 }, U (x0 ) x→x0 > x > x, ρ(x , x0 ) < ρ(x, x0 ) ( ρ(x, y) x, y ). (Riemann) , b . . . a f (x)dx , [a, b] T x0 , a = x0 < x1 < x2 < · · · < xn−1 < xn = b, T = (x0 , x1 , x2 , · · · , xn−1 , xn ). ξi ∈ [xi , xi+1 ] (i = 0, 1, 2, · · · , n − 1) Tξ = (x0 , ξ0 , x1 , ξ1 , x2 , · · · , xn−1 , ξn−1 , xn ). ∆ Tξ , λ(T ) = max{xi+1 − xi }. > Tξ > Tξ , λ(T ) < λ(T ), i ϕ(Tξ ) = n−1 f (ξi )∆xi , Tξ ∈ ∆. i=0 1.4.2 , . ∆ = {(n, m) : n, m = 1, 2, · · ·}, (n, m) > (n , m ), n > n , m m , ∆ . ∆ R2 ϕ(∆; >) ϕ((n, m)) = (1/n, 1/m). , ϕ(∆; >) → (0, 0). ∆ = {(n, 1) : n = 1, 2, · · ·}, ∆ ∆ , “ ” ϕ (∆ ; >) (0, 0), (0, 0) , . , “ ” . ∆ ∆ ∆, δ ∈ ∆, δ ∈ ∆ δ > δ. ∆ ∆ , 1.5 · 21 · ϕ(∆; >) → x ⇒ ϕ(∆ ; >) → x, , , . ∆ . δ ∈ ∆ δ1 < δ2 < · · · < δn = δ, δ1 , δi+1 δi . , δ = ω, 2ω, ω 2 , δ1 ω, 2ω, ω 2 ; δ = ω + 3 , δ1 = ω, δ2 = ω + 1, · · · , δ4 = ω + 3. δ ∈ ∆ n (ω, 2ω, ω 2 1, ω + 3 4), ∆ R ϕ(∆; >) ϕ(δ) = 1/n, n δ . ∆ ∆ ϕ(∆ ; >) → 0. δ ∈ ∆, 0 1.5 ϕ(∆; >) , 1.5.1 X Y f (continuous), Y V f −1 (V ) X . , f : X → Y f −1 (f (A)) ⊃ A, f (f −1 (B)) ⊂ B. f , , f (f −1 (B)) = B. , f −1 (B) ⊂ A B ⊂ Y − f (X − A). ( 0.3), (1.5.1) 1.5.1 (i) X Y f ; −1 (ii) Y F f (F ) ; (iii) B ⊂ Y , f −1 (B) ⊃ f −1 (B); (iv) A ⊂ X, f (A) ⊂ f (A); (v) x ∈ X f (x) V , x U , f (U ) ⊂ V . −1 (i) ⇒ (ii). F Y , (i), f (Y − F ) X , f −1 (F ) = X − f −1 (Y − F ) X . (ii) ⇒ (i). . (ii) ⇒ (iii). f −1 (B) f −1 (B) f −1 (B) f −1 (B) . · 22 · 1 (iii) ⇒ (iv). (iii), f −1 (f (A)) ⊃ f −1 (f (A)) ⊃ A, f (A) ⊃ f (f −1 (f (A))) ⊃ f (A). (iv) ⇒ (ii). F Y , (iv), f (f −1 (F )) ⊂ f (f −1 (F )) ⊂ F = F , f −1 (F ) ⊂ f −1 (F ), f −1 (F ) ( 1.3.1). (i) ⇒ (v). x ∈ X, V f (x) , G, (i), f −1 (G) . U = f −1 (G), U x (v) ⇒ (i). G Y , x ∈ f (v), x U −1 , f (U ) ⊂ G ⊂ V . (G) ⇒ f (x) ∈ G. G f (x) f (U ) ⊂ G, U ⊂ f x ∈ f −1 (G) , f −1 (G) ( f (x) ∈ G ⊂ V , −1 (f (U )) ⊂ f −1 (G), . 1.1.2). 1.5.2 X Y f , Y X , f (homeomorphism) (topological mapping). , X Y (homeomorphic). (topological property) (topological invariant). . R . , (0, 1) R f x → (2x − 1)/x(1 − x); , . , , . 1.5.3 X Y f (closed mapping), X F f (F ) Y ; (open mapping), X U f (U ) Y . 1.5.1 R2 R ( ) f : (x, y) → x , , F = {(x, y) : y = 1/x, x ∈ R − {0}} R2 ( ), f (F ) = R − {0} ( x ) R . , f ( 2.1.3). X [0, 2], Y [0, 1], X, Y . f (x) = 0, x ∈ [0, 1], x − 1, x ∈ (1, 2]. , f . F X = [0, 2] , F1 = F ∩ [0, 1], F2 = F ∩[1, 2], F1 , F2 X , F = F1 ∪F2 , f (F1 ) = {0} Y = [0, 1] 1.5 · 23 · , f (F2 ) = {x − 1 : x ∈ F2 } Y , f (F ) = f (F1 ) ∪ f (F2 ) Y , f . f , X (0, 1) f ((0, 1)) = {0}, Y . X R ( ), Y R , Y R , . f X Y (x → x, x ∈ R). , f , , . , f f () . 1.5.2 (i) X Y f ; (ii) X A, f (A◦ ) ⊂ (f (A))◦ ; (iii) Y B, f −1 (B) ⊂ f −1 (B); (iv) x ∈ X U f (U ) f (x) . (i) ⇒ (ii). f (A◦ ) ⊂ f (A), (i), f (A◦ ) , f (A◦ ) ⊂ (f (A))◦ ( 1.3.2). (ii) ⇒ (iv). U x ∈ X , x ∈ U ◦ , (ii), f (x) ∈ (f (U ))◦ , f (U ) x . (iv) ⇒ (iii). x ∈ f −1 (B), f (x) ∈ B, U x , (iv), f (U ) f (x) , f (U ) ∩ B = ∅, x ∈ U , f (x ) ∈ B, x ∈ f −1 (B), U ∩ f −1 (B) = ∅, x ∈ f −1 (B). (iii) ⇒ (ii). A◦ ⊂ A ⊂ f −1 (f (A)), A◦ , A◦ ⊂ [f −1 (f (A))]◦ . M ◦ = X − X − M ( (1.5.2) 1.3.5), [f −1 (f (A))]◦ = X − X − f −1 (f (A)). (1.5.3) X − f −1 (f (A)) = f −1 (Y − f (A)), (1.5.3) [f −1 (f (A))]◦ = X − f −1 (Y − f (A)). (1.5.4) [f −1 (f (A))]◦ ⊂ X − f −1 (Y − f (A)). (1.5.5) (iii) (1.5.4) (1.5.2) , (1.5.5) N ◦ = Y − Y − N ( 1.3.5), f (A◦ ) ⊂ f (X − f −1 (Y − f (A))) = f (f −1 (Y − Y − f (A))) ⊂ Y − Y − f (A) = (f (A))◦ . · 24 · 1 (ii) ⇒ (i). A , A = A◦ , (ii), f (A) = f (A◦ ) ⊂ (f (A))◦ , f (A) ( 1.3.2), f . . 1.5.3 (i) X Y f ; (ii) X A, f (A) ⊃ f (A). (i) ⇒ (ii). f (A) ⊃ f (A), (i), f (A) , f (A) ⊃ f (A). (ii) ⇒ (i). A X , (ii), f (A) ⊃ f (A) A = A, f (A) ⊃ f (A), f (A) . . . 1.5.4 f X Y , f y ∈ Y X U ⊃ f −1 (y), Y W y∈W −1 f (W ) ⊂ U . f , y ∈ Y X U ⊃ f −1 (y), W = Y − f (X − U ), W Y , 1.5.1 (1.5.1) , y ∈ W , −1 −1 f (W ) ⊂ U . , F X , y ∈ Y − f (F ), f (y) ⊂ X − F , , Y W y ∈ W f −1 (W ) ⊂ X − F , W y (1.5.1) , W ∩ f (F ) = ∅, f (F ) Y . . “” , ( ), . 1.5.1 f X Y , (i) f ; (ii) E ⊂ Y X U ⊃ f −1 (E), X V f −1 (E) ⊂ V ⊂ U V = f −1 (f (V )), f (V ) Y ; (iii) y ∈ Y X U ⊃ f −1 (y), X V f −1 (y) ⊂ V ⊂ U V = f −1 (f (V )), f (V ) Y . (i) ⇒ (ii). X U ⊃ f −1 (E), y ∈ E, f −1 (y) ⊂ U , 1.5.4, Y Wy y ∈ Wy f −1 (Wy ) ⊂ U . V = y∈E f −1 (Wy ), f (V ) = y∈E Wy (f ). V . (ii) ⇒ (iii). . (iii) ⇒ (i). y ∈ Y X U ⊃ f −1 (y), (iii), X V f −1 (y) ⊂ V ⊂ U V = f −1 (f (V )), f (V ) Y . W = f (V ), W Y , y ∈ W f −1 (W ) ⊂ U . 1.5.4, f . . 1.5.1 f . X , Y Sierpiński (Sierpiński space [372] ), Y = {0, 1} {∅, {0}, Y }, 1 · 25 · f : X → Y f (X) = {0}. f . 1.1 (O1)∼(O3) 1.5.1 (ii) (iii), 1 (X, T ), , (X, T ), T = T . 1.2 U (x) (x ∈ X) , 1.3 , U (x). A (N1)∼(N5) (O1)∼(O3) 1.3.4 (1.3.1) 1.4 , , . 1.1.2 . (X, T ), 1.3.1 , (X, T ), T = T . (C1)∼(C4) , 1.3.4 (1.3.1) , 1.3.1 A, A = A A ⊂ X . 1.5 X X , X ( 1.2.1) 0 X ( X ). 1.6 Smirnov 1.7 Niemytzki ( 1.2.2) x 1.3.1 {1/n} . , ? 1.8 R∗ x∗ x ∈ R . 1.9 1.10 A B A ∩ B ⊂ A ∩ B A − B ⊂ A − B, ? . 1.11 X, A1 , A2 , · · ·, ∞ ∞ ∞ ∞ Ai = i=1 1.12 ( Ai+j ∪ . i=1 j=0 . X A, ) 1.13 Ai i=1 . 14 , R 14 . R E 1/2m + 1/3n + 1/5l (m, n, l ) , E d , (E d )d , ((E d )d )d , (E d )d = E d ? 1.14 E E d ? R ( ) ; , . 1.15 X A X U V V ∩ A = ∅. 1.16 D 1.17 A 1.18 A X, X A, D ∩ A B ⊂ X, A B. X, U ⊂ X, U = A ∩ U . A. · 26 · 1 1.19 F x x F 1.20 F 1.21 . 1.22 . 1.23 1.24 1.25 1.26 X Y , F X , f (F ) = {f (F ) : X Y , f ϕ(∆; >) f ◦ ϕ(∆; >) Y x A ⊂ X X f (x). A − {x} x. . δ ∈ ∆, Aδ = {ϕ(δ ) : δ ∈ ∆, δ > δ}, x ϕ(∆; >) x Aδ (δ ∈ ∆) 1.27 . . f x A ⊂ X, A ∈ F X − A ∈ F . f F ∈ F } Y X , . , , Y ? 1.28 f , f 1.29 X ; Y Y . X , f . A FrA (i) FrA ⊂ FrA; (ii) FrA◦ ⊂ FrA; (iii) X = A◦ ∪ FrA ∪ (X − A)◦ . 1.30 Fσ . 2 . , 2.1 (X, T ) , X ⊂ X, X U ∈ T , U = U ∩ X , X U T (O1)∼(O3), T = {U : U = U ∩ X , U ∈ T } X , T (relative topology), (X , T ) (X, T ) (subspace); X X () , (X , T ) (X, T ) (open subspace) ( (closed subspace)). 2.1.1 F ⊂ X ⊂ X, F X X F = F ∩ X , A ⊂ X X A = A ∩ X . F F X , X − F X , U X, X − F = U ∩ X , F = X − (U ∩ X ) = X ∩ (X − U ), X − U X. , A ⊂ X , A = X ∩ F , F X, X − A = X − (X ∩ F ) = X ∩ (X − F ). X − F X, X − A X , A X . A = A ∩ X . A X , A ∩ X X A , A ⊂ A ∩ X . , A X , X F A = F ∩ X , A ⊂ A ⊂ F , A ⊂ F , A ∩ X ⊂ F ∩ X = A. A = A ∩ X . . . A ⊂ X ⊂ X, A X ClX (A), U X , 2.1.1 ClX (A) = Cl(A) ∩ X . Y ⊂ X, U |Y = {U ∩ Y : U ∈ U }, T X , X ⊂ X T |X . f (x) [a, b] , a b , a b [a, b] · 28 · 2 , , [a, b] R . . X, Y , X Y ( 1.5.1), (X, U ), Y X Y f , Y f . “ ”, , . , V Y −1 V V f (V ) X , (O1)∼(O3), V = {V : f −1 (V ) ∈ U } Y , (Y, V ) , V (2.1.1) f Y . X D, , D X ( 0.1 ) (i) D∈D D = X; (ii) D . X Y f , {f −1 (y) : y ∈ Y } X . R X ( xRx, xRx ⇒ x Rx, xRx x Rx ⇒ xRx R), R X , X/R , X . X (X, U ) , X X x , X/R f , X/R (2.1.1) , ( f R : f (x) = f (x ) ⇔ xRx , Y = X/R), (quotient topology), Y (= X/R) (quotient space), f (quotient mapping). X D, Y (= X(D)) (decomposition space), f (natural mapping) (natural quotient mapping). . 2.1.1 1.2.1 Smirnov R+ , {1/n : n = 1, 2 · · ·} , , (2.1.1), Y . , X A, X R A , X − A , X/R (2.1.1), , A X , f : X → X/A X/A. ( 2.1). 2.1.1 , F = {1/n : n ∈ N}, Y R+ /F , f : R+ → R+ /F . 2.1 · 29 · 2.1.2 1.2.2 Niemytzki R , x , , Y . , (2.1.1), , X, (Y, V ) X Y f , X f . “ ”, , . , U X Y V , U (O1)∼(O3), U = {U : U = f −1 (V ), V ∈ V } X , (X, U ) , U f (2.1.2) X . . R2 , {Sε ((x, y)) : > 0} (x, y) , (x, y) R2 (x, y) , , , y . , x, (X, U ), (Y, V ), Z = X × Y , Z = X × Y W , W {U × V : U ∈ U , V ∈ V } , (Z, W ) (X, U ) (Y, V ) . . (Xi , Ti ) (i = 1, 2, · · · , n), X = ni=1 Xi , X W , W n Vi : Vi ∈ Ti , i = 1, 2, · · · , n (2.1.3) i=1 ( (B1) (B2)). , . , , , {(Xγ , Tγ )}γ∈Γ , Γ , X = γ∈Γ Xγ , pγ X Xγ (γ ∈ Γ ) . X pγ , (2.1.2) {W : W = p−1 γ (Vγ ), Vγ ∈ Tγ , γ ∈ Γ } · 30 · 2 X W , W , ⎧ ⎫ ⎨ ⎬ (2.1.4) Vγ : Vγ ∈ Tγ , γ ∈ Γ , γ Vγ = Xγ ⎩ ⎭ γ∈Γ ( (B1) (B2)). , W pγ (γ ∈ Γ ) X [400] , , W (product topology), A. Tychonoff Tychonoﬀ . (X, W ) {(Xγ , Tγ )}γ∈Γ (product space). . ⎧ ⎫ ⎨ ⎬ Vγ : Vγ ∈ Tγ , γ ∈ Γ ⎩ ⎭ γ∈Γ X , (box topology), Tychonoﬀ . , Tychonoﬀ . (2.1.3), (2.1.4), , , . 2.1.3 () n . Rn n R I ω = {(x1 , x2 , · · · , xi , · · ·) : 0 xi 1/i, i = 1, 2 · · ·} [0, 1/i] (i = 1, 2, · · ·) . I ω ( 4.1.1) . (Hilbert cube), {Xγ }γ∈Γ , Γ , Xγ (γ ∈ Γ ) . (2.1.4) , , γ∈Γ Xγ . (2.1.4) . 2.1.2 X Y = γ∈Γ Yγ f Y pγ (γ ∈ Γ ), pγ ◦ f . f . , , , , pγ ◦ f (γ ∈ Γ ) , pγ ◦ f (γ ∈ Γ ) , Uγ Yγ , (pγ ◦ f )−1 (Uγ ) = −1 f −1 (p−1 Y . U Y γ (Uγ )) , pγ (Uγ ) , Y , U p−1 γ (Uγ ) , 2.1 · 31 · −1 (f −1 (U )) (f −1 (p−1 (U ) X γ (Uγ ))) , f . . 2.1.3 f X Y , ϕ Y Z , ϕ ϕ ◦ f . ϕ , ϕ ◦ f . , ϕ ◦ f −1 −1 −1 , U Z , (ϕ ◦ f ) (U ) = f (ϕ (U )) X , , ϕ−1 (U ) Y , ϕ . . 1.4 , . ( 2.1.4) , . 2.1.1 X Y f , x ∈ X X x F , f (F ) = {f (A) : A ∈ F } Y f (x). f , x ∈ X f (x) V , x U f (U ) ⊂ V ( 1.5.1 (v)), F x, x U , F ∈ F , F ⊂ U . , f (F ) , f (F ) f (F ) ⊂ f (U ) ⊂ V , f (F ) f (x). , x ∈ X, V f (x) , x U (x), U (x) x, f (U (x)) f (x) ( F ⊂ V , U ∈ U (x), ), F ∈ f (U (x)) f (U ) ⊂ V , f . . ϕ(∆; >) , ( 2.31). 2.1.4 X = γ∈Γ Xγ F x = {xγ }γ∈Γ F Xγ (γ ∈ Γ ) pγ (F ) Xγ xγ . F → x, pγ , 2.1.1 . , U X x , U U = γ∈Γ p−1 γ (Uγ ), Γ Γ . γ ∈ Γ , pγ (F ) → xγ , pγ (F ) Uγ , F ∈ F pγ (F ) ⊂ Uγ . p−1 γ (Uγ ) ⊃ F , ( 1.4.1 −1 −1 (Fl2)), pγ (Uγ ) ∈ F , γ ∈ Γ , pγ (Uγ ) F ( 1.4.1 (Fl3)). U , U ∈ F , F → x. . ϕ(∆; >) , ( 2.32). 2.1.4, (coordinatewise convergence) (pointwise convergence). , γ ∈ Γ , Xγ = X, X Γ . Γ , X Γ X ω . , I = [0, 1] I ω , I ω , Hilbert [0, 1/i] I (i ∈ N), i∈N [0, 1/i] · 32 · 2 ( 2.1.3) I ω . I ω Hilbert . 2.1.1 X D (upper semicontinuous), D ∈ D U ⊃ D, V , D ⊂ V ⊂ U, V D . 2.1.5 D X X(D) f , . 1.5.1 (iii) . . 2.2 , , . , (axioms of separation). 2.2.1 (T0 ) X x1 , x2 , x2 ∈ / U (x1 )). (, x1 U (x1 ) T0 (T0 -axiom of separation), T0 T0 (T0 -space). . 2.2.1 T0 , T0 (i) X T0 ; (ii) X x1 , x2 , x1 ∈ / {x2 }, x2 ∈ / {x1 }; (iii) X x1 x2 {x1 } {x2 }. (i)⇒ (ii). x1 ∈ {x2 } x2 ∈ {x1 } , x1 x2 , x2 x1 , (i). / {x2 }, x1 (ii)⇒ (i). x1 ∈ U (x1 ) x2 ∈ / U (x1 ). (ii)⇒ (iii). . (iii)⇒ (ii). x1 ∈ {x2 }, x2 ∈ {x1 } , {x1 } ⊂ {x2 }, {x2 } ⊂ {x1 }, {x1 } ⊂ {x2 } = {x2 }, . {x2 } ⊂ {x1 }, {x1 } = {x2 }, (iii). 2.2.2 (T1 ) X x1 , x2 , x1 x2 ∈ / U (x1 ), x2 U (x2 ) x1 ∈ / U (x2 ). U (x1 ) T1 (T1 -axiom of separation), T1 T1 (T1 -space). 2.2 · 33 · X Sierpiński ( 0 {0} T0 , T1 . 1.5.1 ), X = {0, 1} {∅, {0}, X}. 1, 1 0, , . 2.2.2 . X T1 {x} (x ∈ X) 2.2.3 X T1 , A X , x A . x A 2.2.3 (T2 ) X x1 , x2 , x1 U (x1 ), x2 U (x2 ), U (x1 ) ∩ U (x2 ) = ∅. T2 (T2 -axiom of separation) Hausdorﬀ (Hausdorﬀ axiom of separation), T2 (T2 -space) Hausdorﬀ (Hausdorﬀ space). 2.2.1 ( T2 T1 ) 1.3.1 R∗ = R ∪ {x∗ }, R R∗ , {x∗ } R∗ . x∗ x∗ . ∗ ∗ ∗ R T1 , x ∈ R x , R T2 . , T2 ⇒ T1 ⇒ T0 , . T2 , , X T2 . “ ”, . 2.2.4 X T2 , F → x, x = x, 2.2.3, x U (x) x V (x ), U (x) ∩ V (x ) = ∅. F → x, U (x) ∈ F , U (x) ∩ V (x ) = ∅, V (x ) ∈ / F , F x . (), X T2 , x, x U , V , (U, V ), F = {M : M ⊃ U ∩ V, U x F , x F → x . . x , V x }. F , F → x 2.2.4 (T3 ) X F F x, U V , U ⊃ F, x ∈ V U ∩ V = ∅. T3 (T3 -axiom of separation), T3 T3 (T3 -space), T1 T3 (regular space). , T1 + T3 ⇒ T2 , ⇒ Hausdorﬀ, , 2.2.2. · 34 · 2 2.2.2 (Smirnov [372] , T2 ) 1.2.1 Smirnov R+ , T2 . F = {1/n : n = 1, 2, · · ·}, F , 0 ∈ / F , F 0 U V U ∩ V = ∅. 2.2.5 X T3 x ∈ X x x ∈ V ⊂ V ⊂ U. U , V . , U . 2.2.5 T3 , (closed neighborhood base). 2.2.5 (T4 ) X F1 F2 , U1 ⊃ F1 , U2 ⊃ F2 U1 ∩ U2 = ∅. T4 U1 U2 , (T4 -axiom of separation), T4 T4 (T4 -space), T1 T4 (normal space). , T1 + T4 ⇒ T1 + T3 , ⇒ , , 2.2.3. 2.2.3 (Niemytzki [372] , ) 1.2.2 Niemytzki R . , . R , x γ Q, η I, R , Q, I . , U V , U ⊃ Q, V ⊃ I, U ∩ V = ∅. x x ε Sε (x). γ ∈ Q, Sd γ (γ) ⊂ U , I, In = {η : η ∈ I, S1/n (η) ⊂ V }, n ∈ N, (2.2.1) ∞ I= In . (2.2.2) n=1 U ∩ V = ∅, Sd γ (γ) S1/n (η) , (γ − η)2 dγ /n. (2.2.3) γ ∈ Q, εγ > 0, nε2γ = dγ , (2.2.3) (2.2.1) (γ − εγ , γ + εγ ) In (, x R ( )), γ ∈ / I n , Q ⊂ R − I n , Q = R Q ⊂ R − I n , R = R − I n = R − (I n )◦ , (I n )◦ = ∅, In R. (2.2.2) , I , 2.2 · 35 · ( , 1.3.2), , R . R R , (, category method). 2.2.6 X T4 F F U , V F ⊂ V ⊂ V ⊂ U. . T0 , T1 , T2 , T3 T0 , T1 , T2 , T3 (). , , T4 T4 , . 2.2.4 (Tychonoﬀ “” [372] , ) [0, ω1 ] ω1 , [0, ω] ω , . 3 3.2.1 3.1.4 [0, ω1 ] × [0, ω] ( Tychonoﬀ “” (Tychonoﬀ plank), [0, ω1 ] “”, [0, ω] “”) . [0, ω1 ] × [0, ω] − {(ω1 , ω)} ( “” ), Tychonoﬀ “” (deleted Tychonoﬀ plank). A = {(ω1 , y) : y < ω} ( “” ), B = {(x, ω) : x < ω1 } ( “” ). A, B ( , “” ). . U ⊃ A, y < ω, ϕ(y) x > ϕ(y) ⇒ (x, y) ∈ U . ϕ(y) < ω1 , ϕ(y) ω1 , sup{ϕ(y)} < ω1 , x∗ ϕ(y), (x∗ , ω) ∈ B, (x∗ , ω) U . 2.2.6 (T5 ) X A ∩ B = A ∩ B = ∅ A B ( A B (separated)), U V U ⊃ A, V ⊃ B U ∩ V = ∅. T5 (T5 -axiom of separation), T5 T5 (T5 -space), T1 T5 (completely normal space). T5 ⇒ T4 , ⇒ . 2.2.4 . 2.2.7 X . X T5 T4 . X T5 , A X , F1 , F2 A . ( 2.1.1), F1 = F 1 ∩ A, F2 = F 2 ∩ A, F 1 , F 2 F1 , F2 · 36 · X . 2 F 1 ∩ F2 = F 1 ∩ F 2 ∩ A, F1 ∩ F 2 = F 1 ∩ A ∩ F 2 , F 1 ∩ F 2 ∩ A = F1 ∩ F2 = ∅, F 1 ∩ F2 = F1 ∩ F 2 = ∅. T5 , U , V , U ⊃ F1 , V ⊃ F2 U ∩ V = ∅, A U ∩ A ⊃ F1 , V ∩ A ⊃ F2 . U ∩ A, V ∩ A A , A T4 . , X T4 , A, B X A ∩ B = A ∩ B = ∅. G = X − (A ∩ B), G ∩ A, G ∩ B G U ⊃ G ∩ A, V ⊃ G ∩ B, . , G U V U ∩ V = ∅. G X , U , V X . G = X − (A ∩ B) = (X − A) ∪ (X − B), U ⊃ G ∩ A = (X − B) ∩ A ⊃ A ∩ A = A. V ⊃ B, X T5 . . , (hereditarily normal space). P (hereditary property) P T0 , T1 , T2 , T3 . 2.2.4 P. T4 . “ X , X .” T0 (T1 , T2 , T3 , ) T0 (T1 , T2 , T3 , ) (, 2.9 2.10, ). ( 2.3.4). , , T3 , T4 , T5 , T3 , , T3 . 2.3 2.3.1 (weight), w(X). X (cardinal number) ℵ0 (second 2.3 · 37 · axiom of countability) (second countable space), . X x X (character), χ(x, X); X , χ(X). ℵ0 (ﬁrst axiom of countability) (ﬁrst countable space), . X U = {Uα : α ∈ A} X (covering), ∪{Uα : α ∈ A} = X. U (), () ; A () , () ; U U (U ⊂ U ) , U U (subcovering). X , X Lindelöf . X , X (separable space). 2.3.1 (i) X x A ⊂ X , , A − {x} x; (ii) A A (iii) x {xn } A; , {xn } x. (i) , 1.25. (ii) A , X − A , X − A x X − A, x ∈ A, (i), (X − A) − {x} = X − A x , A. (iii) . . 2.3.1 (i), (ii) Fréchet [125] (Fréchet space), [125] (sequential space). ⇒ Fréchet ⇒ . . , . 2.3.2 . U X , U ∈ U , x(U ) ∈ U , A = {x(U ) : U ∈ U } . A X, X − A , U , . . 2.3.3 Lindelöf . V = {Vα }α∈A X , U X . Vα (α ∈ A) U ∈ U , U U (U ⊂ U ) X. U ∈ U , α(U ) ∈ A U ⊂ Vα(U) , V = {Vα(U) : U ∈ U } . . · 38 · 2 R (a, b) , a, b , , R , R Lindelöf . 2.3.1 ( [372] , ) X , , X , (ﬁnite complement topology). () , X. . B, T1 , B x () {x}. de Morgan (X − {x}), X − {x} , X − {x} , . 2.3.2 ( [372] , Lindelöf ) 2.2.4 ω1 B(ω1 ) = {(β, ω1 ] : β < ω1 } , [0, ω1 ], [0, ω1 ] . [0, ω1 ] . U [0, ω1 ] , ω1 ∈ U1 ∈ U , β1 = min{β : (β, ω1 ] ⊂ U1 }, β1 ∈ / U 1 , β1 ∈ U 2 ∈ U ; β2 = min{β : (β, β1 ] ⊂ U2 }, β2 ∈ / U 2 , β2 ∈ U 3 ∈ U . V (ω1 ) = (β1 , ω1 ], V (β1 ) = (β2 , β1 ], V (β2 ) = (β3 , β2 ], · · · . V = {V (ω1 )}∪{V (βn ) : n ∈ N}∪{{0}} [0, ω1 ] , · · · < β3 < β2 < β1 . , {β1 , β2 , β3 , · · ·} , [0, ω1 ] , , U V U . [0, ω1 ), , Lindelöf . α ∈ [0, ω1 ), α = 0, β(α) < α, V = {{0}} ∪ {(β(α), α] : 0 < α < ω1 } . , [0, ω1 ), [0, ω1 ] . , 1.3.1 R∗ , , R∗ . , Lindelöf 2.3.4. 2.3.4 (Tychonoﬀ [399] ) Lindelöf . A, B Lindelöf X . , x ∈ A, U (x), U (x) ∩ B = ∅, U = {U (x) : x ∈ A} A; , y ∈ B, V (y), V (y) ∩ A = ∅, V = {V (x) : x ∈ B} B. U ∪ V ∪ {X − (A ∪ B)} X , X Lindelöf , , A B {Un } {Vn }. Un = Un − ∪{V k : k n}, Vn = Vn − ∪{U k : k n}, 2.3 · 39 · Un ∩ Vm = ∅, Vn ∩ Um = ∅ (m n), Un Vm , ∞ Un , U= ∞ V = n=1 , 2.3.1 Vn n=1 A, B. . . 2.3.3 2.3.4 . . ( ) ( ) . 2.3.2 [0, ω1 ] [0, ω1 ), Lindelöf Lindelöf . ( 2.3.3 2.3.4). ( ) ( ) ( 2.17). Pondiczery [334] , Hewitt [192] Marczewski [276] c ( ) . K. A. Ross A. H. [343] Stone , W. W. Comfort [96] . Lindelöf Lindelöf ( 2.3.3 2.3.4). 2.3.3 ( , Sorgenfrey [371, 372] ) X, [a, b) ( a, b ) B , (half-open interval topology), Sorgenfrey (Sorgenfrey line). B . (a, b), (a, +∞) , (a, b) = ∪{[α, b) : a < α < b}. R , T2 . , x . , x ∈ X, {[x, ai ) : ai } . , {[an , bn )} B , c ∈ X c = an (n = 1, 2, · · ·), c > c, [an , bn ), c ∈ [an , bn ) ⊂ [c, c ). Lindelöf . {Uα }α∈A X , Uα◦ R Uα , R Lindelöf , U = α∈A Uα◦ {Uα◦ }α∈A {Uα◦i }. F = X − U , F , {Uα }α∈A . a ∈ F, a {Uα }α∈A , xa > a (a, xa ) ∩ F = ∅, · 40 · 2 {(a, xa )}a∈F ( F ), ( 2.21), F . . F ∩ H = F ∩ H = ∅ F , H, y ∈ H, ε(x) > 0, ε(y) > 0, H ⊂ X − F, x ∈ F, [x, x + ε(x)) ∩ H = ∅, F ⊂ X − H, [y, y + ε(y)) ∩ F = ∅, [x, x + ε(x)) ∩ [y, y + ε(y)) = ∅. U = ∪{[x, x + ε(x)) : x ∈ F }, V = ∪{[y, y + ε(y)) : y ∈ H}, F , H , U ∩ V = ∅. . 2.2.7, X . 2.3.4 ( , Sorgenfrey [371, 372] ) 2.3.3 X Y = X × X, (half-open square topology), Sorgenfrey (Sorgenfrey plane). . Y E = {(x, y) : x + y = 1}, , , E , Lindelöf . , Lindelöf Lindelöf ( 2.18), Y Lindelöf . . , Lindelöf Lindelöf Y = X × X . E, x + y = 1 Q, I, Q ∪ I = E, Q, I E , Y . 2.2.3 ( , E , 2.2.3 Niemytzki x , ), Q, I . . , . 2.3.5 ( [114] ) ( ) X, . R , Dt (t ∈ R) D = {0, 1}, |R| = c, Dt . c ( 2.3.1 ), D = t∈R Dt , X D . X . , x ∈ X, x 2.4 {Vi }i∈N , x U · 41 · i ∈ N Vi ∩ (X − U ) = ∅. x = {xt }t∈R ∈ X, {Vi }i∈N Ri ⊂ R x x∈X∩ (2.3.1) , , i ∈ N, Wti ⊂ Vi , (2.3.2) t∈R , t ∈ Ri , Wti Dt xt ( {xt }, Dt ); t ∈ R − Ri , Wti = Dt . R , t0 ∈ R − i∈N Ri , U = p−1 x D t0 (xt0 ) , Dt0 , U D , Wti − U (i ∈ N) X t∈R D . , , Wti ∩ (X − U ) = ∅ t∈R (2.3.2) t∈R , (2.3.1), Wti − U = D , X Wti ∩ (D − U ) . t∈R . 2.4 2.2 , (neighborhood separation property). ( ) , (functional separation property). 2.4.1 (Urysohn [403] ) A, B T4 X , f : X → [0, 1] f (x) = 0, x ∈ A; f (x) = 1, x ∈ B. A, B , X − B A , T4 , U1/2 A ⊂ U1/2 ⊂ U 1/2 ⊂ X − B. T4 , U1/4 U3/4 A ⊂ U1/4 ⊂ U 1/4 ⊂ U1/2 ⊂ U 1/2 ⊂ U3/4 ⊂ U 3/4 ⊂ X − B. · 42 · 2 , , k/2n , Γ k/2n , {Uγ }γ∈Γ . , Γ [0, 1], γ < γ A ⊂ Uγ ⊂ U γ ⊂ Uγ ⊂ U γ ⊂ X − B. f : X → [0, 1] f (x) = inf{γ : x ∈ Uγ }, x ∈ 1, Uγ , x∈ / Uγ . , x ∈ B , f (x) = 1; x ∈ A , x ∈ Uγ (γ ∈ Γ ), f (x) = 0. f . x0 ∈ X, f (x0 ) ∈ (0, 1) ( f (x0 ) = 0 1 , ). ε > 0 0 < f (x0 ) − ε < f (x0 ) < f (x0 ) + ε < 1 ( f (x0 ) ε (0,1) ). γ , γ ∈ Γ , 0 < f (x0 ) − ε < γ < f (x0 ) < γ < f (x0 ) + ε < 1. f (x) = inf {γ : x ∈ Uγ }, x0 ∈ Uγ , x0 ∈ / U γ ( f (x0 ) γ ). Uγ − U γ x0 , U (x0 ), f (U (x0 )) ⊂ (f (x0 ) − ε, f (x0 ) + ε). . 2.4.1 “ A, B, f : X → [0, 1] f (x) = 0, x ∈ A; f (x) = 1, x ∈ B” ( ), F4 F4 , 2.4.1 T4 ⇒ F4 . , F4 f , U = f −1 ([0, 1/2)), V = f −1 ((1/2, 1]), U ⊃ A, V ⊃ B U ∩ V = ∅. , , T1 + F4 , “T1 X , A, B, f : X → [0, 1], f (x) = 0, x ∈ A; f (x) = 1, x ∈ B.” f X X Y , X Y g, g(x) = f (x), x ∈ X , g f X ( ) (extension); f g X (restriction). , X = [0, 1], X = (0, 1], (0, 1] f (x) = 1/x [0, 1] ; (0, 1] ϕ(x) = x · sin(1/x) [0, 1] , g(0) = 0, g(x) = ϕ(x), x ∈ (0, 1]. A, B X , f : A ∪ B → [0, 1], f (x) = 0, x ∈ A; f (x) = 1, x ∈ B, f A ∪ B , A, B 2.4 · 43 · A ∪ B . f X ( ) g, −1 −1 U = g ([0, 1/2)), V = g ((1/2, 1]), U , V A, B , X T4 . , . 2.4.2 (Tietze [403] ) X T4 , F , f F R , f X g, sup{|g(x)| : x ∈ X} = sup{|f (x)| : x ∈ F }. µ = sup{|f (x)| : x ∈ F }, F1 = f −1 ([µ/3, ∞)), Urysohn , F2 = f −1 ((−∞, −µ/3]). g0 : X → R µ/3, g0 (x) = x ∈ F1 , −µ/3, x ∈ F2 . sup{|g0 (x)| : x ∈ X} = µ/3, f1 (x) = f (x) − g0 (x), x ∈ F, sup{|f1 (x)| : x ∈ F } = µ1 2µ/3. , gn , fn (i) gn X , fn F ; (ii) f0 (x) = f (x), x ∈ F ; fn (x) = fn−1 (x) − gn−1 (x); (iii) sup{|fn (x)| : x ∈ F } = µn (2/3)n µ, µ0 = µ; (iv) sup{|gn (x)| : x ∈ X} = µn /3. ∞ g(x) = gn (x), x ∈ X. (2.4.1) n=0 n (iii) (iv), |gn (x)| µn /3 (2/3)n · µ/3, ∞ n=0 (2/3) · µ/3 = µ, (2.4.1) ( (i)) , g X . , (ii) f (x) = ∞ gn (x) = g(x), x ∈ F. n=0 g f X sup{|g(x)| : x ∈ X} = µ. . P. Urysohn 1925 . 1915 , H. Tietze ( 4.1.1) [114] . 2.4.1 T1 X (completely regular space) Tychonoﬀ (Tychonoﬀ space), X F x ∈ / F, f : X → [0, 1], f (x) = 0; f (x ) = 1, x ∈ F . · 44 · 2 F3 F3 , F3 T1 . , (F3 ⇒ T3 ), ( F4 ⇒ T4 ), . Tychonoﬀ [400] , , A. Mysior [311] . 2.4.1 ( ) M0 = {(x, y) ∈ R2 : y 0}, z0 = (0, −1) M = M0 ∪ {z0 }, L = R × {0}, Li = [i, i + 1] × {0}, i ∈ N. z = (x, 0) ∈ L, A1 (z) = {(x, y) ∈ M0 : 0 y 2}, A2 (z) = {(x + y, y) ∈ M0 : 0 y 2}. M (i) M0 − L ; (ii) z = (x, 0) ∈ L (A1 (z) ∪ A2 (z)) − B, B z ; (iii) z0 Ui (z0 ) = {z0 } ∪ {(x, y) ∈ M0 : x i}, i ∈ N, M T2 . , z ∈ M0 , z M M . , M F z0 ∈ / F . i0 ∈ N F ∩ Ui0 (z0 ) = ∅, U1 = Ui0 +2 (z0 ), U2 = M − (Ui0 +2 (z0 ) ∪ Li0 ∪ Li0 +1 ), U1 , U2 z0 , F . f : X → [0, 1] f (z0 ) = 1 M . , f (L1 ) = {0}. i ∈ N, Ki = {z ∈ Li : f (z) = 0}. Ki . Kn , Cn ⊂ Kn , z ∈ Cn j ∈ N, F (z, j) = A2 (z)−f −1 ([0, 1/j]), F (z, j) z , F (z, j) , A0 (z) = j∈N F (z, j), A0 (z) A2 (z) A0 (z) = A2 (z) − f −1 (0), f (A2 (z) − A0 (z)) = {0}. A ∪{A0 (z) : z ∈ Cn } L , A . Ln+1 − A ⊂ Kn+1 . t ∈ Ln+1 − A, z ∈ Cn A1 (t) ∩ (A2 (z) − A0 (z)) = ∅, A1 (t) {tm } f (tm ) = 0, m ∈ N. f f (t) = 0, t ∈ Kn+1 . , zi ∈ Ki , zi → z0 , f (z0 ) = 0, . . T1 , F4 ⇒ F3 , ⇒ , . 2.2.3 Niemytzki R , , , . 2.4 · 45 · x p(x, 0) F p (). p Uε (p) Uε (p) ∩ F = ∅, Uε (p) = Sε (p) ∪ {p}, Sε (p) x p ε , q ∈ Sε (p), q x p d(q), R g g(p) = 0, g(q) = d(q), q ∈ Sε (p), ε, q∈ / Uε (p), Sε (p) p ( ). f = g/ε, f : R → [0, 1], f (p) = 0; f (q) = 1, q ∈ F . T2 F2 X x1 , x2 , f : X → [0, 1], f (x1 ) = 0, f (x2 ) = 1, F2 T2 (functional separated T2 -space). , T2 T2 , . Urysohn [403] T2 T2 . Arens (simpliﬁed Arens square[372] ). I = (0, 1) , X = (I × I) ∪ {(0, 0), (1, 0)} ⊂ R2 , (i) I × I X ; (ii) (0, 0) Un (0, 0) = {(0, 0)} ∪ {(x, y) : 0 < x < 1/2, 0 < y < 1/n}, (iii) (1, 0) n ∈ N; Un (1, 0) = {(1, 0)} ∪ {(x, y) : 1/2 < x < 1, 0 < y < 1/n}, n ∈ N. Arens . X T2 . (0, 0), (1, 0) , f : X → [0, 1], f (0, 0) = 0, f (1, 0) = 1, X T2 . , Tychonoﬀ Tychonoﬀ . 2.4.3 Tychonoﬀ Tychonoﬀ . , f , x, U x, f (x) = 0, f (x ) = 1 (x ∈ X − U ) f (x, U ) . f1 , f2 , · · · , fn (x, U1 ), (x, U2 ), · · · , (x, Un ) , g(x) = sup{fi (x) : i = 1, 2, · · · , n}, g (x, ni=1 Ui ) . α∈A Xα (x, U ), f , U , x = {xα }α∈A . x ∈ α∈A Xα , Uα xα Xα , Xα Tychonoﬀ , (xα , Uα ) fα , fα ◦ pα (x, p−1 α (Uα )) , pα Xα , p−1 α (Uα ) . · 46 · 2 , T1 T1 . . X, Y , f : X → Y (, embedding), f : X → f (X) . 2.4.4 (Tychonoﬀ [400] ) X Tychonoﬀ X I = [0, 1] . I = [0, 1] Tychonoﬀ , Tychonoﬀ . 2.4.3, Tychonoﬀ , , X Tychonoﬀ , {fα }α∈A X [0, 1] , A, P = α∈A Iα (= I A ), Iα = [0, 1], α ∈ A. x ∈ X, P f (x) = {fα (x)}α∈A X P f , f : X → f (X) . 2.1.2, f . x, y X . X T1 , , fα : X → [0, 1] fα (x) = 0, fα (y) = 1, f (x), f (y) α , f (x) = f (y), f . f −1 , U X x (, U ). X Tychonoﬀ , fα : X → [0, 1] fα (x) = 0; fα (y) = 1, y ∈ X − U . P f (x) V = Uα × Iα , α =α Uα = [0, 1). x ∈ / U , fα (x ) = 1, f (x ) ∈ / V . f (x ) ∈ V x ∈ U , p ∈ V ∩ f (X) ⇒ f −1 (p ) ∈ U . f −1 (V ∩ f (X)) ⊂ U . f −1 f (X) X . f X f (X) ⊂ P . . 2.5 2.5.1 X (connected space), X ; X X ⊂ X (connected), X . , X , X , X . 2.5.1 R . R = F ∪ G, F , G . J = [a, b] J ∩ F = ∅, J ∩ G = ∅, b F G, b ∈ J ∩ G, c = sup(J ∩F ), J ∩F , c ∈ J ∩F . c < b, c , (c, b] ⊂ J ∩G. 2.5 · 47 · J ∩ G , [c, b] ⊂ J ∩ G, c ∈ F ∩ G, F , G . R . R Q . F = {r : r ∈ Q, r > η} ( . η, G = {r : r ∈ Q, r < η} Q), Q = F ∪ G. , R 2.5.1 {Aγ }γ∈Γ X , γ∈Γ Aγ = ∅, ∪{Aγ : γ ∈ Γ } . γ∈Γ Aγ = F ∪ G, F , G γ∈Γ Aγ G , γ ∈ Γ , . x ∈ γ∈Γ Aγ , x ∈ F x ∈ G. x ∈ F , G ∩ Aγ = ∅. F ∩ Aγ = F , G ∩ Aγ = G . F , G Aγ , Aγ = F ∪ G , F ∩ G = ∅, Aγ , . . 2.5.2 A X A ⊂ B ⊂ A, B . B = F ∪ G, F , G B . , F , G B . x ∈ F , x ∈ A ∩ B (A B , 2.1.1), x F F ∩ A = ∅; G ∩ A = ∅. F = F ∩ A, F , G . . G = G ∩ A, A , A = F ∪ G , A , 2.5.3 x X , P X , P . 2.5.1, P . 2.5.2, P . P ⊂ P , P = P , P . . x x ∈ P , 2.5.2 2.5.3 P X ( , component), ( , maximal connected subset); X , X (totally disconnected space). , , , . 2.5.4 X . Xα (α ∈ A) , X = α∈A Xα F , G , (F G) , X . F , x = {xα }α∈A ∈ F . x F . · 48 · 2 x F , . P = {{xα }α∈A : xα = xα , α = α0 }, P ⊂ X Xα0 , P . , P P ∩ F, P ∩ G . x ∈ P ∩ F , P ∩ F = ∅, P , P ∩ G = ∅, P ∩ F = P , P ⊂ F . x F . Q = {{yα }α∈A : yα = xα α }, , Q ⊂ F , Q ⊂ F . , , Q X, F = X, F , F = X, G = ∅. X . . 2.5.1 2.5.4 , n Rn . 2.5.3 X (locally connected), x ∈ X x U , V , x ∈ V ⊂ U ; X . , (, ). . 2.5.2 ( [372] ) R2 f (x) = sin(1/x) (0 < x 1) (0, 0) , (topologist’s sine curve). 2.5.2, , (0, 0) {(0, 0)}, {(0, 0)} . 2.5.4 X (arcwise connected space), a, b, f : [0, 1] → X, f (0) = a, f (1) = b. , , , 2.5.2 . 2.1 A 2.2 Y T2 2 X , f : X → X/A . ? 2.1.1 ? 2.1.2 Niemytzki R Y . ? ? 2.3 ? T1 Y . 2.4() X {x} (x ∈ X) 2.5(!) “ ” . T0 , T1 . T0 , T1 . 2.6 2.7 2 · 49 · y = sinx (x ∈ R) R [−1, 1] . X T2 x ∈ X} . W , Aγ ⊂ Xγ (γ ∈ Γ ). γ∈Γ Aγ = γ∈Γ Aγ . 2.8 {Xγ }γ∈Γ 2.9 T2 T2 2.10 2.11 2.12 X × X (diagonal) ∆ = {(x, x) : . . , Fσ . 2.13 2.14 X , F , G . F ⊂ W ⊂ G 2.15 W X Fσ , X [0, 1] x ∈ X, f (x) > 0}. 2.16 W X Fσ , W = ∞ i=1 Fi , ◦ Fi+1 ⊂ Fi+1 (i = 1, 2, · · ·). 2.17 ( 2.18 Lindelöf 2.19 2.20 ) Lindelöf Fi Fi ⊂ ( ) . . X , X (countable chain condition), CCC, . complement space [372] , ( (countable )). 2.22 Lindelöf , 2.23 2.24 X ,Y X X − Y = A ∪ B, A B ( 2.2.6), A ∪ Y 2.25 E 2 Lindelöf 2.26 X 2.27 f 2.28 . . R2 , . , E R Y, f (x) = y} W = {x : . , , f X . 2.21 . , G ⊃ F , Fσ . X T2 , X Y . , {(x, y) : x ∈ X, y ∈ X × Y . f X F I n (I = [0, 1], n ∈ N) , f X . 2.29 X (discrete [46] ), x ∈ X . {Fn } {Gn } Gn ⊃ Fn (n = 1, 2, · · ·). X , U (x) · 50 · 2 2.30 (Dowker) {Fγ }γ∈Γ X γ ∈ Γ , Uγ ⊃ Fγ , Fγ ⊂ Vγ ⊂ V γ ⊂ Uγ . 2.31 x 2.32 X Y f , {Uγ }γ∈Γ X {Vγ }γ∈Γ γ ∈ Γ , . , x ∈ X X ϕ(∆; >), f ◦ ϕ(∆; >) Y f (x). X = γ∈Γ Xγ ϕ(∆; >) x = {xγ }γ∈Γ ϕ(∆; >) Xγ (γ ∈ Γ ) pγ ◦ ϕ(∆; >) Xγ xγ . 3 . R Heine-Borel , . n Rn . . 3.1 3.1.1 X (compact space bicompact space), X . , Lindelöf ( 2.3.1), Lindelöf , . 2.3.2 , [0, ω1 ] , [0, ω1 ] . , α, [0, α] . ( 1.4.1) X F = {Fγ }γ∈Γ , F , Γ ⊂ Γ , γ∈Γ Fγ = ∅. 3.1.1 . X “X ” “ X X, X”, , de Morgan “X ”, . . 3.1.1. 3.1.1 ( ) . . X A, X U A ⊂ ∪U , U (cover) A. , . 3.1.2 X A X A A. 3.1.1 3.1.2, · 52 · 3 3.1.2 F1 , F2 , · · · , Fk X , F = ki=1 Fi Fi (i = 1, 2, · · · , k) . 3.1.2 , U = X − A, Fγ = X − Uγ (γ ∈ Γ ). de Morgan , 3.1.2 U ⊃ γ∈Γ Fγ U ⊃ γ∈Γ Fγ . A , U , 3.1.2 A , X , . 3.1.3 X , {Fγ }γ∈Γ X , U . U ⊃ γ∈Γ Fγ , Γ ⊂ Γ , U ⊃ γ∈Γ Fγ . 3.1.3 {Fγ }γ∈Γ X , ( Fγ0 ) , U . U ⊃ γ∈Γ Fγ , Γ ⊂ Γ , U ⊃ γ∈Γ Fγ . Fγ0 3.1.3 X, Fγ0 ∩ Fγ (γ ∈ Γ ) Fγ . . 3.1.4 X T2 , A, B , U, V , U ⊃ A, V ⊃ B. x ∈ B. y ∈ A, x = y, T2 , Vx Uy , x ∈ Vx , y ∈ Uy , {Uy }y∈A X k A ⊂ i=1 Uyi . A, 3.1.2, yi ∈ A (i = 1, 2, · · · , k), k k Vx = i=1 Vyi , Ux = i=1 Uyi , x ∈ Vx , A ⊂ Ux . , {Vx }x∈B X B, {Vxj }jn n n V = j=1 Vxj , U = j=1 Uxj , A ⊂ U, B ⊂ V . B. . 3.1.1, 3.1.4 . 3.1.4 T2 . 3.1.5 T2 . A T2 X . x ∈ X −A, 3.1.4 B = {x}, U , V , A ⊂ U, x ∈ V , V ⊂ X − A. A . . 3.1.4 , X T3 , x , . . 3.1.5 A T3 X , B ⊂ X − A, U ⊃ A, V ⊃ B. U , V , 3.1.5 , . 3.1.6 A Tychonoﬀ X , B ⊂ X − A, f f (x) = 0, x ∈ A; f (x) = 1, x ∈ B. X [0, 1] x ∈ A, X [0, 1] fx fx (x) = 0; fx (x ) = 1, x ∈ B. A ⊂ x∈A fx−1 ([0, 1/2)). 3.1.2, x1 , x2 , · · · , xk ∈ 3.1 A, A⊂ · 53 · k −1 i=1 fxi ([0, 1/2)). g(x) = min{fx1 (x), fx2 (x), · · · , fxk (x)}, g(x) X [0, 1] , A ⊂ g −1 ([0, 1/2)), g(x) = 1 (x ∈ B). f (x) = 2 · max{g(x) − 1/2, 0}, f (x) . . . 3.1.7 . f X Y , {Uγ }γ∈Γ Y −1 X . X , , {f (Uγ )}γ∈Γ −1 {f (Uγi )}i=1,2,···,k , X, {Uγi }i=1,2,···,k {Uγ }γ∈Γ , Y , Y . . 3.1.8 T2 . f X T2 Y , F X . 3.1.1, F . Y T2 , , 3.1.7, f (F ) Y . 3.1.5, f (F ) , f . . 3.1.6 T2 . 3.1.7 X T1 T2 T1 ⊃ T2 , (X, T1 ) , (X, T2 ) T2 , T1 = T2 . . 3.1.9 X (i) X ; (ii) X . ⇒ (i). F X , F = {A : A ∈ F } , 3.1.1, , x A (A ∈ F ), 1.4.3, x F . (i) ⇒ (ii). (i), X x, 1.4.3 , x. (ii) ⇒ . U X , , F = {X − U : U ∈ U } . 1.4.1, F ⊃ F . (ii), F , F x, x F , F · 54 · 3 , U ∈ U , x ∈ X − U = X − U. U X ( de Morgan ), U , X . . ( 3.8). ( 2.3.1) . 3.1.2 X N X (network [14] ), x ∈ X x U , N N , x ∈ N ⊂ U; X (network weight), n(X). X , X {{x} : x ∈ X} . , n(X) X w(X) X |X| , n(X) w(X) n(X) |X|. 3.1.1 , n(X) = w(X) . 3.1.3 {Xγ }γ∈Γ , γ = γ , Xγ ∩ Xγ = ∅. X = γ∈Γ Xγ U ⊂ X X , γ ∈ Γ , U ∩ Xγ Xγ . (O1)∼(O3), X . {Xγ }γ∈Γ (topological sum), γ∈Γ Xγ . 3.1.1 Ii (i = 1, 2, · · ·) I = [0, 1], x ∈ [0, 1], ∞ xi Ii . X = i=1 Ii Ii , R x ∈ (0, 1], xi ∈ Ii ; 0, 0i 0j , 0i , 0∗ , X/R = K, f . , K 0∗ ∞ i=1 f ([0i , xi )), xi (i = 1, 2, · · ·) , K 0∗ ℵℵ0 0 = c, w(K) c. , K − {0∗} (0i , Ii ] , B , B = B ∪ {{0∗}} K , n(X) ℵ0 (). n(K) < w(K). , K . 3.1.10 [14] X T2 , n(X) = w(X). n(X) w(X). n(X) = m, m , X T1 , |X| m, X , . m ℵ0 . N m. N (3.1.1) N1 , N2 U1 ⊃ N1 , U2 ⊃ N2 U1 ∩ U2 = ∅, (3.1.1) U1 , U2 ∈ T1 T1 X . (3.1.1) (N1 , N2 ), T1 (U1 , U2 ) , T1 U1 , U2 B, B 3.1 · 55 · B0 . B0 (B1) (B2) ( 1.2.2). B0 (B1) . x ∈ X, U ∈ T1 x ∈ U , , x ∈ V ⊂ V ⊂ U , N1 ∈ N x ∈ N1 ⊂ V , X − V . V x ∈ X − V , N2 ∈ N , x ∈ N2 ⊂ X − V , (N1 , N2 ) (3.1.1) , T1 (U1 , U2 ) , x ∈ U1 ∈ B ⊂ B0 , B0 (B2). B0 . B0 T2 , T2 ⊂ T1 . (X, T2 ) T2 . x1 , x2 ∈ X, x1 = x2 , (X, T1 ) T2 , U , U ∈ T1 x1 ∈ U , x2 ∈ U U ∩ U = ∅. N , N1 , N2 ∈ N x1 ∈ N1 ⊂ U , x2 ∈ N2 ⊂ U , (N1 , N2 ) (3.1.1) , (U1 , U2 ) , U1 , U2 ∈ T2 . (X, T2 ) T2 , 3.1.7, T2 = T1 . , |B0 | m, w(X) n(X). . 3.1.8 T2 , w(X) |X|. 3.1.9 f X T2 Y , w(Y ) w(X). B X , f , {f (U ) : U ∈ B} Y , 3.1.10 . . X 2.3.5 , X , 3.1.9 T2 . X , X X , X , 3.1.9 T2 . 3.1.2 (Alexandroﬀ [9] ) R2 Ci = {(x, y) : y = i, 0 x 1} (i = 1, 2). Z = C1 ∪ C2 , z ∈ Z B(z) (i) z ∈ C2 , B(z) = {{z}}; (ii) z = (x, 1) ∈ C1 , B(z) = {Uk (z)}∞ k=1 , Uk (z) = {(x , y ) : 0 < |x − x | < 1/k} ∪ {z}. , {B(z)}z∈Z (NB1)∼(NB4) ( 1.2.3). Z Alexandroﬀ (Alexandroﬀ’s double lines space). , Z T2 . C2 c , Z; C1 R , Z. Z ( 3.9). , Z , . , · 56 · 3 Z R : C2 , C1 , Z/R Z/C1 . C1 Z/R z ∗ , . Z/R ( 3.1.7). , Z/R T2 , z∗ . 3.1.9 . 3.2 Tychonoﬀ Tychonoﬀ ( 3.2.1) 2.1 Tychonoﬀ 3.5 ). " , , ( Tychonoﬀ , Tukey , N. Bourbaki [57] . A , A A A A . A 3.2.1 (Tukey ) , A0 ∈ A ∈ A , A ⊃ A0 , A = A0 . 3.2.1 (Tychonoﬀ [400, 401] ) . X = γ∈Γ Xγ , Xγ (γ ∈ Γ ) , F X , . Tukey , F0 ⊃ F . F , , x ∈ X, A ∈ F0 , x ∈ A. F0 , k A1 , A2 , · · · , Ak ∈ F0 , An ∈ F0 ; (3.2.1) n=1 A0 ⊂ X, A0 ∩ A = ∅ A ∈ F0 , F0 A0 ∈ F0 . (3.2.2) , γ ∈ Γ , Xγ {pγ (A)}A∈F0 xγ ∈ Xγ , xγ ∈ A∈F0 pγ (A). Wγ Xγ Xγ , . xγ , A ∈ F0 , Wγ ∩ pγ (A) = ∅, p−1 γ (Wγ ) ∩ A = ∅, −1 (3.2.2), pγ (Wγ ) ∈ F0 . (3.2.1), Γ0 ⊂ Γ , γ∈Γ0 p−1 γ (Wγ ) ∈ F0 , γ∈Γ0 p−1 γ (Wγ ) A ∈ F0 . ⎧ ⎫ ⎨ ⎬ −1 pγ (Wγ ) : Γ0 Γ , Wγ xγ ⎩ ⎭ γ∈Γ0 3.3 · 57 · x = {xγ }γ∈Γ ( ), A ∈ F0 , A ∈ F0 , x ∈ A. . x Tychonoﬀ , Tychonoﬀ . J. L. Kelley[231] Tychonoﬀ , . L. E. Ward[406] Tychonoﬀ ( ) . R T2 , 3.2.1, I = [0, 1] T2 ; , T2 ( 3.1.4), Tychonoﬀ , 2 Tychonoﬀ ( 2.4.4) . 3.2.2 . X Tychonoﬀ X T2 3.2.1 Rn A , J = [a, b] ⊂ R A ⊂ J n ⊂ Rn ; f : X → R , f (X) R . 3.2.3 Rn A A . A Rn , Rn T2 , A ( 3.1.5). ∞ A ⊂ i=1 Kin , Ki = (−i, i) i < j Kin ⊂ Kjn , A . i0 A ⊂ Kin0 . J = [−i0 , i0 ], A ⊂ J n , A . , A Rn , , A ⊂ J n , J = [a, b]. 3.2.1, J n , A J n , A ( 3.1.1). . 3.2.1 3.3 ( 3.1.7), . . 1.5 . —— , 3.3.1 [238] p . . . . X , Y , X × Y Y F X ×Y , p(F ) F Y . y0 ∈ / p(F ), F X × Y , x ∈ X, (x, y0 ) ∈ / F , x X Ux y0 Y Vxy0 , (Ux × Vxy0 ) ∩ F = ∅. {Ux }x∈X X n X , {Ux1 , Ux2 , · · · , Uxn }. Vy0 = i=1 Vxi y0 , Vy0 , · 58 · 3 y0 , (X × Vy0 ) ∩ F = ∅, Vy0 ∩ p(F ) = ∅. p(F ) Y . . ( 3.15), Kuratowski . 3.3.1 " X , Y ( 3.16). 3.3.1 [405] X Y f : X → Y (perfect mapping), y ∈ Y, f −1 (y) X . # ( $, 1993), “perfect mapping” , . 3.3.1 , y ∈ Y , f −1 (y) = X ×{y} X X × Y . 3.3.2 f : X → Y X Y . X T2 T3 , Y T2 T3 . −1 X T2 , y1 , y2 ∈ Y (y1 = y2 ), f (y1 ) ∩ f −1 (y2 ) = ∅. f , f −1 (y1 ), f −1 (y2 ) , 3.1.4, X U , V , U ⊃ f −1 (y1 ), f , V ⊃ f −1 (y2 ) U ∩ V = ∅. 1.5.4, Y U , V , y1 ∈ U , y2 ∈ V f −1 (U ) ⊂ U, f −1 (V ) ⊂ V, U ∩ V = ∅, Y T2 . X T3 , F Y , y ∈ / F , f −1 (y) ∩ f −1 (F ) = ∅, f −1 (y) , f −1 (F ) ⊂ X − f −1 (y). 3.1.5, X U , V , U ⊃ f −1 (y), y ∈ V ⊃ f −1 (F ) U ∩ V = ∅. , Y U , V , −1 −1 U , F ⊂ V f (U ) ⊂ U, f (V ) ⊂ V (V F ), U ∩ V = ∅, Y T3 . X , B X , U B , U X . U ∈ U , U = ∪{f −1 (y) : f −1 (y) ⊂ U }, f (U ) = Y − f (X − U ), f , f (U ) Y , V = {f (U ) : U ∈ U }. V Y . (3.3.1) 3.4 k · 59 · y ∈ Y , V y, f −1 (y) ⊂ f −1 (V ). f −1 (y) , U ∈ U , f −1 (y) ⊂ U ⊂ f −1 (V ). (1), f −1 (y) ⊂ U ⊂ U , y ∈ f (U ) ⊂ V , f (U ) ∈ V . . . 2 2.1.1 , D X , f X D , D , ( 2.1.1) 2.1.5, 3.3.2 X(D), . 3.3.3 X D , D , X(D) T2 T3 . X T2 T3 . . 3.3.4 f : X → Y X Y . Y Lindelöf , X Lindelöf . Lindelöf . , . U = {Uα }α∈Λ X . y ∈ Y , f −1 (y) U Uα , Uα Uy , f −1 (y) ⊂ Uy . 1.5.4, Y Uy , y ∈ Uy , f −1 (Uy ) ⊂ Uy . {Uy }y∈Y Y , Y Lindelöf , {Uy i }i∈N , {f −1 (Uy i )}i∈N X , f −1 (Uy i ) ⊂ Uyi (i ∈ N), Uyi U Uα , Uα ( Lindelöf . . 3.3.5 Lindelöf 3.3.1 i ∈ N) U , X Lindelöf . , 3.3.3 . . 3.4 k R , R , . R {(a, b) : a, b }. , R {[a, b] : a < b} ( ), , R , . 3.4.1 . X (locally compact), x∈X · 60 · 3 , (localization). R , . . , ( 3.1.1), . 3.4.1 . , . 3.4.2 T3 X . U x ∈ X , , x C. T3 , x V , x ∈ V ⊂ U ∩ C, V ⊂ C, C , V ( 3.1.1). . % . 3.4.3 T2 X Tychonoﬀ . U x0 ∈ X . , X → [0, 1], f (x0 ) = 0; f (x) = 1, x ∈ X − U . f : , C x0 , V = U ∩ C ◦ . X T2 , C ( 3.1.5), V ⊂ C, V ( 3.1.1). T2 V g : V → [0, 1], ( 3.1.4), Tychonoﬀ , g(x0 ) = 0; g(x) = 1, x ∈ V − V . X f : X → [0, 1], f (x) = g(x), x∈V, 1, x∈X −V. f X . F [0, 1] . 1 ∈ / F , f −1 (F ) = g −1 (F ), g , f −1 (F ) X . 1 ∈ F , f −1 (F ) = . V ⊂ U , f (X − V ) = g −1 (F ) ∪ (X − V ) X . f {1} ⇒ f (X − U ) = {1}, f . . 3.4.2, . 3.4.1 T2 T2 . , T2 . ( 3.1.7). , ( ). . 3.4.4 f X . Y . y ∈Y, 3.4 k · 61 · x ∈ f −1 (y), X , x 3.1.7, f (C) y . . C, f , . 3.4.1 f X Y , A X A = f −1 (B), B ⊂ Y . f A f |A : A → B . E [ 0.2 (i)] A , X F , f (A ∩ F ) = f (A) ∩ f (F ). E = A ∩ F. (3.4.1) f X Y , F X, f (F ) Y . (3.4.1) , f (E) = f (A ∩ F ) f (A). f |A A f (A) = B . . 3.4.5 f X Y . Y , X X . y ∈ Y, f −1 (y) X , X , x ∈ f −1 (y), x Cx . {Cx◦ }x∈X f −1 (y) , {Cx◦1 , Cx◦2 , · · · , Cx◦n }, Cy = ni=1 Cxi n f −1 (y) ⊂ U ⊂ Cy , U = i=1 Cx◦i . f , 1.5.1, X V f −1 (y) ⊂ V ⊂ U , f (V ) Y , y ∈ f (V ) ⊂ f (Cy ). f , f (Cy ) Y ( 3.1.7), f (Cy ) y . Y . , Y , x ∈ X, f (x) = y Uy . 3.4.1, f f −1 (Uy ) f |f −1 (Uy ) f −1 (Uy ) Uy , . 3.3.3 f −1 (Uy ) X , x . X . . , . , . 3.4.2 X , X A X C, A ∩ C C. A , C, A ∩ C C. , . A , A x ∈ / A. X , x C, A ∩ C = ∅. A ∩ C C x, x ∈ / A ∩ C, A ∩ C C. . 3.4.2 [128] X k (k-space), X A X C, A ∩ C C. 3.4.2, 3.4.6 . k . · 62 · 3 3.4.7 k . . A , A x ∈ / A, X , A {xi }, x ( 2.3.1). C = {xi : i ∈ N} ∪ {x}, C , A ∩ C = {xi : i ∈ N} C. . 3.4.8 f k X Y , Y k , k k . A ⊂ Y , Y K, A ∩ K K, A Y . f , f −1 (A) X . C X , f , f (C) Y ( 3.1.7). , A ∩ f (C) f (C). g = f |C : C → f (C) (f C ⊂ X ). g , g −1 (A ∩ f (C)) C, g −1 (A ∩ f (C)) = f −1 (A) ∩ C, f −1 (A) ∩ C C. C X , X k , f −1 (A) X . . 3.4.9 [94] X k X . k ( 3.4.6), 3.4.8, k . , X k , X . X S, S X . X {Kα }α∈Λ . () (& Kα Kα × {α}), , S = α∈Λ Kα . ( 3.1.3), S A ⊂ S S () α ∈ Λ, A ∩ Kα () Kα . S , S X f α ∈ Λ, f |Kα Kα X Kα , f (obvious mapping[171] ). f S X . , F X , f −1 (F ) S . (i) F X , f −1 (F ) ∩ Kα = F ∩ Kα Kα , Kα , f −1 (F ) S; (ii) f −1 (F ) S, f −1 (F ) ∩ Kα Kα , (S ) Kα , f −1 (F ) ∩ Kα = F ∩ Kα Kα (X ) Kα , X k , F X . . k A. Arhangel’skiı̌[18, 20] . 3.5 ( ). , Lindelöf 3.5 · 63 · , , . 3.5.1 X (countably compact space), X . ( 3.1.1), . , [0, ω1 ) ( 3.10), . 3.1.1 , ( 3.18). 3.5.1 X . 3.1.3 , . {Fγ }γ∈Γ X , , U . U ⊃ γ∈Γ Fγ , Γ ⊂ Γ , U ⊃ γ∈Γ Fγ . ( 3.1.9) . 3.5.2 x A ω (ω-accumulation point), x A . , A ω . 2.2.3, T1 A ω . 3.5.2 X (i) X ; (ii) X ω . ⇒ (i). {xn } X , Fn = {xn+i : i = 0, 1, 2, 3, · · ·} (n = 1, 2, · · ·), F = {F n } . 3.5.1, , x F n , x {xn } ( 1.4.7). (i) ⇒ (ii). A X , A {xn }, {xn } A ω . (ii) ⇒ . {Fn } X , n F = Fi : n = 1, 2, · · · . i=1 xn ∈ ni=1 Fi (n = 1, 2, · · ·). {xn }, , x , x {xn } ; , , ω x . {xn } x. x ∈ Fn (n = 1, 2, · · ·). 3.5.1, X . . · 64 · 3 2.2.3, . 3.5.1 X T1 , X . 3.5.3 X (sequentially compact), X . , [0, ω1 ) ( 3.10), . , 3.5.1 3.6.2 βN. 3.5.1 ( [372] ) I = [0, 1] . α ∈ I, Dα = {0, 1}, , Dα . X = α∈I Dα . Tychonoﬀ ( 3.2.1), X . X . n ∈ N, xn ∈ X pα (xn ) = α n , X , X {xn } α ∈ I, pα : X → Dα . , {xnk } {xn } , x ∈ X. α ∈ I, 2.1.4, Dα {pα (xnk )}k∈N pα (x). β ∈ I k , pβ (xnk ) = 0; k , pβ (xnk ) = 1. {pβ (xnk )}k∈N 0, 1, 0, 1, · · ·, , . X . , 3.5.2 . 3.5.3 . X , X [ 2.3.1 (iii)], . 3.5.4 . 3.5.4 X (pseudo-compact), X . 3.5.5 . f X , Un = {x : |f (x)| < n} (n = 1, 2, · · ·), {Un : n = 1, 2, · · ·} X . X , {Un1 , · · · , Unk }. |f (x)| < max{n1 , n2 , · · · , nk }, x ∈ X . X . . 3.5.5 , 3.5.2. 3.5.2 ( [372] [372] , ) X , x0 ∈ X. X ∅ x0 X , (particular point topology). X , (, X x ∈ X ) . 3.5.2 X . X , X . X 3.5 · 65 · . Sierpiński 1.5.1 ) . T0 , T4 . T4 . R {(a, +∞) : a ∈ R} R , Rr . (right order topology), R , Rr T0 . Rr , Rr T4 . Rr , Rr . Rr {(−n, +∞)}n∈N , Rr . 3.5.6 [193] . , Tietze ( 2.4.2) . 3.5.1 (Tietze ) X T4 , F , f F R , f X . arctan f F , ( | arctan f | < π/2. Tietze , arctan f X ( π/2. G = {x : |Φ(x)| = π/2}, ) Φ, |Φ| G X F , Urysohn X [0, 1] g(F ) ⊂ {1}, g(G) ⊂ {0}. Φ = g · Φ, X Φ , g, |Φ | < π/2 Φ (x) = arctan f (x), x ∈ F. ϕ = tan Φ f X ( ) . . 3.5.6 X , 3.5.2, {xn } , . {xn : n = 1, 2, · · ·}. X, X . f f (xn ) = n(n = 1, 2, · · ·). 3.5.1, f X ϕ, ϕ . X . . 3.5.2 Rr , 3.5.6 T4 . , , . , , ( ), . . · 66 · 3 (i) T2 , T2 , T2 ( 3.10), [0, ω1 ] T2 , T2 ( 3.23), ( 3.11). [0, ω1 ) [0, ω1 ) × [0, ω1 ] (ii) ( 3.2.1), . E. Čech “ ?” J. Novák[321] Tychonoﬀ , , [114] 3.10.19. , , ( [114] p. 208). . . J. Dieudonné 1944 . X U = {Uα }α∈A (locally ﬁnite[2] ), x ∈ X, x U (x) U (x) ∩ Uα = ∅, α ∈ A . V = {Vβ }β∈B U = {Uα }α∈A (reﬁnement), V Vβ U Uα ; V , V U (reﬁnement of a covering). 3.5.5 [106] X (paracompact), X . , , . , , , . 3.5.7 . F X , V = {Vα }α∈A F . F Vα , X Uα , Vα = Uα ∩ F . U = {Uα }α∈A , U ∪ {X − F } X , , {Wβ }β∈B , {Wβ ∩ F }β∈B F V , F . . T2 . . 3.5.2 ∪{U : U ∈ U }. U (), U ⊂ U , ∪{U : U ∈ U } = ∪{U : U ∈ U } ⊂ ∪{U : U ∈ U } , . x∈ / ∪{U : U ∈ U }, U , x V (x) U U , U1 , U2 , · · · , Un , x∈ / ∪{U : U ∈ U } ⇒ x ∈ / X− n i=1 U i n U i, i=1 x U1 , U2 , · · · , Un , x W (x) = 3.5 V (x) ∩ (X − · 67 · n i=1 U i ) ∪{U : U ∈ U } , x ∈ / ∪{U : U ∈ U }. ∪{U : U ∈ U } ⊃ ∪{U : U ∈ U }. . 3.5.2 U (), U ⊂ U , ∪{U : U ∈ U } . 3.5.3 X , A, B , x ∈ B, Ux , Vx , A ⊂ Ux , x ∈ Vx Ux ∩ Vx = ∅, U , V , A ⊂ U, B ⊂ V U ∩ V = ∅. {X − B} ∪ {Vx }x∈B X , , {Ws }s∈S . S1 = {s : s ∈ S, Ws ⊂ Vx x ∈ B }. Vx ⊂ X − Ux , V x ⊂ X − Ux ⊂ X − A, A∩V x = ∅, A∩W s = ∅ (s ∈ S1 ) B ⊂ s∈S1 Ws . 3.5.2, Ws = s∈S1 Ws . s∈S1 U = X − s∈S1 W s . U V = s∈S1 Ws 3.5.3 . . 3.5.8 [106] T2 . 3.5.3 A , T2 , 3.5.3 , T2 . , . . 3.5.7, 3.5.8 ( 3.1.1 3.1.4). , . 3.5.9 Lindelöf . , . 3.5.10 . . , X , X U , X , V U, V , V V1 , V2 , · · · , Vn , · · ·, n−1 Vn − Vi = ∅ (n = 2, 3, · · ·). i=1 x1 ∈ V1 , n (n = 2, 3, · · ·), xn ∈ Vn − n−1 i=1 Vi , {xn } (, x , x , Vn , V ). 3.5.2 X . . · 68 · 3 , , ( [60] [417] [265] ), 3.5.10 . 6.5 . 3.6 , . , . , , (i) “−∞” “+∞”, , . , (−1, 1) ( f : x → x/(1 + |x|)), −1, 1, [−1, 1]. −∞ +∞ −1 1 f . (ii) , , “∞” . , . “∞” , , , “∞” . 3.6.1 (compactiﬁcation) (f, Y ), Y ,f X Y (f (X) = Y ), Y T2 , (f, Y ) T2 . , X f (X) , Y X . X Y , f : X → f (X) ⊂ Y , (f, f (X)) X . T2 , Tychonoﬀ ) ( 2.4.4) 3.2.2 (Tychonoﬀ X Tychonoﬀ X T2 . 3 f −1 f (X) ⊂ Y f h X ϕ q ϕ(X) ⊂ Z 3.6 · 69 · 3.6.2 (f, Y ), (ϕ, Z) X , (f, Y ) (ϕ, Z), Y Z h, h ◦ f = ϕ, −1 h : Y → Z. ϕ ◦ f : f (X) → Z , ( T2 ) . 3.6.3 [2] X (one point compactiﬁcation) X ∗ = X ∪ {∞}, (i) X ; (ii) X ∗ U X ∗ − U X . . . 3.6.1 (Alexandroﬀ [2] ) (i) X X ∗ , X ; (ii) X ∗ T2 X T2 ; (iii) X (). 3.6.3 (i), (ii) . ∞ , () (i) , (i) . ∞ , X , X , (ii) . ∞ , (ii) (i) ((i) (i) ), X X , X , (ii) . X ∗ , X . U X ∗ , ∞ U ∈ U , X ∗ − U , , X ∗ . (ii) X ∗ T2 , X T2 T2 ( 3.4.1). X T2 , X ∗ T2 , x ∈ X, x ∞ , 3.4.1, x U , X ∗ − U ∞ . (iii) X ∗ , Y ∗ X , Y ∗ − X = ∞Y . f : X ∗ → Y ∗, f (∞) = ∞Y ; f (x) = x, x ∈ X. f . f, f −1 , f , ∞ Y ∗ . X ∗ − C ∞ , C X , f (C) , f , f (X ∗ − C) = Y ∗ − f (C) Y ∗ . . X X ∗ (i, X ∗ ), i X . 3.6.2, T2 , . 3.6.1 X T2 , A X , A . A , x ∈ A, x A V, X T2 , V , W V A , X · 70 · U 3 W = U ∩ A. A = X, U =U ∩X =U ∩A⊂U ∩A=W ⊂V ( W ⊂ V , V ). V x X , A . . 3.6.2 (f, Y ) X T2 , X X ∗ T2 , (f, Y ) (i, X ∗ ). 3.6.2, f −1 : f (X)(⊂ Y ) → X ∗ ∗ Y . h : Y → X , h(y) = f −1 (y), y ∈ f (X), ∞, y ∈ Y − f (X). h Y X∗ . X ∗ U X , U X X ∗ T2 , 3.6.1, X , h−1 (U ) = f (U ) f (X) . , f (X) . Y T2 f (X) = Y , 3.6.1, −1 f (X) Y , h (U ) Y . X ∗ U X ( ∞ ∈ U ), U = X ∗ − C, C X , h−1 (C) = f (C) f (X) , Y . Y T2 , h−1 (C) Y , h−1 (U ) Y . . . h Y 3.6.1 , ∞ . (0, 1], 0 [0, 1] . (0, 1] f : x → sin(1/x), (0, 1] [0, 1] . “ ” , f ——Stone-Čech . A , |A| A , I = [0, 1] , I A |A| [0, 1] . q ∈ I A q = {xα }α∈A , xα ∈ [0, 1] (α ∈ A), q A [0, 1] , α ∈ A, q(α) = xα . pα I A α (α ∈ A) , pα ◦ q = xα = q(α). 3.6.2 (3.6.1) f A B , y ∈ I B , f ∗ (y) = y ◦ f, (3.6.2) f ∗ I B I A . B y ∈ I B I , y ◦ f A I , f ∗ (y) = y ◦ f I B I A (). 3.6 · 71 · f ∗ (y) = y ◦ f I B I A , I A ( α ∈ A), pα ◦ f ∗ (y) I B ( 2.1.2). (3.6.2) (3.6.1), pα ◦ f ∗ (y) = pα (y ◦ f ) = (y ◦ f )(α) = y ◦ f (α), f (α) ∈ B, y ∈ I B . (3.6.1), y ◦ f (α) = pf (α) ◦ y. I B y f (α) , f∗ = y ◦ f . . y 1 I = [0, 1] y ? IA x y ∈ IB A f - B F (X) X I = [0, 1] . |F (X)| F (X) F (X) , I |F (X)| [0, 1] . Tychonoﬀ F (X) I . , I F (X) T2 . 3.6.4 X I F (X) e (evaluation mapping), x ∈ X, α ∈ F (X), pα ◦ e(x) = α(x). (3.6.3) α ∈ F (X) X [0, 1] , . 3.6.3 . . e X I (3.6.3) F (X) . , pα ◦ e = α, α X , 2.1.2, e X F X [0, 1] . F X (separate points), x, y ∈ X, x = y, f ∈ F f (x) = f (y); X (separate points and closed sets), X A x∈ / A, f ∈ F f (x) ∈ / f (A). , T2 ( 2.4.1) [0, 1] , [0, 1] (, 3.24). · 72 · 3 3.6.4 e X I F (X) , (i) F (X) , e X e(X) ; (ii) F (X) , e . (i) x ∈ X U e(U ) e(x) F (X) I e(X) [ 1.5.2 (iv)], F (X) , x X − U , f ∈ F (X), f (x) ∈ / f (X − U ). V = {y : y ∈ I F (X) , pf (y) ∈ / f (X − U )}. , V I F (X) , e(x) ∈ V , V ∩ e(X) ⊂ e(U ). (ii) x , x ∈ X, x = x , α ∈ F (X), α(x ) = α(x ). 3.6.4 (3.6.3), pα ◦ e(x ) = pα ◦ e(x ), e(x ) = e(x ). . 3.6.3 3.6.4 , . 3.6.3 X Tychonoﬀ , e X I F (X) e(X) . e(X) T2 I F (X) T2 , e(X) = βX. , (e, βX) X T2 . 3.6.5 (J. Dieudonné, 1949) X Tychonoﬀ , X T2 (e, βX) Stone-Čech (Stone-Čech compactiﬁcation), βX e(X) I F (X) . 3.6.4 (Stone-Čech [79, 379] ) X Tychonoﬀ , Y T2 , f X Y , (e, βX) X Stone-Čech , e(X) −1 Y f ◦ e βX Y . X Y f , F (Y ) F (X) f ∗ f ∗ (α) = α ◦ f, α ∈ F (Y ). (3.6.4) I F (X) I F (Y ) f ∗∗ f ∗∗ (q) = q ◦ f ∗ , q ∈ I F (X) . (3.6.5) e X I F (X) , g Y I F (Y ) , X Tychonoﬀ , 3.6.3, e X e(X) ⊂ βX, Y T2 , g Y F (X) T2 , g(Y ) ). g(Y ) = g(Y ) = βY ( Y , g(Y ) , I f ∗∗ e(X) ⊂ e(X) = βX ⊂ I F (X) −→ I F (Y ) ⊃ βY = g(Y ) = g(Y ) e 6 X 6 g f - Y 3 · 73 · 3.6.2, f ∗∗ , , f ∗∗ ◦ e = g ◦ f , g −1 ◦ f ∗∗ f ◦ e−1 . f ∗∗ ◦ e = g ◦ f . x ∈ X, h ∈ F (Y ), (3.6.1), (3.6.5), (3.6.4), (3.6.3) (1) (5) (4) (3),(1) ph (f ∗∗ ◦ e(x)) = (f ∗∗ ◦ e(x))(h) = (e(x) ◦ f ∗ )(h) = e(x) ◦ f ∗ (h) = e(x)(h ◦ f ) ==== (h ◦ f )(x) = h ◦ f (x) (3),(1) (1) ==== g(f (x))(h) = (g ◦ f (x))(h) = ph (g ◦ f (x)). . 3.6.1 T2 , Stone-Čech . 3.6.5, X Tychonoﬀ . (f, Y ) X T2 , f Tychonoﬀ X T2 Y , −1 3.6.4, f ◦ e h βX Y , 3.6.2 . . 3.6.1 ( Stone-Čech ) [0, 1] (0, 1] , (0, 1] Stone-Čech , (0, 1] T2 [−1, 1] x → sin(1/x), [0, 1] . , Stone-Čech , x → arctan x . 3.6.2 (βN, ) Stone-Čech , N ( ) , T2 , Tychonoﬀ . B. Pospı́šil[338] N Stone-Čech |βN| = 2c (c ), J. Novák[321] βN 2c . βN , βN , N ∪ {x} (x ∈ βN − N) ( [112] p. 244). , [0, ω1 ) [0, ω1 ], [0, ω1 ) Stone-Čech [0, ω1 ] ( 3.25). 3.1 3 . 3.2 {xn } x0 , {x0 } ∪ {xn : n ∈ N} . 3.3 3.4 T1 3.5 . 3.6 3.7 X . , , . , K T2 X . . X {Ui : i = 1, 2, · · · , k} K, · 74 · 3 X {Ki : i = 1, 2, · · · , k} k Ki , K= Ki ⊂ Ui , i k. i=1 3.8 3.1.9 3.9 Alexandroﬀ 3.10 [0, ω1 ) 3.11 [0, ω1 ), [0, ω1 ] , [0, ω1 ) × [0, ω1 ] . . ( 3.1.2) . . ( A = [0, ω1 ) × {ω1 } B = {(α, α) : α < ω1 } 3.12 T2 3.13 A T2 .) . , A , ∩{A : A ∈ A } . 3.14 f X , f , ε > 0, x ∈ X, f (x) > ε. 3.15[310] X , X [178] 3.16 . Y , f : X × Y → Y , f : X × Y → Y . X , Y . 3.17 T1 3.18 . X ). 3.19 3.20 T2 X {Fn } X ( ∞ n=1 Fn = ∅. ( , ). 3.21 X X . 3.22 . 3.23 3.24 Φ T1 3.25 [0, ω1 ) Stone-Čech 3.26 k X X [0, 1] . , , X . β[0, ω1 ) [0, ω1 ]. Y f f X . 3.27 3.28 3.29 X X T2 ( 2.3.1 ) k (f, Y ). . . U = {Uα }α∈A (point-finite), x ∈ X, x ∈ Uα α ∈ A . 3.30 3 · 75 · f : X → Y X Y , Y K, f −1 (K) X . 3.31 f :X →Y X Y (quasi-perfect[306] ) y ∈ Y , f −1 (y) ), Y K, f −1 (K) ( X . 3.32(Wallace [232] ) X1 , X2 , Ai ⊂ Xi (i = 1, 2), W , W ⊃ A1 × A2 , Xi Ui ⊃ Ai (i = 1, 2), ( X1 ×X2 W ⊃ U1 × U2 ⊃ A1 × A2 . 7.2.4). 4 . R, n Rn , , . , , . 4.1 4.1.1 X , x, y ∈ X, ρ(x, y) (M1) ρ(x, y) = 0 x = y; (M2) ρ(x, y) = ρ(y, x); (M3) ρ(x, y) ρ(x, z) + ρ(z, y), z ∈ X (), ρ(x, y) X (, metric), X ρ (, metric space), (X, ρ), X. (M1)∼(M3) (, metric axioms). (M1) () , () ; (M2) ρ(x, y) , x, y ; (M3) , , . 4.1.1 , (M1) (M1 ) ρ(x, y) = 0, x = y, ρ(x, y) X (, pseudo-metric), (X, ρ) (, pseudo-metric space). , , , , , , , . R x, y ρ(x, y) = | x − y |, , R . R (usual metric). , n Rn x = (x1 , x2 , · · · , xn ), y = (y1 , y2 , · · · , yn ) ρ(x, y) = (x1 − y1 )2 + (x2 − y2 )2 + · · · + (xn − yn )2 , 4.1 · 77 · ρ(x, y) ( ), Rn . (Euclidean metric). X, ρ∗ (x, x) = 0, ρ∗ (x, y) = 1, x = y, , (X, ρ∗ ) , (discrete metric space). , R n Rn , ( 4.1), (Lebesgue) ( 4.2) ( 4.1.1) . ∞ 2 4.1.1 () i=1 xi < ∞ x = (x1 , x2 , · · · , xn , · · ·) , (y = (y1 , y2 , · · · , yn , · · ·)) ∞ ρ(x, y) = (xi − yi )2 . (4.1.1) i=1 ρ(x, y) . 2 (4.1.1) , ∞ i=1 (xi − yi ) , (Cauchy) n 2 n n ai b i a2i · b2i , i=1 i=1 . (4.1.2) i=1 ai , bi . (4.1.2) n (ai + bi )2 = i=1 n a2i + 2 i=1 n ai b i + i=1 a2i + 2 i=1 ⎛ =⎝ n n a2i · 1/2 a2i b2i i=1 i=1 n n + i=1 n 1/2 b2i i=1 n + 1/2 ⎞2 b2i (4.1.3) b2i i=1 ⎠ . i=1 , xi ai , −yi bi , n n n 2 (xi − yi )2 xi + yi2 . i=1 n i=1 (4.1.3) i=1 n . n → ∞, , (4.1.3) , n → ∞ , (4.1.3) , . ∞ 2 . i=1 (xi − yi ) · 78 · 4 , ρ(x, y) (M1) (M2), (M3). , (4.1.3) −yi yi , n → ∞, ∞ ∞ ∞ 2 (xi + yi )2 x + y2. (4.1.4) i=1 i=1 i i=1 i a, b, c , a = (a1 , a2 , · · · , an , · · ·), b = (b1 , b2 , · · · , bn , · · ·), c = (c1 , c2 , · · · , cn , · · ·). (4.1.4) ai − ci xi , ci − bi yi , ∞ ∞ ∞ (ai − bi )2 (ai − ci )2 + (ci − bi )2 . i=1 i=1 i=1 ρ(x, y) (M3). ρ(x, y) , (Hilbert space), l2 (l2 -space). 4.1.2 ([38] ) (n1 , n2 , · · · , nk , · · ·) . x = (n1 , n2 , · · · , nk , · · ·), y = (m1 , m2 , · · · , mk , · · ·) ρ(x, y) = 0, x = y, 1/λ, x = y, λ nλ = mλ . ρ(x, y) , , (Baire’s zero-dimensional space). ρ(x, y) (M1) (M2), (M3). x, y, z . ρ(x, y) = 0 , (M3). ρ(x, y) = 1/λ0 , x = (n1 , n2 , · · · , nλ0 −1 , nλ0 , · · ·), y = (n1 , n2 , · · · , nλ0 −1 , mλ0 , · · ·), nλ0 = mλ0 . z = (l1 , l2 , · · · , lλ0 −1 , lλ0 , · · ·). 4.1 · 79 · (i) z λ0 − 1 x , λ1 < λ0 , lλ1 = nλ1 , ρ(x, z) 1/λ1 > 1/λ0 , , ρ(z, y) , ρ(x, y) ρ(x, z) + ρ(y, z). (ii) z λ0 − 1 x, y , λ < λ0 lλ = nλ . ρ(x, z) = 1/λ0 , ρ(x, y) ρ(x, z) + ρ(y, z); ρ(x, z) = 1/λ2 , λ2 > λ0 , lλ0 = nλ0 , mλ0 = nλ0 , lλ0 = mλ0 , ρ(z, y) = 1/λ0 , ρ(x, y) ρ(x, z) + ρ(y, z). ρ(x, y) (M3). , A, (generalized Baire’s zero-dimensional space), N (A). . (X, ρ) , Sε (x0 ) = {x : x ∈ X, ρ(x, x0 ) < ε} x0 (open ball) ( ε (ε-open ball)). R , Sε (x0 ) = (x0 − ε, x0 + ε) x0 ; R2 , Sε (x0 ) x0 , ε ( ). 4.1.1 (X, ρ) . x ∈ X, U (x) = {Sε (x) : ε > 0}, U = ∪{U (x) : x ∈ X} = {Sε (x) : ε > 0, x ∈ X} X . (B1) (B2) ( 1.2.2), (B2) (∪{U : U ∈ U } = X). (B1). Sr (x) ∩ Ss (y) = ∅, z ∈ Sr (x) ∩ Ss (y), t = min{r − ρ(x, z), s − ρ(y, z)}, St (z). St (z) ⊂ Sr (x) ∩ Ss (y). w ∈ St (z), ρ(w, z) < t, (M3), ρ(w, x) ρ(w, z) + ρ(z, x) < t + ρ(z, x) r − ρ(x, z) + ρ(z, x) = r. St (z) ⊂ Sr (x), St (z) ⊂ Ss (y), z ∈ St (z) ⊂ Sr (x) ∩ Ss (y). . , {Sε (x) : ε > 0, x ∈ X} X . ρ X (metric topology). , . R ρ(x, y) =| x−y | R . , Rn Rn (Euclidean topology), Rn . · 80 · 4 4.1.3 () R2 P1 = (x1 , y1 ), P2 = (x2 , y2 ), ρ(P1 , P2 ) = (x1 − x2 )2 + (y1 − y2 )2 , ρ (P1 , P2 ) = max{| x1 − x2 |, | y1 − y2 |}, ρ (P1 , P2 ) = | x1 − x2 | + | y1 − y2 | . ρ, ρ , ρ , R2 . , . O = (0, 0), A = (1, 1), √ ρ(O, A) = 2, ρ (O, A) = 1, ρ (O, A) = 2. ρ R2 , . ρ , ρ , (M1), (M2) , (M3). P3 = (x3 , y3 ). ρ (P1 , P2 ) = max{| x1 − x2 |, | y1 − y2 |} = max{| x1 − x3 + x3 − x2 |, | y1 − y3 + y3 − y2 |} max{| x1 − x3 |, | y1 − y3 |} + max{| x3 − x2 |, | y3 − y2 |} = ρ (P1 , P3 ) + ρ (P3 , P2 ), ρ (P1 , P2 ) =| x1 − x2 | + | y1 − y2 | =| x1 − x3 + x3 − x2 | + | y1 − y3 + y3 − y2 | | x1 − x3 | + | y1 − y3 | + | x3 − x2 | + | y3 − y2 | = ρ (P1 , P3 ) + ρ (P3 , P2 ), , ρ , ρ R2 . ρ R2 . () Sε (P ) = {P : ρ(P, P ) < ε} . Sε (P ) ρ Sε (P ), Sε (P ) ⊂ Sε (P ) ⊂ Sε/2 (P ), ρ . Sε (P ) ⊂ Sε (P ). P = (x , y ) ∈ Sε (P ), | x − x| + | y − y| < ε, a2 + b2 | a| + | b| (x − x)2 + (y − y)2 | x − x | + | y − y |< ε, 4.1 · 81 · ρ(P, P ) < ε, P ∈ Sε (P ). Sε (P ) ⊂ Sε (P ). , Sε (P ) ⊂ Sε/2 (P ). , ρ R2 . , ρ R2 . , Sε (P ) ρ Sε (P ), √ Sε/ (P ) ⊂ Sε (P ) ⊂ Sε (P ). 2 , . . R2 , , R2 (X, ρ∗ ) (ρ∗ (x, x) = 0, ρ∗ (x, y) = 1, x = y), ρ∗ , U (x) = {{x}, X} ( Sε (x) = {x}, 0 < ε 1), . 4.1.1 , {S1/n (x) : n = 1, 2, · · ·} {Sε (x) : ε > 0} U (x), U (x) X . . (M1), , ρ(x, y) = r > 0, x = y. Sr/2 (x), Sr/2 (y) x, y . 4.1.5). T2 . ( 4.1.2 (X, ρ) , , ρ(x, y) X × X . (M3), ρ(x, y) ρ(x, x0 ) + ρ(x0 , y), x0 X , ρ(x, y) − ρ(x0 , y) ρ(x, x0 ). x, x0 , ρ(x0 , y) − ρ(x, y) ρ(x0 , x). (M2), , | ρ(x, y) − ρ(x0 , y) | ρ(x, x0 ). ε > 0, δ(ε) = ε, x ∈ Sε (x0 ) | ρ(x, y) − ρ(x0 , y) |< ε, lim ρ(x, y) = ρ(x0 , y). ρ(x, y) x , x→x0 δ(ε) ε , , y y , x y . . ρ(x, y) X × X . . · 82 · 4.1.2 4 (X, ρ) . x ∈ X A, B ⊂ X D(A, x) = D(x, A) = inf {ρ(x, y)} y∈A x A (distance from a point to a set); D(A, B) = inf x∈A,y∈B {ρ(x, y)} A B (distance from a set to a set). D(x, ∅) = D(∅, x) = 1, D(A, ∅) = D(∅, A) = 1. 4.1.3 A (X, ρ) , , D(A, x) X . (M3), ρ(x, z) ρ(x, y) + ρ(y, z), inf {ρ(x, z)} ρ(x, y) + inf {ρ(y, z)}, z∈A z∈A D(A, x) ρ(x, y) + D(A, y). D(A, x) − D(A, y) ρ(x, y). x, y , D(A, y) − D(A, x) ρ(x, y). , | D(A, y) − D(A, x) | ρ(x, y). y ∈ Sε (x) | D(A, y) − D(A, x) | < ε. D(A, x) X . . , , . 4.1.4 (X, ρ) , A ⊂ X, A = {x : D(A, x) = 0}. 4.1.3, D(A, x) X , {0} , , {0} {x : D(A, x) = 0} X. , {x : D(A, x) = 0} ⊃ A. {x : D(A, x) = 0} ⊃ A. , y ∈ A, Sε (y) Sε (y) ∩ A = ∅, D(A, y) ε, y ∈ {x : D(A, x) = 0}. {x : D(A, x) = 0} ⊂ A. . 4.1.5 . T2 . A, B (X, ρ) . 4.1.4, A = {x : D(A, x) = 0}, B = {x : D(B, x) = 0}. 4.1 · 83 · 4.1.3, D(A, x), D(B, x) X , D(x) = D(A, x) − D(B, x), D(x) X . U = {x : D(x) < 0}, V = {x : D(x) > 0}. U = D−1 ((−∞, 0)), V = D−1 ((0, +∞)) R (−∞, 0), (0, +∞) D , U, V . U ⊃ A V ⊃ B. A ∩ B = ∅, x ∈ A ⇒ x ∈ B, D(A, x) = 0 ⇒ D(B, x) > 0, D(A, x) = 0 ⇒ D(x) < 0 ⇒ x ∈ U, A ⊂ U . B ⊂ V . . . (X, ρ) , A ⊂ X, ρ , (A, ρ) ( , 2.2.7 ), A X . 4.1.5, ( ). 4.1.6 Gδ . 4.1.4, (X, ρ) F = {x : D(F, x) = 0}. n = 1, 2, · · ·, Gn = {x : D(F, x) < 1/n}, D(F, x) X [0, +∞) , [0, 1/n) [0, +∞), Gn . , {x : D(F, x) = 0} = ∞ {x : D(F, x) < 1/n}, n=1 F= ∞ . n=1 Gn , F Gδ . 4.1.3 (perfect), Gδ ; (perfectly normal), , . 4.1.1 . , ( ). ∗ (X, ρ ) , X , . 4.1.7 (X, ρ) , (i) ; (ii) Lindelöf ; (iii) ; · 84 · 4 (iv) ; (v) ; (vi) . (i) ⇒ (ii). 2.3.3. (ii) ⇒ (iii). A X . x ∈ A X Ux Ux ∩ A = {x}. U = {Ux }x∈A ∪ {X − A}. A , U . A X , (ii), X Lindelöf , U . (iii) ⇒ (iv). B X , B . 3.6.1 T2 , B B . 4.1.6), B Gδ ( ∞ B Fσ , B = i=1 Ai , Ai B, X. Ai X , (iii), Ai , B . (iv) ⇒ (v). U = {Uα }α∈A X . α ∈ A, xα ∈ Uα , B = {xα : xα ∈ A} , (iv), B , U . (v) ⇒ (vi). i = 1, 2, · · ·, Fi 1/i . Fi . Tukey ( 3.2.1), Fi Ai (i = 1, 2, · · ·). x, y ∈ Ai , ρ(x, y) > 1/i, Ai S1/2i (x), {S1/2i (x) : x ∈ Ai } . (v) . A , Ai , A = ∞ i=1 Ai X, A = X. , x ∈ X − A, 4.1.4, D(A, x) > 0, i0 D(A, x) > 1/i0 , D(Ai0 , x) D(A, x) > 1/i0 . , x Ai0 1/i0, , Ai0 Fi0 . (vi) ⇒ (i). A = {x1 , x2 , · · · , xn , · · ·} X . U = {Sr (xn ) : r , n = 1, 2, · · ·}. U X . , A , , U . x ∈ X U x, Sr (x) ⊂ U . A X , xi ∈ A ρ(xi , x) < r/3, V = S2r/3 (xi ). ρ(xi , x) < r/3, x ∈ V , x ∈ V, ρ(xi , x ) < 2r/3, ρ(x, x ) ρ(x , xi ) + ρ(xi , x) < 2r/3 + r/3 = r, V ⊂ Sr (x), x ∈ V ⊂ U , U X . . 4.1 · 85 · . X ℵ1 (ℵ1 -compact space), X ; X . (iii), (v) X ℵ1 ( 2.21). (ii) ⇒ (iii) Lindelöf ⇒ ℵ1 ; ⇒ ℵ1 . ( , (X, ρ∗ ) X ). . 4.1.8 (X, ρ) , (i) X (ii) X ; (iii) X ω ; ( (iv) X ; (v) X (vi) X (vii) X ); ( ); ( ); ( ). , (iii), (iv), (v) ( 3.5.2). T1 , (ii), (iii), (iv), (v) ( 3.5.1). , (i), (ii), (iii), (iv), (v) ( 3.5.5 3.5.6). , (iii), (iv), (v), (vi) ( 3.5.3 3.5.4). ( 4.1.5) , , (i) (vi) . , (vii) (i) (vi) . Lindelöf , ( 3.5.9, (vii) (v) ). : , (ii) ⇒ Lindelöf , . A X , A , (ii) A , A , X , A , A . 4.1.7, X Lindelöf . . . () R r Q ( η I), Sε (r) ( Sε (η)) , Q ( I) ( 3.4.1). , ( (X, ρ∗ ) X ). 4.1.4 A (X, ρ) , d(A) = supx,y∈A{ρ(x, y)} A (diameter); , , d(A) = ∞. d(∅) = 0. · 86 · 4 4.1.9 (X, ρ) , ρ (x, y) = min{1, ρ(x, y)}, (X, ρ ) , ρ, ρ . , ρ (M1) (M2). ρ (M3), , x, y, z ∈ X 1 ρ (x, z) > ρ (x, y) + ρ (y, z), ρ (x, y) < 1, ρ (y, z) < 1, ρ (x, y) + ρ (y, z) = ρ(x, y) + ρ(y, z) ρ(x, z), ρ (x, z) > ρ(x, z), . (X, ρ ) . X ρ, ρ . ε > 0, x ∈ X, Sε (x) = {x : x ∈ X, ρ(x , x) < ε}, Sε (x) = {x : x ∈ X, ρ(x , x) < ε}. 0 < ε < 1 , Sε (x) = Sε (x), ρ, ρ . . 1 , , 1. 4.1.5 (X, ρ) (X , ρ ) f (isometry mapping), X x, y ρ(x, y) = ρ (f (x), f (y)). (x = x ⇒ f (x) = f (x )). f , f , . , f X Sr (x) X Sr (f (x)), . , . (metric invariant). ( ). , R R I = (0, 1) , d(R) = ∞, d(I) = 1, , . 4.1.10 {(Xn , ρn )} , (Xn , ρn ) x = {xn }, y = {yn }, 1. X = ∞ n=1 Xn ρ(x, y) = ∞ 1 n=1 2n · ρn (xn , yn ), ρ X , ρ X ρn Xn (n = 1, 2, · · ·) . ρ , X , ρ () ρn Xn ( ). V , U . 4.1 · 87 · V = Sr (x) V , n0 , U 1 U = y : ρn (xn , yn ) < n0 +1 , n n0 + 1 , 2 y ∈ U , 1/2n0 < r. n 0 +1 ∞ 1 1 + . ρ(x, y) n0 +1 n 2 2 2n n=1 n=n +2 1 0 n 0 +1 ∞ ∞ 1 1 1 1 < = 1, = n0 +1 , n n n 2 2 2 2 n=1 n=1 n=n +2 0 ρ(x, y) < 1 1 1 + n0 +1 = n0 . 2n0 +1 2 2 U ⊂ V . V U . , U ∈ U U , U {x : xn ∈ W }, (n) (n) W Xn . (Xn , ρn ) , Sr (xn ) Sr (xn ) ⊂ W (n) ( Sr (xn ) = {yn : ρn (yn , xn ) < r}). ρ , ρ(x, y) = ∞ 1 k=1 1 ρk (xk , yk ) n ρn (yn , xn ). 2k 2 , ρn (xn , yn ) < r, ρ(x, y) < r/2n . x Sr/2n (x) ∈ V , Sr/2n (x) ⊂ U ∈ U . U , U U x ∈ U , V V x ∈ V ⊂ U . U V . , ρ X = ∞ n=1 Xn ρn Xn (n = 1, 2, · · ·) . . ω I ( 2.1.3) ( 4.1.1) ; 4.1.10 . I ω . . , , “ {Xα }α∈A (A ) ”. X = α∈A Xα B = {B} B pα (B) α , pα (B) = Xα . : B Xα Xα , xα ∈ Xα Vα (Vα = Xα ). x = {xα } ∈ α∈A Xα . , {Un (x)} x · 88 · 4 . Un (x) B , A , α0 ∈ A Un (x) (n = 1, 2, · · ·), pα0 (Un (x)) = Xα0 . xα0 Vα0 ⊂ Xα0 , x = {xα } p−1 Un (x). {Un (x)} α0 (Vα0 ) x . . 4.1.11 {(Xα , ρα )}α∈A , X = α∈A Xα . 4.1.9, α ∈ A, x, y ∈ Xα , ρα (x, y) 1. x, y ∈ X, ρ ρ(x, y) = ρα (x, y), 1, x, y Xα . ρ (M1) (M2). , ρ (M3) ρ(x, z) ρ(x, y) + ρ(y, z). , x, y, z ρ(x, z) > ρ(x, y) + ρ(y, z), ρ(x, y) < 1, ρ(y, z) < 1, α ∈ A x, y, z ∈ Xα , ρ(x, y) + ρ(y, z) = ρα (x, y) + ρα (y, z) ρα (x, z) = ρ(x, z), . (M3) . ρ X , (X, ρ) , ( 3.1.3), ρ X ρα Xα . . , . , . 4.1.4 (J. Dieudonn´e [132] , ) E ⊂ R2 , y ( Y ) (1/n, k/n2 ) , n , k . E (i) {(1/n, k/n2)} ; (ii) Y (0, y0 ) {Un (y0 ) : n = 1, 2, · · ·}, Un (y0 ) = {(x, y) : x 1/n, |y − y0 | x}. , E , Y E . , Y , ( , Y , Y ρ (ρ(x, x) = 0, ρ(x, y) = 1, x = y)). {(1/n, k/n2)} , E . 4.2 · 89 · E . (1/n, k/n2 ) E , E , Y E , 4.1.7, E . , E . , X , X . [376] , A. H. Stone . , , , 4.1.5. 4.1.5 ( ) R S = {0} ∪ {1/n : n ∈ N} (). α < ω, Sα S. α<ω Sα , X. X A ( ) X/A Sω , (sequential fan). X ( 4.1.11), f : X → Sω ( 2.1), Sω . f (A) = {a}, Sω a , Sω , Sω . a Sω {Uα }α<ω , f −1 (Uα ) = (Uα − {a}) ∪ A X A . α < ω, Sα A , xα ∈ Sα ∩ (Uα − {a}), {xα : α < ω} X , X − {xα : α < ω} X , U = Sω − f {xα : α < ω} Sω , a, Uα ⊂ U , Sω . 4.2 4.2.1 (X, ρ) (totally bounded), ε > 0, ε X, ε F (ε) X = x∈F (ε) Sε (x). F (ε) ε (ε-dense) X . ∗ (X, ρ ) , X . R () . R [a, b], (a, b) . , . l2 A = {xn : n ∈ N}, xn = (0, 0, 0, · · · , 1, 0, · · ·) ( n 1, 0), ( 4.1.1), A √ ( A d(A) < +∞), A 2; , √ ε 2, F (ε) ε A ( A = x∈F (ε) Sε (x)). . 4.2.1 . · 90 · 4 (X, ρ) , M ⊂ X. ε > 0, F (ε/2) = {x1 , x2 , · · · , xk } ε/2 X. x ∈ M , xi ∈ F (ε/2) ρ(x, xi ) < ε/2. xi {xm1 , xm2 , · · · , xml }, j l, xj ∈ M ρ(xj , xmj ) < ε/2. F = {x1 , x2 , · · · , xl }, F ε M . x ∈ M , M ⊂ X, xi ∈ F (ε/2) ρ(x, xi ) < ε/2, xi xmj , ρ(x, xj ) ρ(x, xmj ) + ρ(xmj , xj ) < ε/2 + ε/2 = ε. . 4.2.2 (X, ρ) , M ⊂ X. (M, ρ) , (M , ρ) . ε/2 M ε M . . 4.2.3 {(Xn , ρn )} , (Xn , ρn ) 1. X = ∞ 4.1.10 ρ, (X, ρ) n=1 Xn (Xn , ρn ) . (X, ρ) . m ∈ N, ∞ ∗ Xm = n=1 An , Am = Xm , An = {x∗n } Xn , n = m. ∗ Xm ( 4.2.1). ρ ( 4.1.10), x∗ , y ∗ ∈ ∗ ⊂ X, ρ(x∗ , y ∗ ) = ρm (x, y)/2m , x = pm (x∗ ), y = pm (y ∗ ). , Xm ∗ F ε/2m (Xm , ρ), pm (F ) ε (Xm , ρm ). (Xm , ρm ) . (Xn , ρn ) . ε > 0, k 1/2k < ε/2, n k, {xn1 , xn2 , · · · , xnm(n) } ε/2 Xn ; n > k, xn0 ∈ Xn . F = {y = (x1j1 , x2j2 , · · · , xkjk , xk+1 , xk+2 , · · ·) : 1 jn m(n), n k}, 0 0 F . F ε (X, ρ). x = (x1 , x2 , · · · , xn , · · ·) X , n k, jn m(n) ρn (xn , xnjn ) < ε/2, F y = (x1j1 , x2j2 , · · · , xkjk , xk+1 , xk+2 , · · ·), 0 0 ρ(x, y) = k 1 n=1 2 ρ (x , xnjn ) + n n n ∞ n=k+1 1 ρ(xn , xn0 ) 2n < ε/2 + ε/2 = ε. F ε X. . . , . {(Xα , ρα )}α∈A , 4.2 (Xα , ρα ) 1, α∈A Xα α∈A Xα (Xα , ρα ) · 91 · 4.1.11 ρ, , A . (X, ρ) ω , (X, ρ) 4.2.4 . ε > 0 F (ε) X = x∈F (ε) Sε (x). , ε0 > 0, , F (ε0 ), X = x∈F (ε0 ) Sε0 (x). 2 x1 ∈ X, X = Sε0 (x1 ), x2 ∈ X − Sε0 (x1 ), X = i=1 Sε0 (xi ), , {x1 , x2 , · · · , xn , · · ·}. , ρ(xi , xj ) ε0 (i = j). ω x0 ∈ X, Sε0 /2 (x0 ) {x1 , x2 , · · · , xn , · · ·} . xn , xm ∈ Sε0 /2 (x0 ), ρ(xn , xm ) ρ(xn , x0 ) + ρ(x0 , xm ) < ε0 /2 + ε0 /2 = ε0 . . (X, ρ) . . 4.2.1 . 4.2.2 (X, ρ) {xn } (Cauchy sequence), ε > 0, k, m, n k , ρ(xn , xm ) < ε. , (X, ρ) . . X ρ, ρ , ρ, ρ , ρ , ρ ( 4.2.1). . 4.2.1 ( ) R {n}, R ρ(x, y) = |x − y| , {n} . R x y . − ρ (x, y) = 1 + |x| 1 + |y| f (x) = x/(1 + |x|) R (−1, 1) , ρ, ρ R . n+l l n 1 = − < , ρ (n + l, n) = 1+n+l 1+n (1 + n + l)(1 + n) n ε > 0, n > [1/ε] = k ([1/ε] 1/ε ), l = 1, 2, · · ·, ρ (n + l, n) < ε. {n} ρ . 4.2.5 (X, ρ) x0 , x0 . X {xn } x0 , Tk {xk , xk+1 , · · ·} , d(Tk ) Tk . x0 ε Sε (x0 ). · 92 · 4 {xn } , x0 . Tk . 4.2.5, 4.2.2 d(Tk ) < ε/2. x0 {xn } , x0 , Sε/2 (x0 ) ∩ Tk = ∅, Tk ⊂ Sε (x0 ), {xn } k 4.1.8, . , . 4.2.3 (X, ρ) (complete metric space), . R, Rn . R , {n/(n + 1)} R , (−1, 1) . R , (−1, 1) , R (−1, 1) , . . 4.2.2 () C[0, 1]. [0, 1] x(t), y(t), ρ(x, y) = max |x(t) − y(t)| , C[0, 1] 0t1 ( 4.1). {xn } C[0, 1] , , , {xn } {xn (t)} [0, 1] : “ ”, {xn (t)} x0 , x0 (t), x0 ∈ C[0, 1]. {xn } C[0, 1] . 4.2.3 () . 4.1.1 l2 , {x(n) } l2 , (n) (n) (n) x(n) = (x1 , x2 , · · · , xk , · · ·). ε > 0, i, (m) (ρ(x m, n > i (n) ,x 2 )) = ∞ , k=1 (m) (n) (m) (xk (n) − xk )2 < ε. (n) (4.2.1) k, (xk − xk )2 < ε, {xk } R , R , {x(n) xk . x = (x1 , x2 , · · · , xk , · · ·), k } 2 (i) ∞ k=1 xk < ∞, x ∈ l2 ; (ii) lim ρ(x(n) , x) = 0. n→∞ 4.2 · 93 · (4.2.1) ∞ k=1 (m) (n) (xk − xk )2 = j k=1 (m) (n) (xk − xk )2 + ∞ (m) k=j+1 (xk (n) − xk )2 < ε, j . , ε, j (m) (xk k=1 , n, (n) − xk )2 < ε. m → ∞, j k=1 j , (n) (xk − xk )2 ε. j → ∞, ∞ k=1 (n) (xk − xk )2 ε. (4.2.2) ( (a + b)2 2(a2 + b2 )) k0 k=1 x2k = k0 (n) (n) (xk − xk + xk )2 k=1 2 k0 (n) (xk − xk )2 + 2 k=1 k0 k=1 (n) (xk )2 , (n) 2 k0 , k0 → ∞ , (4.2.2) ∞ ( k=1 (xk ) ∞ 2 (n) (n) (n) (n) x = (x1 , x2 , · · · , xk , · · ·) ∈ l2 ), k=1 xk , x ∈ l2 , (i). , (4.2.2) ε , ∞ (n) lim ρ(x, x(n) ) = lim (xk − xk )2 = 0. n→∞ n→∞ k=1 , l2 ρ , x(n) → x, , 4.2.6 (n) l2 {x (ii). } , l2 (X, ρ) (X, ρ) . . . . (X, ρ) , 4.2.1, (X, ρ) 4.2.5, (X, ρ) , . . · 94 · 4 ( , (X, ρ) . ), . (X, ρ) , , (X, ρ) , ( 4.1.8). {xn } (X, ρ) . , {xn } {xnk } , (X, ρ) {xn } {xnk }, {xnk } . (X, ρ) , 1 X. S1 {xn } xn , S1 xn n N1 , N1 , n ∈ N1 , xn ∈ S1 . S2 N1 1/2 X. N2 (N2 ⊂ N1 ) ( N1 ), n ∈ N2 , xn ∈ S2 . , Nk , 1/(k + 1) Sk+1 Nk+1 ⊂ Nk , n ∈ Nk+1 , xn ∈ Sk+1 . , nk , nk+1 ∈ Nk+1 , nk+1 > nk . Nk , . i, j k, ni , nj Nk ( N1 ⊃ N2 ⊃ · · · ⊃ Nk ⊃ · · ·), xni , xnj 1/k . {xnk } . . n1 ∈ N 1 , n2 ∈ N 2 , n2 > n1 . [237] 4.2.7 (Cantor ) , (i) Fn+1 ⊂ Fn (n ∈ N); (ii) lim d(Fn ) = 0 {Fn }, ∞ n=1 Fn . n→∞ . (X, ρ) , {Fn } (i), (ii) . n ∈ N, xn ∈ Fn , {xn } . (ii), ε > 0, nε , n > nε , d(Fn ) < ε; (i), n m > nε , xn ∈ Fn ⊂ Fm , xm ∈ Fm , ρ(xn , xm ) d(Fm ) < ε. {xn } . (X, ρ) , {xn } x0 ∈ X, x0 Fn (n = 1, 2, · · ·) . Fn , x0 ∈ ∞ n=1 Fn . ∞ ∞ x0 n=1 Fn , n=1 Fn = {x0 }. y ∈ ∞ n=1 Fn , (ii), ε > 0, nε n > nε , d(Fn ) < ε, x0 , y ∈ Fn , ρ(x0 , y) d(Fn ) < ε. ε , ρ(x0 , y) = 0, y = x0 . 4.2 · 95 · . {xn } (X, ρ) . k, nk , n nk ρ(xnk , xn ) < 1/2k , nk , nk nk+1 (k = 1, 2, · · ·). {Fk }, Fk = S1/2k−1 (xnk ) (k = 1, 2, · · ·), S1/2k−1 (xnk ) = {y : ρ(y, xnk ) < 1/2k−1 }. d(Fk ) 1/2k−2 (k = 1, 2, · · ·), lim d(Fk ) = 0 ( (ii)). k→∞ y ∈ Fk+1 , nk , ρ(y, xnk+1 ) 1/2k , ρ(xnk , xnk+1 ) < 1/2k , ρ(y, xnk ) ρ(y, xnk+1 ) + ρ(xnk , xnk+1 ) < 1/2k + 1/2k = 1/2k−1 , y ∈ Fk , Fk+1 ⊂ Fk (k = 1, 2, · · ·) ( (i)). , ∞ x0 . n=1 Fn = {x0 } . {xn } 1/2k−2 < ε, n > nk , ρ(xnk , xn ) < 1/2k . , ε > 0, k x0 ∈ Fk , ρ(xnk , x0 ) 1/2k−1 , ρ(x0 , xn ) ρ(x0 , xnk ) + ρ(xnk , xn ) 1 1 1 < k−1 + k < k−2 < ε, 2 2 2 lim xn = x0 . (X, ρ) n→∞ . . 4.2.8 (X, ρ) , M ⊂ X (M, ρ) , M X. x ∈ M , Fk = M ∩ S1/k (x)(k = 1, 2, · · ·), {Fk } M Cantor (i), (ii). (M, ρ) , ∞ ∞ , Cantor , k=1 Fk . , k=1 Fk = {x}, x ∈ M . . M = M. 4.2.9 (X, ρ) (M, ρ) M X. 4.2.8 . . M , (M, ρ) (X, ρ) , x ∈ X. M X, x ∈ M. . 4.2.10 {(Xn , ρn )} , (Xn , ρn ) 1. ∞ 4.1.10 ρ, (X, ρ) n=1 Xn . (Xn , ρn ) · 96 · 4 ∞ ∗ (X, ρ) . m ∈ N, Xm = n=1 An ( Am = Xm , An = {x∗n } Xn , n = m) ( 4.2.9). ∗ ∗ ∗ pm = pm |Xm : Xm → Xm , (Xm , ρm ) ∗ ∗−1 {xn }, {p∗−1 m (xn )} Xm . {pm (xn )} {xn } . (Xm , ρm ) . (Xn , ρn ) . {(xin )n∈N }i∈N (X, ρ) , X i (xin )n∈N , n = 1, 2, · · · , {xin }i∈N (Xn , ρn ) , x0n ∈ Xn . 2.1.4 ( 2.32), i 0 0 {(xn )n∈N }i∈N x = (xn )n∈N ∈ X. (X, ρ) . . , , . 4.2.11 {(Xα , ρα )}α∈A , (Xα , ρα ) 1. α∈A Xα 4.1.11 ρ, α∈A Xα . (Xα , ρα ) . [181] 4.2.12 ( ) . A = ∞ (X, ρ) , n=1 An , An A X , : U X , A∩U = ∅. , Cantor ( 4.2.7) (i), (ii) {Fn }, Fn ⊂ An ∩ U, n ∈ N, Cantor ∞ n=1 Fn = ∅, ∞ ∞ ∞ A∩U = An ∩ U = (An ∩ U ) ⊃ Fn = ∅, n=1 n=1 n=1 A X. {Fn }. A1 X, U , A1 ∩ U = ∅. x1 ∈ A1 ∩ U , A1 ∩ U , ε1 0 < ε1 < 1/22 , Sε1 (x1 ) ⊂ A1 ∩ U . A2 X, Sε1 (x1 ) , A2 ∩ Sε1 (x1 ) = ∅. x2 ∈ A2 ∩ Sε1 (x1 ), A2 ∩ Sε1 (x1 ) , ε2 0 < ε2 < ε1 /2, Sε2 (x2 ) ⊂ A2 ∩Sε1 (x1 ). , Sε2 (x2 ) ⊂ Sε1 (x1 ) Sε2 (x2 ) ⊂ A2 ∩U . , {Fn } = {Sεn (xn )} Fn+1 ⊂ Fn d(Fn ) 1/2n (n = 1, 2, · · ·), Cantor (i), (ii). , Fn ⊂ An ∩ U (n = 1, 2, · · ·), {Fn } . . . , T2 . ( 4.18), . , : X (Baire space), X 4.3 · 97 · . 4.2.12 . 4.2.3 . (X, ρ) ( 1.3.4). A = ∞ . n=1 An , An ∞ ∞ ∞ (X − An ), X −A=X − An ⊃ X − An = n=1 n=1 X − An ( 4.10), ∞ n=1 , (X − An ) = ∅. n=1 X − A = ∅. . : . 4.3 (X, ρ) ρ X T , (X, T ). , . , , . , : , ? (metrizable problem). , , . 4.3.1 X (metrizable), X ρ, ρ X . [404] 4.3.1 (Urysohn ) ω I ( 2.1.3) , . U X , U (U, V ), U ⊂ V. (U, V ) , {(U1 , V1 ), (U2 , V2 ), · · · , (Un , Vn ), · · ·}. X ( n ∈ N, 2.3.3 2.3.4), Urysohn ( fn : X → [0, 1] fn (x) = 0, x ∈ U n ; fn (x) = 1, x ∈ X − Vn . 2.4.1), · 98 · 4 f : X → Iω 1 1 f (x) = f1 (x), f2 (x), · · · , fn (x), · · · . 2 n , f ( 2.1.2). f ( x = x ⇒ f (x) = f (x )). X T1 , x W (x), x . , U (Un , Vn ) x ∈ Un ⊂ U n ⊂ Vn ⊂ W (x), fn (x) = 0, fn (x ) = 1. f (x) = f (x ). f −1 . W (x) X x . / W (x), n ∈ N ρ(f (x), f (x )) , x ∈ 1/n , f −1 (S1/n (f (x))) ⊂ W (x), S1/n (f (x)) = {f (x ) : ρ(f (x), f (x )) < 1/n}. f −1 . f X I ω . I ω , X . . Smirnov ( 1.2.1 2.2.2) Urysohn . 4.3.2 X, : (i) X ; (ii) X I ω ; (iii) X . (i) ⇒ (ii) 4.3.1; (ii) ⇒ (iii), I ω , [0, 1/n] , ( 2.17), I ω 2.3.2); (iii) ⇒ (i), ( 4.1.5), , ( ( 4.1.7). . Urysohn . , . , , . , Urysohn . Nagata, Bing . Smirnov σ σ Urysohn . 3.5 ( 3.5.5) , . 2.9 . , U = {Uα }α∈A ( U (x) ∩ Uα = ∅ α ∈ A ( ), x ∈ X x U (x) , 4.1.10 , , ρ(f (x), f (x ) 1/(2n n); 4.1.10 . 4.3 · 99 · α ∈ A) . , . ( ) , U = n∈N Un , Un ( ), U = n∈N Un σ (σ ). , σ . , [374] . 1948 , A. H. Stone , —— Stone . 4.3.3 Stone . Stone . [374] 4.3.3 (Stone ) , σ . {Uα }α∈A (X, ρ) . α ∈ A, Uα,n = {x : D(x, X − Uα ) 1/2n }, n ∈ N, Uα = ∞ n=1 Uα,n . x ∈ Uα,n , y∈ / Uα,n+1 (4.3.1) , (4.3.1) D(x, X − Uα ) − D(y, X − Uα ) > 1/2n − 1/2n+1 = 1/2n+1 , ( |D(x, A) − D(y, A)| ρ(x, y)) x ∈ Uα,n , y ∈ / Uα,n+1 ⇒ ρ(x, y) 1/2n+1 . A ( 0.3 Zermelo (4.3.2) ), ∗ Uα,n = Uα,n − ∪{Uβ,n+1 : β < α, β ∈ A}, α ∈ A; n ∈ N. (4.3.3) α, α ∈ A, α < α α < α, (4.3.3) ∗ Uα∗ ,n ⊂ X − Uα,n+1 Uα,n ⊂ X − Uα ,n+1 . (4.3.4) ∗ ∗ x ∈ Uα,n , y ∈ Uα∗ ,n , α < α , (4.3.3), x ∈ Uα,n ⇒ x ∈ Uα,n , (4.3.4) ∗ , y ∈ Uα ,n ⇒ y ∈ / Uα,n+1 ; α < α , [ (4.3.3) (4.3.4) ] y ∈ Uα ,n , x ∈ / Uα ,n+1 . α < α α < α, (4.3.2) ρ(x, y) 1/2n+1 , ∗ D(Uα,n , Uα∗ ,n ) 1/2n+1. (4.3.5) , ∗ ∪{Uα,n : α ∈ A, n ∈ N} = X. + ∗ Uα,n = {x : D(x, Uα,n ) < 1/2n+4 }, (4.3.6) · 100 · 4 ∼ ∗ Uα,n = {x : D(x, Uα,n ) < 1/2n+3 }. (4.3.7) ∗ + +− ∼ Uα,n ⊂ Uα,n ⊂ Uα,n ⊂ Uα,n ⊂ Uα . (4.3.8) , (4.3.5), (4.3.7) , α, α ∈ A, α = α , ∼ D(Uα,n , Uα∼ ,n ) 1/2n+2. ∼ , (4.3.9) (4.3.6) n∈N {Uα, U σ n : α ∈ A} . σ , Fn = +− Uα,n , n ∈ N. (4.3.9) (4.3.10) α∈A ∼ n ∈ N, {Uα,n : α ∈ A} (4.3.8) Fn , ∼ ∼ Wα,1 = Uα,1 , Wα,n = Uα,n − n−1 Fi (n 2), (4.3.11) i=1 Wα,n (n = 1, 2, 3, · · ·) . ∪{Wα,n : α ∈ A, n ∈ N} = X. +− (4.3.6) (4.3.8), ∪{Uα,n : n ∈ N, α ∈ A} = X. x ∈ X, m +− ∼ x ∈ Uα,m . (4.3.8), x ∈ Uα,m , (4.3.10), x ∈ / Fn (n = 1, 2, · · · , m − 1), (4.3.11), x ∈ Wα,m . {Wα,n : α ∈ A, n ∈ N} σ . x ∈ X, (4.3.6), x ∈ Uα∗0 ,n0 , (4.3.7) S1/2n0 +4 (x) ⊂ Uα+0 ,n0 ⊂ Uα+− ⊂ Fn0 . 0 ,n0 n > n0 , S1/2n0 +4 (x) ∩ Wα,n = ∅ (α ∈ A); n n0 , (4.3.9), S1/2n+3 (x) ∼ Uα,n , ∼ S1/2n0 +4 (x) ⊂ S1/2n+3 (x), Wα,n ⊂ Uα,n , n n0 , S1/2n0 +4 (x) Wα,n . W = n∈N {Wα,n : α ∈ A} σ . S1/2n0 +4 (x) W n0 , W . . , Wα,n ⊂ Uα , W U . 4.3.3 . 4.3 4.3.4 · 101 · ( 3.5.5), . . , 4.3.4, . . 5 , . (X, ρ) , n ∈ N, Un = {S1/n (x) : x ∈ X} X. 4.3.3, Un σ Vn , V = n∈N Vn X σ ( 4.3.5). V , x ∈ X x U , S1/n (x) ⊂ U , Vn x V , V ⊂ S1/n (x), Vn Un . . U X (), x ∈ X, st(x, U ) = ∪{U : U ∈ U , x ∈ X}; A ⊂ X, st(A, U ) = ∪{U : U ∈ U , U ∩ A = ∅}. 4.3.2 U X . V [184] (point-star reﬁnes) U , {st(x, V ) : x ∈ X} U ; [398] (star reﬁnes) U , {st(V, V ) : V ∈ V } U . 4.3.5 σ . (X, ρ) . n ∈ N, Un = {S1/2n (x) : x ∈ X}. 4.3.3, Un σ , x ∈ X, st(x, Un ) ⊂ S1/n (x) . Vn . Vn Un , x ∈ X, st(x, Vn ) ⊂ st(x, Un ). V = n∈N Vn , x ∈ S1/n (x) ⊂ U , Vn x ∈ X x U , n ∈ N x V , V ⊂ st(x, Vn ) ⊂ st(x, Un ) ⊂ S1/n (x), x ∈ V ⊂ U . V X σ . . Uyrsohn ( 4.3.1) , Tychonoﬀ ( 2.3.4) , Nagata-Smirnov ( 4.3.6), Tychonoﬀ . 4.3.1 σ . X , B X σ . A, B . , x ∈ A, B Ux x ∈ Ux U x ∩ B = ∅, U = {Ux : x ∈ A}, U A. V = {Vy : y ∈ B} B, y ∈ Vy , Vy ∈ B V y ∩ A = ∅. U , V σ B, U = n∈N Un , V = n∈N Vn , Un , Vn , Un = ∪{U : U ∈ Un }, Vn = ∪{V : V ∈ Vn }. Un = {S1/n (x) : x ∈ X}, Un Un . · 102 · 4 ( 3.5.2), U n = ∪{U : U ∈ Un }, V n = ∪{V : V ∈ Vn }. U n B , V n A (n ∈ N) ( 2.3.4), Un = Un − ∪{V k : k n}, Vn = Vn − ∪{U k : k n}, U= n∈N A, B Vn n∈N . Un , V = . . 4.3.2 (X, T ) T0 , {ρn }n∈N X . ρn (x, y) 1 (x, y ∈ X) (i) ρn : X × X → R ( X T ); (ii) x ∈ X, A⊂X x∈ / A, n ∈ N Dn (x, A) = inf{ρn (x, a) : a ∈ A} > 0, X X ρ(x, y) = ∞ 1 n=1 2n ρn (x, y) T . ρ(x, y) X . (M2) (M3) ρ(x, x) = 0 (x ∈ X). X T0 , x, y ∈ X, x ∈ / {y}, y∈ / {x} ( 2.2.1). x ∈ / {y}, (ii) n ∈ N Dn (x, {y}) = inf{ρn (x, a) : a ∈ {y}} > 0. ρn (x, y) > 0, ρ(x, y) > 0. ρ X . ρ T . 4.1.4, D(x, A) = 0 x ∈ A, 4.1.4, A = {x : D(x, A) = 0} A X ρ , A (X, T ) . 4.3 · 103 · x∈ / A, (ii) n ∈ N Dn (x, A) = r > 0, D(x, A) D(x, A) r/2n > 0. , (i) ρn : X × X → R ( (X, T )) , ρ ). 4.1.3 f (x) = D(x, A) (X, T ) . ( , x ∈ A, f (x) ∈ f (A) ⊂ f (A) = {0} ( x ∈ A ⇒ D(x, A) = 0, f (A) = {0}). D(x, A) = 0. . [315, 364] 4.3.6 (Nagata-Smirnov ) X , X σ . 4.1.5 4.3.5 , . . X B = n∈N Bn , Bn = {Bαn }αn ∈An . n, m αn ∈ An , Vαn ,m = ∪{Bαm : Bαm ∈ Bm , B αm ⊂ Bαn }. (4.3.12) , V αn ,m ⊂ Bαn . 4.3.1, X . Urysohn , fαn ,m : X → [0, 1] fαn ,m (X − Bαn ) ⊂ {0}, fαn ,m (V αn ,m ) ⊂ {1}. Bn , x ∈ X U (x) An (x) ⊂ An U (x)∩Bαn = ∅, αn ∈ An −An (x). X ×X {U (x)×U (y)}x,y∈X , U (x) × U (y) gn,m : U (x) × U (y) → R gn,m (x1 , x2 ) = {|fαn ,m (x1 ) − fαn ,m (x2 )| : αn ∈ An (x) ∪ An (y)}, (x1 , x2 ) ∈ U (x) × U (y). αn ∈ / An (x) ∪ An (y) , fαn ,m U (x) U (y) , gn,m (x1 , x2 ) = {|fαn ,m (x1 ) − fαn ,m (x2 )| : αn ∈ An }, (x1 , x2 ) ∈ U (x) × U (y). , gn,m : U (x ) × U (y ) → R , (x1 , x2 ) ∈ (U (x) × U (y)) ∩ (U (x ) × U (y )), gn,m (x1 , x2 ) = gn,m (x1 , x2 ). ρn,m : X × X → R , ρn,m (x1 , x2 ) = gn,m (x1 , x2 ), (x1 , x2 ) ∈ U (x) × U (y). ρn,m , ρn,m (x1 , x2 ) = min{1, ρn,m(x1 , x2 )}, · 104 · 4 ρn,m X ρn,m (x1 , x2 ) 1 (x1 , x2 ∈ X). X {ρn,m }n,m∈N . , 4.3.2 (i). (ii). x ∈ X A ⊂ X, x ∈ / A, B, B ∈ B x ∈ B ⊂ B ⊂ B, A ⊂ X − B. , B = Bαn ∈ Bn , B = Bαm ∈ Bm , αn ∈ An , αm ∈ Am , (4.3.12), Bαm ⊂ Vαn ,m , fαn ,m fαn ,m (x) = 1; fαn ,m (a) = 0 (a ∈ A). gn,m (x, a) 1, ρn,m (x, a) 1 ρn,m (x, a) = 1(a ∈ A). inf {ρn,m (x, a)} = a∈A 4.3.2 (ii). 4.3.2, X . . 4.3.5 4.3.6, . [46] 4.3.7 (Bing ) X X σ . 4.3.6 4.3.7 Bing-Nagata-Smirnov (Bing-Nagata-Smirnov metrization theorem). 1. 4.4 . , 4.1.5 , . , . , . 4.4.1 ( ) 1 1 1 X = {(0, 0)} ∪ 0, + · : i ∈ N, i j ∈ N i i j 1 1 1 1 ∪ 1, :i∈N ∪ 1, + · : i ∈ N, i j ∈ N . i i i j , 1 1 1 0, + · : i ∈ N, i j ∈ N i i j 1 1 1 1, + · : i ∈ N, i j ∈ N i i j 4.4 , q . · 105 · , , Y . A ⊂ X . q −1 (q(A)) , ( ) q(A) . A , q −1 (q(A)) Y T2 . Y . Y F = {(1, 1/i) : i ∈ N}, Y (0, 0) ∈ / F . U, V Y (0, 0) ∈ U, F ⊂ V , U ∩ V = ∅. (0, 0) ∈ U , i0 ∈ N j i i0 , (0, 1/i + 1/i · 1/j) ∈ q −1 (U ). F ⊂ V , j0 i0 j j0 , (1, 1/i0 + 1/i0 · 1/j) ∈ q −1 (V ). y = 1/i0 + 1/i0 · 1/j0 , (0, y) ∈ q −1 (U ) (1, y) ∈ q −1 (V ). q(0, y) = q(1, y) ∈ U ∩ V . . X, Y , f : X → Y (ﬁnite-to-one), y ∈ Y , f −1 (y) . , 4.4.1 q, y ∈ Y, q −1 (y) , . 4.4.1[278] X , X , X . U X , A = {As }s∈S U . x ∈ X V (x) A , F , {V (x)}x∈X . s ∈ S, Ws = X − ∪{F : F ∈ F , F ∩ As = ∅}. , Ws F ∈ F, As ( F ). , s ∈ S, ∅. Ws ∩ F = ∅ As ∩ F = (4.4.1) s ∈ S, U (s) ∈ U As ⊂ U (s) Vs = Ws ∩ U (s), {Vs }s∈S U . x ∈ X F , F ( V (x) ) A , (4.4.1) Ws , Vs . {Vs }s∈S . . : X {As }s∈S , X {Vs }s∈S As ⊂ Vs , s ∈ S. · 106 · 4 U = {Uα }α∈A (point-ﬁnite), x ∈ X, x ∈ Uα α ∈ A . , . 4.4.2[106] V = {Vα }α∈A U = {Uα }α∈A X , V α ⊂ Uα (α ∈ A). T X , Φ f : A → T f (α) = X α ∈ A, f (α) = Uα f (α) ⊂ Uα . α∈A Φ “<” f1 , f2 ∈ Φ, f1 < f2 , α ∈ A, f1 (α) ⊃ f2 (α) f1 (α) ⊂ Uα , f1 (α) = f2 (α). Φ . Φ Ψ = {fs : s ∈ S}. Ψ . f0 (α) = s∈S fs (α), α ∈ A. f0 ∈ Φ. < , f0 (α) f0 (α) = Uα f0 (α) ⊂ Uα . α∈A f (α) = X. x ∈ X, {Uα }α∈A , x Uα . {α1 , · · · , αk } = {α ∈ A : x ∈ Uα }. αi (1 i k), f0 (αi ) = Uαi , x ∈ f0 (αi ). , i (1 i k), f0 (αi ) ⊂ Uαi , < , si ∈ S, s > si , f0 (αi ) = fs (αi ). s0 = max{s1 , · · · , sk }, f0 (αi ) = fs0 (αi )(1 i k). {fs0 (α) : α ∈ A} X, x ∈ Uα α ∈ {α1 , · · · , αk } fs0 (α) ⊂ Uα , k k x ∈ i=1 fs0 (αi ) = i=1 f0 (αi ). α∈A f (α) = X. f0 ∈ Φ. f0 Ψ . Zorn , Φ f . α ∈ A, f (α) ⊂ Uα , 4.4.2 . , α0 ∈ A f (α0 ) ⊂ Uα0 . , f (α0 ) = Uα0 . F = X − ∪{f (α) : α = α0 }, F F ⊂ f (α0 ). , V , F ⊂ V ⊂ V ⊂ f (α0 ) = Uα0 . f : A → T f (α) = f ∈ Φ f > f , f V, α = α0 , f (α), α = α0 , . . 4.4.3 . f X Y , U X . V = {f (U ) : U ∈ U } Y . y ∈ Y , x ∈ f −1 (y), U (x) U ∈ U , f −1 (y) , U (x) f −1 (y), U (x) Vy , Vy ⊃ f −1 (y). f , 1.5.1, Wy Vy ⊃ Wy ⊃ f −1 (y), f (Wy ) Y Wy = f −1 (f (Wy )), Wy U ∈ U . f (Wy ) 4.4 · 107 · f (U ) Wy U , y f (Wy ) f (U ) . . X U (locally countable), x ∈ X, x U ∈ U . f −1 , f (y) Lindelöf , . 4.4.4 . f X Y , Y −1 −1 ( 2.12). V Y , f (V ) = {f (V )}V ∈V X . , U = {Us }s∈S f −1 (V ). 4.4.2, F = {Fs }s∈S Fs ⊂ Us , s ∈ S. F , 4.4.3, {f (Fs )}s∈S Y V , 4.4.1, Y . . 4.4.5 K (X, ρ) , U ⊃ K, r > 0 Sr (K) ⊂ U , Sr (K) = ∪{Sr (x) : x ∈ K}. f (x) = D(x, X − U ). 4.1.3, f : X → [0, ∞) . 4.1.4 K f (x) > 0, K , r > 0 f (x) r, x ∈ K. Sr (K) ⊂ U . . [299] 4.4.1 X {Wi }i∈N X (development), x ∈ X, x U , i ∈ N, st(x, Wi ) ⊂ U . (developable space). . i ∈ N, Wi = {S1/2i (x) : x ∈ X}, {Wi }i∈N . 4.4.6 X σ , X σ . {Wi }i∈N X . i ∈ N, σ Bi = j∈N Bi,j Wi , Bi,j . B = i∈N Bi , B σ . B X . x ∈ X, x U , 4.4.1, i ∈ N, st(x, Wi ) ⊂ U . Bi Wi , st(x, Bi ) ⊂ st(x, Wi ) ⊂ U . B ∈ Bi x ∈ B ⊂ U. . , . S ⊂ X f : X → Y (saturated set), S = f −1 (f (S)), f −1 (y) ∩ S = ∅, f −1 (y) ⊂ S. 4.4.1[308, 375] . f (X, ρ) Y . y ∈ Y , · 108 · 4 i ∈ N, Ui (y) = S1/i (f −1 (y)) = S1/i (x) x∈f −1 (y) = {x : x ∈ f −1 (y), ρ(x, x ) < 1/i}, (4.4.2) Wi (y) = Y − f (X − Ui (y)), (4.4.3) Vi (y) = f −1 (Wi (y)) ⊂ Ui (y). (4.4.4) f −1 (y) ⊂ Ui (y), y ∈ Y − f (X − Ui (y)) = Wi (y). f , Wi (y) Y y , Vi (y) X f −1 (y) , (4.4.3), (4.4.4) Vi (y) Ui (y) , f −1 (z) ⊂ Ui (y) ⇒ f −1 (z) ⊂ Vi (y). (4.4.2), (4.4.3), (4.4.4), y ∈ Y , j i, Uj (y) ⊂ Ui (y), Wj (y) ⊂ Wi (y), Vj (y) ⊂ Vi (y). (4.4.5) i ∈ N, Wi = {Wi (y)}y∈Y Y . {Wi }i∈N Y . , y ∈ Y , {Wi (y)}i∈N y V y , f −1 (y) ⊂ f −1 (V ). . (4.4.6) f −1 (y) , 4.4.5, i ∈ N, S1/i (f −1 (y)) = Ui (y) ⊂ f −1 (V ). (4.4.4), Vi (y) ⊂ f −1 (V ), Wi (y) = f (Vi (y)) ⊂ V , (4.4.6) . ∃j∈N (4.4.3), (4.4.4), f −1 (y) ⊂ V2i (y). st(y, Wj ) ⊂ Wi (y). f −1 (y) , (4.4.7) 4.4.5, j 2i Uj (y) ⊂ V2i (y). y Wj Wj (z), (4.4.8) Wj (z) ⊂ Wi (y), (4.4.7) . (4.4.4), f −1 (y) ⊂ f −1 (Wj (z)) = Vj (z) ⊂ Uj (z). x ∈ f −1 (y) ⊂ Uj (z), (4.4.2), x ∈ f −1 (z), f −1 (z) ∩ Uj (y) = ∅. (4.4.8) V2i (y) , f −1 (z) ⊂ V2i (y). ρ(x, x ) < 1/j, (4.4.9) 4.4 · 109 · t ∈ Wj (z), f (Vj (z)) = Wj (z), f −1 (t) ∩ Vj (z) = ∅. f −1 (t) ⊂ Vj (z) ⊂ Uj (z). (4.4.2), f −1 (z) ⊂ Uj (z), Vj (z) , ∀ x ∈ f −1 (t), ∃ x ∈ f −1 (z) ⇒ ρ(x, x ) < 1/j 1/2i. (4.4.10) (4.4.9) (4.4.4), f −1 (z) ⊂ V2i (y) ⊂ U2i (y), x ∈ f −1 (z), x ∈ f −1 (y) ρ(x , x ) < 1/2i. (4.4.10) ρ(x, x ) < 1/i. f −1 (t) ⊂ Ui (y). Vi (y) Ui (y) , f −1 (t) ⊂ Vi (y), t ∈ Wi (y). t , Wj (z) ⊂ Wi (y). (4.4.7) . (4.4.6), (4.4.7) {Wi }i∈N Y . ( 4.3.4), 4.4.4, Y . 4.4.6 4.3.6, Y . . 4.4.7 [283] f T1 X Y , X ∂f −1 (y) (y ∈ Y ) . : “ x ∈ X, x {Un (x)}, xn ∈ Un (x), {xn } x .” ( 4.13). . , h : X → R y ∈ Y , h ∂f −1 (y) , {xn } ⊂ ∂f −1 (y) |h(xn+1 )| > |h(xn )| + 1, n ∈ N. Vn = {x : x ∈ X, |h(x) − h(xn )| < 1/2}, {Vn }n∈N , xn ∈ Vn . {Un (y)} y . zn ∈ Vn ∩ f −1 (Un (y)) f (zn ) . z1 = x1 , z2 , z3 , · · · , zn−1 , Wn = (Vn ∩ f −1 (Un (y))) − n−1 f −1 (f (zk )). k=2 f , X T1 , f −1 (f (zk )) , Wn xn . xn f −1 (y) , Wn − f −1 (y) . zn ∈ Wn − f −1 (y). zn . Z = {zn : n ∈ N}, Z . f , f (Z) = {f (zn ) : n ∈ N} Y , {f (zn )} . f (zn ) ∈ Un (y) (n ∈ N), Y . . “ ” “”, ( 3.5.2 ). · 110 · 4 X, Y , f : X → Y (compact mapping) ( (boundary-compact mapping)), y ∈ Y , f −1 (y) (∂f −1 (y)) X . ( 3.3.1). 4.4.8 f T1 X Y , f |F F Y . F ⊂ X f , Y T1 , f −1 (y) X, ∂f −1 (y) ⊂ f −1 (y). y Y , {y} , f −1 (y) , ∂f −1 (y) = ∅ ( 1.3.10), py ∈ f −1 (y). Y E, F = ∪{{py } : y ∈ E} ∪ (∪{∂f −1 (y) : y ∈ Y − E}). F . x ∈ / F , x f −1 (y). , y ∈ E, f −1 (y) , f −1 (y) − {py } x ( {py } ) F ; y ∈ Y − E, f −1 (y) − ∂f −1 (y) x ( A − ∂A = A◦ ) F . F X . f F f |F F Y , y ∈ Y f |F ∂f −1 (y) {py }, f |F . . ( ): f T1 f |F : F → Y X Y , F ⊂ X −1 y ∈ Y , (f |F ) (y) , ∂f −1 (y). [308, 375] 4.4.2 (Morita-Hanai-Stone ) f (X, ρ) Y , (i) Y ; (ii) Y ; (iii) f . (i) ⇒ (ii), ; (ii) ⇒ (iii), 4.4.7 3.5.6 ∂f −1 (y) , 4.1.8 ∂f −1 (y) ; (iii) ⇒ (i), 4.4.8 4.4.1 . . 4.4.9 . f X Y . f , y ∈ Y , x ∈ f −1 (y), {Un (x)} x . {f (Un (x))} y . . . , “ x ∈ f −1 (y)”, A. Arhangel’skiˇı[17] , (almost open mapping): y ∈ Y , x ∈ f −1 (y), x 4.5 · 111 · U (x), f (U (x)) y . , ( 4.19), . 4.4.3 [39] . 4.4.2 4.4.9 . . [177, 335] 4.4.4 (Hanai-Ponomarev ) T0 . X T0 . {Uα }α∈A X . A N (A), , ρ 4.1.2. , N (A) (α1 , α2 , · · ·) = {αn }. N (A) S = {{αn } : {Uαn }n∈N x ∈ X }. f : S → X, f (α) = x, α = {αn } ∈ S {Uαn }n∈N x ∈ X ( X T0 , f ). X , f . f , α = {αn } ∈ S, k ∈ N, f (S1/k (α)) = k Uαn , (4.4.11) n=1 S1/k (α) = {α ∈ S : ρ(α, α ) < 1/k}. α = {αn } ∈ S ρ(α, α ) < , αn = αn , S , f (α ) 1/k, ρ , n k {Uαn }n∈N , f (α ) ∈ kn=1 Uαn = kn=1 Uαn , f (S1/k (α)) ⊂ kn=1 Uαn . , x ∈ kn=1 Uαn , {Uβj : j k + 1} x , α = (α1 , α2 , · · · , αk , βk+1 , βk+2 , · · ·) ∈ S, f (α ) = x ρ(α, α ) < 1/k ( ρ ), x ∈ f (S1/k (α)). (4.4.11) . f . U f (α) = x , α = {αn }. f , {Uαn }n∈N x , Uαi x ∈ Uαi ⊂ U . (4.4.11), f (S1/i (α)) ⊂ Uαi ⊂ U , f . f . {S1/k (α) : k ∈ N, α ∈ S} S , α ∈ S, k ∈ N, (4.4.11), f (S1/k (α)) , f . , f N (A) X . . 4.4.9 4.4.4 T0 , . 4.5 . 1938 A. · 112 · 4 Weil [409] , , . N. Bourbaki [57] . . , 1940 J. W. Tukey[398] , . X X . ( 4.3.2). U X . x ∈ X, st(x, U ) = ∪{U : U ∈ U , x ∈ U }; A ⊂ X, st(A, U ) = ∪{U ∈ U : U ∩ A = ∅}. U , V X , V V U ∈ U V ⊂ U , V U , V < U ; {st(V, V ) : V ∈ V } U , V ∗ U , V < U . U , V X , U ∧ V = {U ∩ V : U ∈ U , V ∈ V } . . 4.5.1 {Uα : α ∈ A} X (U1) X U , α ∈ A (U2) α, β ∈ A, γ ∈ A (U3) α ∈ A, β ∈ A U,V U , Uα < U , U ∈ {Uα : α ∈ A}; Uγ < Uα , Uγ < Uβ ; ∗ Uβ < Uα ; (U4) x, y ∈ X (x = y), α ∈ A Uα x y, {Uα : α ∈ A} X (uniformity), X {Uα : α ∈ A} (uniform space), (X, {Uα : α ∈ A}). X. {Uα : α ∈ A} {Uβ : β ∈ B} (B ⊂ A) (basis of uniformity), α ∈ A, β ∈ B Uβ < Uα . 4.5.1 X {Uβ : β ∈ B} X 4.5.1 (U2)∼(U4). Φ = {Uβ : β ∈ B} X Φ = {Uα : α ∈ A} . β, β ∈ B, Φ ⊂ Φ, β, β ∈ A, Φ (U2), γ ∈ A Uγ < Uβ , Uγ < Uβ . Φ Φ , γ ∈ B Uγ < Uγ , Uγ < Uβ , Uγ < Uβ . Φ (U2). Φ (U3), (U4). , Φ = {Uβ : β ∈ B} (U2)∼(U4). Φ = {U : ∃ β ∈ B Uβ < U }, U X . Uα , Uα ∈ Φ, Φ , β, β ∈ B, Uβ < Uα , Uβ < Uα . Φ (U2), γ ∈ B Uγ < Uβ , Uγ < Uβ , 4.5 · 113 · Uγ ∈ Φ Uγ < Uα , Uγ < Uα . Φ (U2). (U4). , Φ (U1), Φ X . . Φ (U3), 4.5.1 . 4.5.1 (X, ρ) , Un = {S1/3n (x) : x ∈ X}, n ∈ N. ∗ Un+1 < Un (n ∈ N). Φ = {Un : n ∈ N} (U2), (U3). X T1 , x, y ∈ X, x = y, Sε (x) y ∈ / Sε (x). n ∈ N 1/3n < ε, y ∈ / S1/3n (x), st(S1/3n+1 (x), Un+1 ) ⊂ S1/3n (x), Un+1 x y, Φ (U4). 4.5.1, Φ , Φ = {U : Un ∈ Φ Un < U } X . (X, ρ) . 4.5.2 . 4.5.2 () (group) G , x, y ∈ G, xy ∈ G (xy x, y ) (G1) (xy)z = x(yz), x, y, z ∈ G; (G2) e ∈ G xe = ex = x x ∈ G −1 (G3) x ∈ G, x −1 e G , x ∈G −1 xx −1 =x ; x = e. x . . (topological group) G T1 (TG1) f (x, y) = xy G × G → G (TG2) g(x) = x−1 G → G −1 ; . −1 A, B ⊂ G, A = {x : x ∈ A}, AB = {xy : x ∈ A, y ∈ B}, A B {x} {y} , xB Ay. B e , B ∈ B, UB = {xB : x ∈ G} G . Φ = {UB : B ∈ B}. G , Φ G . (NB3) ( B ∈ B, B1 ∈ B 1.2.3), (U2). Φ (U3), st(xB1 , UB1 ) ⊂ xB, x ∈ G. (4.5.1) f (x1 , x2 , x3 ) = x1 x−1 2 x3 . (TG1), (TG2) f : G × G × G → G , f (e, e, e) = e, f (e, e, e) , e B ∈ B, e B1 ∈ B f (B1 × B1 × B1 ) = B1 B1−1 B1 ⊂ B. xB1 x1 B1 ∈ UB1 . x1 B1 ⊂ xB, st(xB1 , UB1 ) ⊂ xB, (4.5.1) . xB1 ∩ x1 B1 = ∅, · 114 · 4 b0 , b1 ∈ B1 ), xb0 = x1 b1 , x1 = xb0 b−1 1 , x1 B1 x1 b (b B1 −1 x1 b = xb0 b−1 1 b ∈ xB1 B1 B1 ⊂ xB. x1 B1 ⊂ xB, (4.5.1) . Φ (U3). G x, y, x−1 y = e. G T1 , e B ∈ B −1 x y ∈ / B. (TG1), (TG2), g(x1 , x2 ) = x−1 , 1 x2 G × G → G g(e, e) = e, g (e, e) , e B, e B1 ∈ B g(B1 × B1 ) = B1−1 B1 ⊂ B. UB1 (U4) . , x, y UB1 x0 B1 , b1 , b2 ∈ B1 x = x0 b1 , y = x0 b2 , −1 x−1 y = (x0 b1 )−1 (x0 b2 ) = b−1 1 x0 x0 b2 −1 = b−1 1 b2 ∈ B1 B1 ⊂ B. . Φ (U4). . 4.5.1, Φ G , G . . 4.5.2 X , {Uα : α ∈ A} (). T = {U : U ∈ X, ∀ x ∈ U, ∃ α ∈ A st(x, Uα ) ⊂ U }, (4.5.2) T X . , ∅ ∈ T , X ∈ T , T ( (O1), (O3)). (O2). U1 , U2 ∈ T , U1 ∩ U2 = ∅. x ∈ U1 ∩ U2 , U1 ∈ T , α1 ∈ A st(x, Uα1 ) ⊂ U1 ; U2 ∈ T , α2 ∈ A st(x, Uα2 ) ⊂ U2 . (U2), γ ∈ A Uγ < Uα1 , Uγ < Uα2 . st(x, Uγ ) ⊂ st(x, Uα1 ) ∩ st(x, Uα2 ) ⊂ U1 ∩ U2 . . 4.5.2 4.5.2 (4.5.2) T X (topology induced by an uniformity). , , T . 4.5.3 T X {Uα : α ∈ A} , (i) {st(x, Uα ) : α ∈ A} (X, T ) x ; (ii) . (i) U x ∈ U , 4.5.2 (4.5.2), α ∈ A st(x, Uα ) ⊂ U , st(x, Uα ) x . V = {x : ∃ β ∈ A st(x , Uβ ) ⊂ st(x, Uα )}. (4.5.3) 4.5 · 115 · , x ∈ V ⊂ st(x, Uα ), V V . (4.5.2) x , γ ∈ A st(x , Uγ ) ⊂ V. (4.5.4) V x , (4.5.3), β ∈ A st(x , Uβ ) ⊂ st(x, Uα ). (U3), ∗ γ ∈ A Uγ < Uβ . x ∈ st(x , Uγ ), Uγ ∈ Uγ , x , x ∈ Uγ . ∗ Uγ < Uβ x ∈ Uγ , st(x , Uγ ) ⊂ st(Uγ , Uγ ) ⊂ Uβ ∈ Uβ . x ∈ Uγ ⊂ Uβ , st(x , Uγ ) ⊂ st(x , Uβ ). st(x , Uγ ) ⊂ st(x, Uα ). (4.5.3), x ∈ V , x , st(x , Uγ ) ⊂ V , (4.5.4) . (ii) Uα (α ∈ A), Uα◦ = {U ◦ : U ∈ Uα }. {Uα◦ : α ∈ A} {Uα : α ∈ A} . ∗ Uα◦ ∈ {Uα : α ∈ A}. (U3), β ∈ A Uβ < Uα , Uβ < Uα◦ , (U1) . Uβ < Uα , V ∈ Uβ , U ∈ Uα st(V, Uβ ) ⊂ U , x ∈ V , st(x, Uβ ) ⊂ U . (i), st(x, Uβ ) x ( x ), ◦ ◦ ◦ x U , x ∈ U . V ⊂ U , Uβ < Uα . . ∗ 4.5.4 (X, T ) T X {Uα : α ∈ A} , . 4.5.3 (i), {st(x, Uα ) : α ∈ A} x ∈ X , (U4) x, y ∈ X, x = y, α ∈ A y ∈ / st(x, Uα ). (X, T ) T1 / F. 4.5.3 (ii), . x ∈ X, F X , x ∈ {Uα : α ∈ A} : U0 , U1 , U2 , · · ·, ∗ st(x, U0 ) ⊂ X − F, Un < Un−1 (n ∈ N). U (k/2n ) (k = 1, 2, · · · , 2n − 1; n ∈ N) U (1/2) = st(x, U1 ), U (1/22 ) = st(x, U2 ), U (3/22 ) = st(U (1/2), U2 ), ······ , U (k/2n )(k = 1, 2, · · · , 2n − 1) , U (k /2n+1 ) ⎧ n ⎪ k = 2k, ⎪ ⎨ U (k/2 ), U (k /2n+1 ) = k = 1, st(x, Un+1 ), ⎪ ⎪ ⎩ st(U (k/2n ), U n+1 ), k = 2k + 1, k > 0. · 116 · 4 ∗ Un+1 < Un , U ⊂ X, st(st(U, Un+1 ), Un+1 ) ⊂ st(U, Un ), Un , st(U, Un+1 ) ⊂ st(st(U, Un+1 ), Un+1 ). x ∈ U (k/2n ) ⊂ U (k/2n ) ⊂ U ((k + 1)/2n ) ⊂ U ((k + 1)/2n ) ⊂ X − F. , U (1) = X, f (x) = inf{r : x ∈ U (r)}. Urysohn ( 2.4.1) , f X [0, 1] f (x) = 0, f (F ) ⊂ {1}, X . . 4.5.5 (X, T ) , X T . x ∈ X x U , f : X → f (x) = 0, f (X − U ) ⊂ {1}, n ∈ N, [0, 1] U (n, x, U ) = {f −1 (Sn (r)) : r ∈ [0, 1]}, Sn (r) = [0, 1] ∩ (r − 1/3n , r + 1/3n ). 4.5.1 f , U (n, x, U ) X ∗ U (n + 1, x, U ) < U (n, x, U ) (n ∈ N). (4.5.5) st(x, U (1, x, U )) ⊂ U. (4.5.6) , U (1, x, U ) = {f −1 (S1 (r)) : r ∈ [0, 1]}, S1 (r) = [0, 1] ∩ (r − 1/3, r + 1/3). r ∈ [0, 1/3) ⇔ 0 ∈ (r − 1/3, r + 1/3) ⇔ x ∈ f −1 (S1 (r)), st(x, U (1, x, U )) = ∪{f −1 (S1 (r)) : r ∈ [0, 1/3)}. f [0, 1), U . (4.5.6) . Φ = {U (n, x, U ) : x ∈ X, U x , n ∈ N}, Ψ = {U1 ∧ · · · ∧ Uk : Ui ∈ Φ, i = 1, 2, · · · , k; k ∈ N}. (4.5.5) Ψ (U2), (U3) . X T1 , (4.5.6), (4.5.5) Ψ (U4), Ψ X . (4.5.2) ( 4.5.2) (4.5.6) Ψ ∗ ∗ , Vi < Ui (i = 1, 2, · · · , k), V1 ∧ · · · ∧ Vk < U1 ∧ · · · ∧ Uk . 4.5 · 117 · T . , Ψ T , U ∈ T , x ∈ U , (4.5.6) (4.5.2) U ∈ T ; , U ∈ T , x ∈ U , (4.5.2) U ∈ Ψ T , U ∈ T . . st(x , U ) ∈ T , , . 4.5.3 (X, T ) (uniformizable), X , T . 4.5.4 4.5.6 4.5.5 , . (X, T ) . X, X × X ∆ = {(x, x) : x ∈ X} X × X (diagonal); X × X D ∆ “ x, y ∈ X, (x, y) ∈ D ⇒ (y, x) ∈ D” ∆ (symmetric entourage). D ∆ , D ◦ D = {(x, y) : z ∈ X, (x, z), (z, y) ∈ D}. 4.5.7 {Uα : α ∈ A} X (), Uα (α ∈ A), Dα = ∪{U × U : U ∈ Uα }, D = {Dα : α ∈ A}, Dα ∆ D (i) Dα , Dβ ∈ D, Dγ ∈ D Dγ ⊂ Dα ∩ Dβ ; (ii) Dα ∈ D, Dβ ∈ D (iii) α∈A Dα = ∆. Dβ ◦ Dβ ⊂ Dα ; Dα ∆ . D Uβ < Uα ⇒ Dβ ⊂ Dα ; (4.5.7) ∗ Uβ < Uα ⇒ Dβ ◦ Dβ ⊂ Dα . (4.5.7) . (4.5.8) (4.5.8) , (x, y) ∈ Dβ ◦ Dβ , z ∈ X, (x, z), (z, y) ∈ Dβ , Uβ ∈ Uβ , Uβ ∈ Uβ , (x, z) ∈ Uβ × Uβ , x, z ∈ Uβ . z, y ∈ Uβ , Uβ ∩ Uβ = ∅. ∗ Uβ < Uα , Uβ ∪ Uβ ⊂ Uα ∈ Uα . x, y ∈ Uα , (x, y) ∈ Uα × Uα ⊂ Dα . (4.5.8) . Dα , Dβ ∈ D, (U2), α, β ∈ A, γ ∈ A, Uγ < Uα , Uγ < Uβ . (4.5.7), Dγ ⊂ Dα , Dγ ⊂ Dβ , Dγ ⊂ Dα ∩ Dβ . D (i). (U3) (4.5.8) D (ii). D (iii). , x = y (x, y) ∈ α∈A Dα , (x, y) Dα , Uα (α ∈ A) Uα x, y ∈ Uα , (U4) . . · 118 · 4 4.5.8 ( ) D0 , D1 , · · · , Di , · · · X × X ∆ D0 = X × X, Di+1 ◦ Di+1 ◦ Di+1 ⊂ Di (i ∈ N), (4.5.9) X ρ Di ⊂ {(x, y) : ρ(x, y) 1/2i } ⊂ Di−1 (i ∈ N). f : X × X → [0, 1] ⎧ ∞ ⎪ ⎨ 0, (x, y) ∈ Di , f (x, y) = i=0 ⎪ ⎩ 1/2i , (x, y) ∈ Di − Di+1 (i = 0, 1, · · ·), f (x, x) = 0, f (x, y) = f (y, x). ρ(x, y) ki=1 f (xi−1 , xi ) (4.5.10) x, y ∈ X, , x0 , x1 , · · ·, xk X x0 = x, xk = y. (4.5.10), ρ(x, x) = 0 ρ(x, y) = ρ(y, x), ρ . ρ X (, {Di }∞ ∞ i=0 i=0 Di = ∆, f (x, y) , ρ X ). , 1 f (x, y) ρ(x, y) f (x, y). (4.5.11) 2 (4.5.11) ( “”) ρ “ ” . (4.5.11) ( “”) (4.5.12) k 1 f (x, y) f (xi−1 , xi ) 2 i=1 (4.5.12) X x0 , x1 , · · · , xk ( x0 = x, xk = y) . , (4.5.11) (4.5.12) ρ ( “” ) . (4.5.12) . , k = 1, (4.5.12) . k < m , (4.5.12) , k=m m . x0 , x1 , · · · , xm , x0 = x, xm = y, a = i=1 f (xi−1 , xi ), a 1/2, f (x, y) 1, (4.5.12) k = m . a < 1/2. (i) a > 0. , f (x0 , x1 ) a/2, f (xm−1 , xm ) a/2. x, y , f (x0 , x1 ) a/2, j j i=1 f (xi−1 , xi ) a , 2 4.5 · 119 · j+1 f (xi−1 , xi ) > i=1 m a , 2 f (xi−1 , xi ) i=j+2 a . 2 , (4.5.12) k < m , j 1 a f (x0 , xj ) f (xi−1 , xi ) , f (x0 , xj ) a, 2 2 i=1 m 1 a f (xj+1 , xm ) f (xi−1 , xi ) , f (xj+1 , xm ) a. 2 2 i=j+2 , a = m 1/2l a i=1 f (xi−1 , xi ), f (xj , xj+1 ) a. l , a < 1/2, l 2. f , f (x0 , xj ) 1/2i , l f (x0 , xj ) 1/2l , f (xj , xj+1 ) 1/2l , f (xj+1 , xm ) 1/2l . (x0 , xj ) ∈ Dl , (xj , xj+1 ) ∈ Dl , (xj+1 , xm ) ∈ Dl , f (x, y) 1/2i ⇔ (x, y) ∈ Di ). (4.5.9), (x0 , xm ) = (x, y) ∈ Dl−1 , f (x, y) 1/2l−1 2a, f (x, y)/2 a. (4.5.12) a > 0 k=m . (ii) a = 0. i = 1, 2, · · · , m, f (xi−1 , xi ) = 0. f , (xi−1 , xi ) ∈ Dj (j = 0, 1, 2, · · ·). (x, y) ∈ Dj ◦ Dj ◦ · · · ◦ Dj (m Dj ) j = 0, 1, 2, · · · ∞ . (4.5.9), (x, y) ∈ i=0 Di . f (x, y) = 0. (4.5.12) a = 0 k = m . (4.5.12) . (4.5.11) . , E = {(x, y) : ρ(x, y) 1/2i }. (x, y) ∈ E, ρ(x, y) 1/2i , (4.5.11) , f (x, y)/2 1/2i , f (x, y) 1/2i−1 , (x, y) ∈ Di−1 , E ⊂ Di−1 . , (x, y) ∈ Di , f (x, y) 1/2i, (4.5.11) , ρ(x, y) f (x, y) 1/2i , (x, y) ∈ E, Di ⊂ E. . 4.5.4 X (metrizable), X ρ Un = {S1/n (x) : x ∈ X} {Un : n ∈ N} . 4.5.9 ( ) . ( · 120 · 4 ( 4.5.1). . {Uα }α∈A X , {Ui : i ∈ N} . ∗ Dα = ∪{U × U : U ∈ Uα }, Uβ < Uα ⇒ Dβ ◦ Dβ ⊂ Dα ( 4.5.7 ∗ (4.5.8) ), {Uα : α ∈ A} Uαi (i ∈ N), Uαi+1 < Uαi Di = ∪{U × U : U ∈ Uαi }, Di+1 ◦ Di+1 ◦ Di+1 ⊂ Di Uαi < Ui . 4.5.8, X ρ ( {Ui : i ∈ N} , 4.5.7 (iii) Uαi < Ui i∈N Di = ∆), 1 (4.5.13) (x, y) : ρ(x, y) i+1 ⊂ Di (i ∈ N). 2 {S1/2i+2 (x) : x ∈ X} < Ui (i ∈ N). (4.5.14) y ∈ S1/2i+2 (x), ρ(x, y) < 1/2i+2 , (4.5.13), (x, y) ∈ Di+1 . x, y ∈ ∗ U ∈ Uαi+1 . S1/2i+2 (x) ⊂ st(x, Uαi+1 ). Uαi+1 < Uαi , st(x, Uαi+1 ) ⊂ U ∈ Uαi , {st(x, Uαi+1 ) : x ∈ X} < Uαi < Ui . (4.5.14) . , {{S1/n (x) : x ∈ X}}n∈N {Uα : α ∈ A} . 4.5.4 . . 4.4.9, Alexandroﬀ-Urysohn . [8] 4.5.10 (Alexandroﬀ-Urysohn ) X , X T0 {Un }n∈N ∗ (i) Un+1 < Un (n ∈ N); (ii) {st(x, Un )}n∈N x ∈ X . . . (X, T ) , (i), {Un }n∈N (U2), (U3). X T0 (ii), {Un }n∈N (U4), 4.5.1). 4.5.2 (4.5.2) , {Un }n∈N X ( X T , (X, T ) ( 4.5.3). 4.5.9 . . , . 4.5.11 . (X, T ) {Uα : α ∈ A} T , X ⊂ X. Uα = {U ∩ X : U ∈ Uα }, {Uα : α ∈ A} X , X . . 4.5.12 . γ ∈ Γ , (Xγ , Tγ ) , {Uαγ : α ∈ Aγ } Tγ . Φ = {Uαγ11 × · · · × Uαγkk × {Xγ : γ = γi , i = 1, 2, · · · , k} : αi ∈ Aγi , γi ∈ Γ , i = 1, 2, · · · , k; k ∈ N}, 4.5 · 121 · Uαγ11 × · · · × Uαγkk = {Uαγ11 × · · · × Uαγkk : Uαγii ∈ Uαγii , i = 1, 2, · · · , k}. Φ γ∈Γ Xγ . . [398] 4.5.5 U (normal covering), ∗ {Un }n∈N Un+1 < Un (n ∈ N) U1 < U . (X, T ) , Φ = {Uα : α ∈ A} X T . (U3) 4.5.3 (ii), Uα (α ∈ A) . Φ (X, T ) . , Φ (U1), (U3). Φ ⊃ Φ(Φ (U4)), Φ (U4). 4.5.5 , Φ ( ) , Φ , Φ (U2), Φ X . (X, T ) , , 4.5.5 Φ T . Φ ⊂ Φ , Φ T . . 4.5.13 , . : “ Φ = {Uα : α ∈ A} (X, T ) T Ψ .” , : , X “ U , α ∈ A Uα < U ”. , (U1), Ψ ⊂ Φ, , 4.5.3 (ii), Ψ Φ . . , α ∈ A, Uα Uα U . xα ∈ Uα , st(xα , Uα ) U . α, β ∈ A, α > β Uα < Uβ , A . ϕ(α) = xα (α ∈ A), ϕ(A; >) , ϕ(α) = xα (α ∈ A) (X, T ) , x ( 3.8). U , x ∈ U ∈ U . . Φ = {Uα : α ∈ A} T , α0 ∈ A st(x, Uα0 ) ∈ U ∈ U ( 4.5.3 ∗ (i)). α∈A Uα < Uα0 , st(st(x, Uα ), Uα ) ⊂ st(x, Uα0 ). x ϕ(A; >) , st(x, Uα ) x , 1.4.7, β > α xβ = ϕ(β) ∈ st(x, Uα ). Uβ < Uα (β > α), st(xβ , Uβ ) ⊂ st(xβ , Uα ) ⊂ st(st(x, Uα ), Uα ) ⊂ st(x, Uα0 ) ⊂ U. xβ . . 4.5.6 X Y f (uniformly continuous), Y ( ) V , f −1 (V ) 4.5.5 Ψ , Ψ ⊂ Φ . Φ T , , st(x , U ) T , st(x , U ) x T , . U , U1 < U , · 122 · 4 X ( ). . X Y f , f −1 , f (uniform isomorphism). X, Y (uniformly isomorphic). 4.5.14 X Y . X, Y {Uα : α ∈ A}, {Vβ : β ∈ B}. V Y , U = f −1 (V ) X . x ∈ U, f (x) ∈ V , 4.5.2 −1 (4.5.2) , β ∈ B, st(f (x), Vβ ) ⊂ V . , f (Vβ ) ∈ {Uα : α ∈ A}. st(y, V ) ⊂ V ⇔ st(f −1 (y), f −1 (V )) ⊂ f −1 (V ), st(x, f −1 (Vβ )) ⊂ st(f −1 (f (x)), f −1 (Vβ )) ⊂ f −1 (V ) = U . (4.5.2) , U X . . 4.5.15 f X Y , X , f . V Y , V ( 4.5.5), ∗ Vn+1 < Vn (n ∈ N) V1 < V , f −1 (V ) Y {Vn }n∈N ∗ X , f −1 (Vn ) (n ∈ N) X f −1 (Vn+1 ) < f −1 (Vn ) (n ∈ N) f −1 (V1 ) < f −1 (V ). f −1 (V ) . , f −1 (V ) X . f . . 4.5.1 X Y . 4.5.15 X , 4.5.13 X , . . Tukey . 4.5.7 Weil , . 4.5.16 D = {Dα : α ∈ A} X × X ∆ D 4.5.7 (i)∼(iii). U (Dα ) = {Dα [x] : x ∈ X}, , Dα [x] = {y : y ∈ X, (x, y) ∈ Dα }. Φ = {U (Dα ) : α ∈ A} (U2)∼(U4), Φ X . α, β ∈ A, Dα , Dβ ∈ D, (i) γ ∈ A Dγ ⊂ Dα ∩ Dβ , x ∈ X, Dγ [x] ⊂ (Dα ∩ Dβ )[x] = Dα [x] ∩ Dβ [x], U (Dγ ) < U (Dα ) U (Dγ ) < U (Dβ ). Φ (U2). α ∈ A, (ii) β ∈ A Dβ ◦ Dβ ◦ Dβ ◦ Dβ ⊂ Dα . U (Dβ ) = {Dβ [x] : x ∈ X} Dβ [x]. Dβ [x] ∩ Dβ [y] = ∅, z ∈ Dβ [x] ∩ Dβ [y], (x, z ) ∈ Dβ , (z , y) ∈ Dβ , (x, y) ∈ Dβ ◦ Dβ . z ∈ Dβ [y], (y, z) ∈ Dβ , 4.5 · 123 · (x, z) ∈ (Dβ ◦ Dβ ) ◦ Dβ ⊂ Dβ ◦ Dβ ◦ Dβ ◦ Dβ ⊂ Dα , ∗ z ∈ Dα [x], Dβ [y] ⊂ Dα [x]. U (Dβ ) < U (Dα ), Φ (U3). x, y ∈ X, x = y, (iii), (x, y) ∈ / α∈A Dα . α ∈ A (x, y) ∈ / Dα . (ii) β∈A Dβ ◦ Dβ ⊂ Dα , (x, y) ∈ / Dβ ◦ Dβ . z ∈ X, x ∈ Dβ [z], y ∈ Dβ [z] . U (Dβ ) x y, Φ (U4). Φ X . . Weil “ D X × X ∆ D , : (i) ∆ D, Dα ∈ D Dα ⊂ D, D ∈ D; (ii) Dα , Dβ ∈ D, Dα ∩ Dβ ∈ D; (iii) Dα ∈ D, Dβ ∈ D Dβ ◦ Dβ ⊂ Dα ; (iv) ∩D = ∆, D X . X D , (X, D). D D D (), D ∈ D, D ∈ D D ⊂ D.” X × X ∆ D D X D Weil (iii), (iv) (ii ) Dα , Dβ ∈ D , Dγ ∈ D Dγ ⊂ Dα ∩ Dβ . , (ii ), (iii), (iv) 4.5.7 (i), (ii), (iii). , 4.5.7 “ {Uα : α ∈ A} Tukey ( 4.5.1), Dα = ∪{U × U : U ∈ Uα }, D = {Dα : α ∈ A} Weil .” , 4.5.16 “ D = {Dα : α ∈ A} Weil , Dα [x] = {y : y ∈ X, (x, y) ∈ Dα }, U (Dα ) = {Dα [x] : x ∈ X}, {U (Dα ) : α ∈ A} Tukey .” . Weil , ( 4.5.6 ) “ X Y f (uniformly continuous), Y ( ∆ ) D, φ−1 (D) X ( ∆ ), φ(x, y) = (f (x), f (y)).” (X, ρ), Weil {{(x, y) : ρ(x, y) < 1/n}}n∈N · 124 · 4 . (X , ρ ) f , ε > 0, δ > 0, ρ(x, y) < δ , ρ (f (x), f (y)) < ε.” “ (X, ρ) 4.1 4 [a, b] C. f, g ∈ C, ρ(f, g) = max{|f (x) − g(x)| : x ∈ [a, b]}. ρ C . (continuous function space), C[a, b], C. 4.2 [a, b] L2 . f, g ∈ L2 , ρ(f, g) = b (f (x) − g(x))2 dx. a ρ L2 . (Lebesgue integrable function space), L2 [a, b], L2 . 4.3 A (X, ρ) , d(A) A . (i) d(A) = d(A); (ii) A , x, y ∈ A, 4.4 (X, ρ) {xn } x ∈ A A 4.5 (X, ρ) δ ∈ ∆} d(A) = ρ(x, y). x ∈ X, {ρ(x, xn )} . x. {ϕ(δ) : δ ∈ ∆} x {ρ(ϕ(δ), x) : . 4.6 (X, ρ), (Y, σ) . , X Y f x ∈ X, ε > 0, δ > 0, ρ(x, x ) < δ ⇒ σ(f (x), f (x )) < ε. 4.7 X ρ1 , ρ2 (equivalent), ρ1 , ρ2 X . X ρ1 , ρ2 , x ∈ X, {xn }, lim ρ1 (x, xn ) = n→∞ 0 ⇔ lim ρ2 (x, xn ) = 0. n→∞ 4.8 (X, ρ), ρ1 (x, y) = ρ(x, y) 1 + ρ(x, y) X , ρ1 ρ . 4.9 f : (i) f x = 0; (iii) f (x + y) f (x) + f (y). (X, ρ) , (X, ρ ) ρ, ρ . ; (ii) f (x) = 0 ρ (x, y) = f (ρ(x, y)). 4 4.10 A X X − A X. 4.11 (X, ρ) · 125 · ε > 0, ε . 4.12 4.13 , . (i) X ; (ii) x ∈ X, x {Un (x)}n∈N xn ∈ Un (x) ⇒ {xn } → x; (iii) x ∈ X, x {Un (x)}n∈N xn ∈ Un (x) ⇒ {xn } x . 4.14 σ ( Gδ ). 4.15 (X1 , ρ1 ), (X2 , ρ2 ), · · · , (Xk , ρk ) . X1 × · · · × Xk x = (x1 , x2 , · · · , xk ), y = (y1 , y2 , · · · , yk ), ρ(x, y) = ρ1 (x1 , y1 ) + ρ2 (x2 , y2 ) + · · · + ρk (xk , yk ). ρ . 4.16 T2 . 4.17 (Moore {Un }, [15, 300, 378] ) T0 X X x ∈ X, {st(st(x, Un ), Un ) : n ∈ N} 4.18 T2 x . ( T2 ). 4.19 ( ). 4.20 X : Y (i) y ∈ Y , x ∈ f f −1 , (y), x U = {U }, f (U ) (U ∈ U ) Y ; (ii) U X 4.21 , {Intf (U ) : U ∈ U } Y X totally (totally normal Fσ {Gα } , {Gα } G U (x) {Gα } (i) ). [111] . ), X G ( x ∈ G, x totally ; (ii) T2 ( ) totally [111] ; (iii) totally totally ( totally ). 4.22 X Y f (quasi-open), X U f (U ) ( Intf (U ) = ∅). f , −1 : f :X→Y (E) X . 4.23 . , E Y , · 126 · 4.24 4 X ρ1 , ρ2 (uniformly equivalent), x, x ∈ X ε > 0, δ1 > 0 δ2 > 0 ρ1 (x, x ) < δ1 ⇒ ρ2 (x, x ) < ε ρ2 (x, x ) < δ2 ⇒ ρ1 (x, x ) < ε. , . , , . 4.25 4.1.9 ρ ρ, 4.8 ρ1 ρ. 4.26 ρi , σi Xi (i = 1, 2, · · ·) 1( x, x ∈ Xi , ρi (x, x ) 1, σi (x, x ) 1). ρ(x, y) = ρ, σ 4.27 ∞ i=1 Xi . (Y, σ) . (X, ρ) , , ρ X σ, , X (totally bounded uniform space), {Uα : α ∈ A} Uα . X (Cauchy ﬁlter), Uα F ∈ F U ∈ Uα U . X (complete uniform space), X X E (ii) X X E (iv) 4.30(Birkhoﬀ-Kakutani ) . : ; X [47, 226] X; (iii) X F F ⊂ U . X ϕ(∆; >) (Cauchy net), Uα U ∈ Uα (i) . (X, ρ) (X, σ) 4.29 ∞ 1 ρ (x , y ), σ(x, y) = σ (xi , yi ). i i i i i i 2 2 i=1 i=1 f (X, ρ) (Y, σ) 4.28 ∞ 1 ; . . 4.31(Weil . [409] ) X X Weil 5 3.5 ( 3.5.5), . 3.5 ( 3.5.7∼ 3.5.10). 4.3 Bing-Nagata-Smirnov ( 4.3.6 4.3.7) 4.4 MoritaHanai-Stone ( 4.4.2) , . , (covering property). . 6 . 5.1 5.1.1 [278] X , (i) X ; (ii) X σ ; (iii) X ; (iv) X . 4.4.1 . 5.1.1 σ . U = n∈N Un σ , Un (n ∈ N) . V1 = U1 , Vn = {U − kα , {Cα,1 }α∈A {Uα }α∈A . i = 1, 2, · · · , n {Uα }α∈A {Cα,i }α∈A (5.1.9) (5.1.10). · 132 · {Cα,n+1 }α∈A . 5 α ∈ A, Uα,n+1 = Uα − Cβ,n − , (5.1.11) β<α {Uα,n+1 }α∈A X , x ∈ X, U x Uα , − {Cα,n }α∈A {Uα }α∈A , ( β<α Cβ,n ) ⊂ β<α Uβ , x ∈ Uα,n+1 . , {Uα,n+1 }α∈A {Cα,n+1 }α∈A , (5.1.11) − ( β<α Cβ,n ) ∩ Cα,n+1 = ∅ ( Cα,n+1 ⊂ Uα,n+1 ). (5.1.9), (5.1.10), (5.1.11) Cα,n β > α Uβ,n+1 , ( β>α Cβ,n+1 )− ⊂ β>α Uβ,n+1 , Cα,n ∩ ( β>α Cβ,n+1 )− = ∅, (5.1.10). α i, − Vα,i = X − Cβ,i , β=α Vα,i α = β, Vα,i ∩ Vβ,i = ∅. {Cα,i }α∈A U , Vα,i ⊂ Cα,i ⊂ Uα . {Vα,i : α ∈ A, i ∈ N} X , , {Vα,i : α ∈ A, i ∈ N} U σ . x ∈ X, {Cα,i }α∈A (i ∈ N) X , αi = min{α ∈ A : x ∈ Cα,i }, k αk = min{αi : i ∈ N}. x ∈ Vαk ,k+1 . αk x ∈ Cαk ,k . (5.1.10) ( − Cβ,k+1 . x∈ / i = k), (5.1.12) β>αk {Cα,k+2 }α∈A , αk , α αk x ∈ Cα,k+2 . (5.1.9) ( i = k+1), x ∈ / ( β<α Cβ,k+1 )− . ( β<αk Cβ,k+1 )− ⊂ ( β<α Cβ,k+1 )− , − x∈ / Cβ,k+1 . (5.1.13) β<αk (5.1.12) (5.1.13), x ∈ Vαk ,k+1 . {Vα,i : α ∈ A, i ∈ N} U = {Uα }α∈A σ . , , {Vα,i : α ∈ A, i ∈ N} {Dα,i : α ∈ A, i ∈ N}. i ∈ N, − Dα,i ⊂ Vα,i , α∈A α∈A 5.1 X · 133 · , Gi − Dα,i ⊂ Gi ⊂ Gi ⊂ Vα,i . α∈A α∈A Wi = {Vα,i ∩ Gi : α ∈ A}, Wi (, 2.30). W = i∈N Wi U σ . . 5.1.4 (i) ⇒ (ii) ⇒ (iii) ⇒ (v) ⇒ (i) (ii) ⇒ (iv) ⇒ (v). (i) ⇒ (ii). 5.1.1 (ii) . (ii) ⇒ (iii). X σ V = i∈N Vi , Vi . X , {V : V ∈ V } U . , {V : V ∈ Vi } . i ∈ N, Hi = ∪Vi , Hi , Wi = V − Hn : V ∈ V i , W = Wi . n m), W ∩Hm = ∅. x ∈ (∪(W ∩( im Wi )))− . im Wi , W ∈ W ∩( im Wi ) ⊂ W , x ∈ W = W . W U . (iii) ⇒ (v). U . (v) ⇒ (i). 5.1.6 . (ii) ⇒ (iv). , (iv) ⇒ (v) (ii) ⇒ (iii) (). . T2 , T4 . . [46] 5.1.4 T1 X (collectionwise normal space), X {Fα }α∈A , {Uα }α∈A , α ∈ A, Fα ⊂ Uα . , . 2.30 “ ” “”. 5.1.6 T4 , . 5.1.5 T1 X , X . {Fα }α∈A X . α = β, Fα = Fβ . α ∈ A, Uα = X − β=α Fβ , Fα ⊂ Uα , α = β, Fα ∩Uβ = ∅. {Uα }α∈A X · 134 · 5 , , {Cα }α∈A . Vα = X − ( β=α Cβ )− , {Vα }α∈A . ( β=α Cβ )− ⊂ β=α Uβ , ( β=α Uβ )∩Fα = ∅, Fα ⊂ Vα , α ∈ A. , α, α ∈ A, ( β=α Cβ ) ∪ ( β=α Cβ ) = X, Vα ∩ Vα = ∅. X . . 5.1.2 [46] T2 . 3.5.8 5.1.4 (v) . . ( 5.9). 5.1.3 ( 5.1.1) . 5.1.5 [278] X I = [0, 1] Φ = {ϕs }s∈S X (partition of unity), x ∈ X, s∈S ϕs (x) = 1, s∈S ϕs (x) = 1 s ∈ S ϕs (x) = 0, 1. , {ϕ−1 s ((0, 1])}s∈S X . U , {ϕ−1 s ((0, 1])}s∈S U . 5.1.6 ( [278] ) X T2 , (i) X ; (ii) X U U ; (iii) X U U . . 5.1.7 U . . X U U = {Uα }α∈A X . , {Fα }α∈A α ∈ A, Fα ⊂ Uα ( 4.4.2). Urysohn ( 2.4.1), α ∈ A, fα : X → [0, 1] fα (x) = 0, x ∈ X − Uα ; fα (x) = 1, x ∈ Fα . f (x) = α∈A fα (x), U fα f X R . {Fα }α∈A X , f X . ϕα = fα /f , Φ = {ϕα }α∈A X . {ϕ−1 α ((0, 1])}α∈A U , Φ U . . 5.1.8 X U X Φ U , U σ . Φ = {ϕα }α∈A X X U , {ϕ−1 α ∈ A, ϕ−1 α ((0, 1])}α∈A X U . α ((0, 1]) 5.1 · 135 · −1 ϕ−1 α ((0, 1]) = ∪{ϕα ((1/i, 1]) : i 2} (α ∈ A). Vα,i = ϕ−1 α ((1/i, 1]), V = {Vα,i : i 2, α ∈ A}, V U . i 2, {Vα,i }α∈A , V σ . x0 ∈ X, i 2 . x0 U (x0 ) U (x0 ) . Φ = {ϕα }α∈A , α∈A ϕα (x0 ) = 1, {Vα,i }α∈A A A , x0 α∈A ϕα (x0 ) > 1 − 1/i. U (x0 ), x ∈ U (x0 ), α∈A ϕα (x) > 1 − 1/i. U (x0 ) Vα,i (α ∈ A ) . , U (x0 ) ∩ Vβ,i = ∅, β ∈ / A , x ∈ U (x0 ) ∩ Vβ,i , α∈A ϕα (x ) + ϕβ (x ) > 1. . . 5.1.6 (i) ⇒ (ii). (ii) ⇒ (iii) T2 , (iii) ⇒ (i) ⇒ (ii). ( 3.5.8), 5.1.7 . (iii) ⇒ (i). 5.1.8 X U σ . (iii) X , 5.1.1 T1 . . , (iii) x0 ∈ X F ⊂ X x0 ∈ / F , U = {X − F, X − {x0 }} Φ = {ϕα }α∈A U . α0 ∈ A, ϕα0 (x0 ) = a > 0, −1 ϕα0 ((0, 1]) ⊂ X − F ϕα0 (x) = 0, x ∈ F . f (x) = 1 − min{1, ϕα0 (x)/a}, f (x0 ) = 0; f (x) = 1, x ∈ F . X . . ( 5.1.1 5.1.2 5.1.4 5.1.6) T2 . . , Mack[271] Junnila[219, 220] . . [219], [220], [271]. 5.1.6 X U (), (). U (interiorpreserving), U ⊂ U , ∩U . 5.1.7 [271] (i) X ; (ii) X ; (iii) X ; (iv) X . · 136 · 5 (iv) 5.1.1 (iv), . 5.1.8 [219, 220] (i) X ; (ii) X σ , () X; (iii) X , () X; (iv) X V x ∈ X, x ∈ Int(st(x, V )). 5.2 5.2∼5.6 5 . 5.2.1 (Michael [281] ) 5.8 5.1.4 (iii) Michael T2 . . . T2 , . 5.1.4 (iii). 5.2.1 “ T2 ”, . 5.1 ? , . , , Mack Junnila . Mack[271] [ 5.1.7 (iv)] ” [ , , 5.1.1 (iv) “ [414] ]. Worrell “ ” ( 6.2.2) , “ ” “T1 ”. [136] “ ()? ” Worrell , “” “”, 5.2.2 . 5.2.1 X Y f [285] (bi-quotient), y ∈ Y X U , ∪U ⊃ f −1 (y), U U ⊂ U y ∈ Intf (∪U ) ( U X , f [359] (countably bi-quotient)); f [19] (pseudo-open), y ∈ Y X U , U ⊃ f −1 (y), y ∈ Intf (U ); f [306] (quasi-perfect), 5.2 f , y ∈ Y , f −1 (y) ( , 3.3.1). · 137 · f −1 (y) , . 5.2.2 f : X → Y X Y ,Y . V Y , f −1 (V ) = {f −1 (V ) : V ∈ V } X . 5.1.8 (iii), U f −1 (V ) {IntU : U ∈ U } X. f −1 (V ) , U f , f (U ) = {f (U ) : U ∈ U } Y ( . 5.8), V . {IntU : U ∈ U } X . f , y ∈ Y , U U ⊂ U y ∈ Intf (∪{IntU : U ∈ U }) ⊂ Intf (∪U ). , U ∈ U U = ∪U , y ∈ Intf (U ). Y V f (U ) = {f (U ) : U ∈ U } Y . 5.1.8 (iii), Y . . , . 5.2.1 [136] . [414] 5.2.2 . [271] 5.2.3 . Worrell “ T1 ”, 5.2.3 , 4.4.8 , 5.2.3 . [136] ()? [244] . , T1 ( 6.2.1), . , . , 5.2.2 , , . (). U · 138 · 5 , . . X, Y , f : X → Y Lindelöf (Lindelöf mapping), y ∈ Y , f −1 (y) X Lindelöf . 5.2.3 [132] X . f X Y Lindelöf , U = {Uα }α∈A X , y ∈ Y , f −1 (y) Lindelöf . f −1 (y) U . f −1 (y) ⊂ i∈N Uy,i = Uy . 1.5.1, Vy f −1 (y) ⊂ Vy ⊂ Uy Vy = f −1 (f (Vy )) f (Vy ) Y . Wy = f (Vy ), {Wy }y∈Y Y . Y , {Wy }y∈Y {Wy }y∈Y Oβ Wy(β) , {Oβ }β∈B . −1 −1 f (Oβ ) ⊂ f (Wy(β) ) = Vy(β) ⊂ Uy(β) . {f −1 (Oβ )}β∈B . Vi = {f −1 (Oβ ) ∩ Uy(β),i }β∈B . V = i∈N Vi U σ . X , 5.1.1 . . 5.2.4 [176] . f X Lindelöf . . 5.2.5 Y , X 5.2.3 5.2.4 , X 5.2.3 X X . . . 5.3 2.2 , T0 , T1 , T2 , T3 () , ( 2.2.4). P , P () P, P P (closed hereditary property) ( (open hereditary property)). ( 2.11) ⇒ ( 2.2.7 ). ( 3.5.7), , ? 2.2.4, [0, ω1 ], [0, ω] [0, ω1 ] × [0, ω] T2 , . T2 , ( 3.5.8). , ⇒ . 5.3.1 [106] X , X ( X ). 5.3 · 139 · M ⊂ X X , V = {Vα }α∈A M M . X U = {Uα }α∈A Vα = Uα ∩ M , α ∈ A. G = ∪U , G ⊃ M . U G, , U G W , {W ∩ M : W ∈ W } M , V . M . . 5.3.1 [278] T2 Fσ ( T2 Fσ (Fσ -hereditary property)). M T2 X Fσ . M = n∈N Fn , Fn (n ∈ N) . V = {Vα }α∈A M , X U = {Uα }α∈A Vα = Uα ∩ M, α ∈ A. n ∈ N, {Uα }α∈A ∪ {X − Fn } X , Wn . Bn = {W ∩ M : W ∈ Wn , W ∩ Fn = ∅}, B= Bn . n∈N B M σ , V . T2 , M . 5.1.1 (ii) M . . 5.3.1 Bn M [157] X. , T1 Fσ . 4.1 ( Fσ , 4.1.3) , 5.3.1 5.3.1 . 5.3.1 [108] ). T2 ( . 5.3.1 [241] {Fα }α∈A (hereditarily closure-preserving), α ∈ A, Hα ⊂ Fα , {Hα }α∈A . σ (σ-hereditarily closure-preserving). , ⇒ ⇒ . 5.5.1. 5.3.2 [130, 428] (i) X X T2 , ; (ii) X G Fσ , G ; (iii) X G Fσ , G σ . (i) ⇒ (ii) (iii) ⇒ (i). · 140 · 5 (i) ⇒ (ii). G T2 X . X ( 3.5.8). x ∈ G, x U (x), U (x) ⊂ G. {U (x)}x∈G G. (i), X , G {Uα }α∈A {U (x)}x∈G . G T2 , ( 3.5.8), G {Fα }α∈A Fα ⊂ Uα , α ∈ A ( 4.4.2), G Fσ {Gα }α∈A Fα ⊂ Gα ⊂ Uα , α ∈ A ( 2.14). {Uα }α∈A G , {Gα }α∈A G . G Gα ⊂ Uα ⊂ U α ⊂ U (x) ⊂ G, Gα X Fσ , G = ∪{Gα : α ∈ A}. (iii) ⇒ (i). X G , 5.3.1 . X G G = i∈N ( αi ∈Ai Xαi ), Xαi X Fσ {Xαi }αi ∈Ai G . U G . Uαi = {U ∩ Xαi : U ∈ U } (αi ∈ Ai ; i ∈ N). Uαi Fσ Xαi . 5.3.1 , Uαi X σ , G σ Vαi = j∈N Vαi ,j , j ∈ N, Vαi ,j G , . Vα∗i ,j = ∪{V : V ∈ Vαi ,j } ⊂ Xαi , {Xαi }αi ∈Ai G , V = Vαi ,j i,j∈N αi ∈Ai G σ , U . T2 , G . 5.1.4 (ii), G , X . . Dowker[111] totally ( 4.21). 5.3.2 (ii) . . 5.3.2 [130] T2 totally . 5.3.2 “ ” “ ” ( 5.19). 5.4 1944 , Dieudonné[106] , Sorgenfrey[371] 1947 . 2 2.3.3 (Sorgenfrey ) 2.3.4 (Sorgenfrey ). 2 Lindelöf ( ) Lindelöf ( ) , . Sorgenfrey Lindelöf , ( 5.1.1), Sorgenfrey (Sorgenfrey ) , , ( 3.5.8). 5.4 · 141 · , . , [106] . Dieudonné 5.4.1 ( 5.20), . 5.4.1 [106] . X , Y . X × Y X p : X × Y → X ( 3.3.1). 5.2.4, X × Y . . X σ (σ-compact), . 5.4.2 [278] T2 σ . X T2 , Y σ . σ Lindelöf , Y ( 5.1.1), Tychonoﬀ , Y Stone-Čech βY . 5.4.1, X × βY . T2 ( 3.1.5), Y βY Fσ , X × Y X × βY Fσ , X × Y T2 , 5.3.1 X × Y . . . ( 5.4.1), ? , 5.4.1. 5.4.1 (Michael [282] ) T2 , . [0, 1], Q, I. [0, 1] , I . X. X . X . U X . U U Q, U ∪ {{x} : x ∈ X − ∪U } σ , U . X , 5.1.2 X . I , . X × I . X × I A = Q × I, B = {(x, x) : x ∈ I}. U , A ∩ U = ∅. n ∈ N, B In = {x : x ∈ I, {x} × S1/n (x) ⊂ U }, S1/n (x) = {y : y ∈ I, ρ(x, y) < 1/n}, ρ . , I = n∈N In , I ( 1.3.2), Int(I n ) = ∅ ( [0, 1] , ), I n [0, 1] , Q ∩ I n = ∅. · 142 · 5 x ∈ Q ∩ I n y ∈ I ρ(x, y) < 1/2n. (x, y) ∈ A, V y I W , (V × W ) ∩ U = ∅. x X X , V = G ∪ K, G , K ⊂ I, x ∈ Q, , x K G, x I n , x ∈ G ∩ In ρ(x , x) < 1/2n, (x , y) ∈ V × W , ρ(x , y) ρ(x , x) + ρ(x, y) < 1/2n + 1/2n = 1/n. In (x , y) ∈ U . (V × W ) ∩ U = ∅. . 5.4.1 , . R Michael (Michael line). X Lindelöf , Lindelöf [0, 1] I [0, 1] I , I [239] p. 422, (Y. . , Bernstein , [233] Kodama) (K. Nagami) 13.5. . X Lindelöf . X , X × I . X, X , Michael[278] ( 5.22). 5.3.2 X, X totally . totally . () . 5.4.3 (Tamano [383, 384] ) X Tychonoﬀ , (i) X ; (ii) X T2 cX, X × cX (iii) X × βX ; (iv) X cX X × cX ; (i) ⇒ (ii) 5.4.1 . . (ii) ⇒ (iii) ⇒ (iv) . (iv) ⇒ (i). {Uα }α∈A X . α ∈ A, cX Vα Uα = X ∩ Vα . Z = cX − α∈A Vα cX − X , ∆ ⊂ X × X X × Z X × cX , X × cX I = [0, 1] f : X × cX → I f (∆) ⊂ {0}, f (X × Z) ⊂ {1}. ρ(x, y) = sup {|f (x, z) − f (y, z)|}. z∈cX (5.4.1) 5.5 · 143 · ρ X . ρ T1 X T2 , T1 ⊂ T2 . x0 ∈ X, x ∈ cX ε > 0, f X × cX G × H , (x0 , x ) f (G × H) ⊂ (f (x0 , x ) − ε/3, f (x0 , x ) + ε/3), d(f (G × H)) < ε (d(D) D ⊂ I cX, cX , H , 4.1.4). G X, H cX, d(f (Gi × Hi )) < ε (i = 1, 2, · · · , k), {x0 } × cX ⊂ k i=1 (Gi × Hi ), x0 ∈ k i=1 Gi . x ∈ (5.4.2) k i=1 Gi , (5.4.1), ρ(x, x0 ) = sup {|f (x, x ) − f (x0 , x )|}. x ∈cX x ∈ cX, x ∈ Hi , (5.4.2) ρ(x, x0 ) < ε, k Gi ⊂ Sε (x0 ) = {y : y ∈ X, ρ(x0 , y) < ε}. i=1 ρ T2 , T1 ⊂ T2 . Stone 4.3.3 , X {S1/2 (x)}x∈X T1 {Wβ }β∈B , X T2 . x ∈ X, y ∈ S1/2 (x), f (x, y) = |f (x, y) − f (y, y)| ρ(x, y) < 1/2. y ∈ S1/2 (x) ( cX ), f (x, y) 1/2. β ∈ B, x ∈ X Wβ ⊂ S1/2 (x), z ∈ Z, f (x, z) = 1, W β ∩ Z = ∅, W β ⊂ α∈A Vα . W β , A(β) ⊂ A, W β ⊂ α∈A(β) Vβ , Wβ ⊂ X ∩ W β ⊂ α∈A(β) Uβ . {Wβ ∩ Uα : β ∈ B, α ∈ A(β)} X , {Uα }α∈A . X . . 5.4.1 5.4.3 . 5.4.4 X T2 T2 Y , X × Y . 5.5 , ( 3 , 3.1.3). · 144 · 5 5.5.1 {Xα }α∈A , Xα (α ∈ A) , α∈A Xα . U = {Uβ }β∈B α∈A Xα . α ∈ A, {Xα ∩ Uβ }β∈B Xα , Xα Vα {Xα ∩ Uβ }β∈B . V = α∈A Vα , V α∈A Xα , U . . 5.5.1 “ ”. “”, , . 5.5.2 [278] {Fα }α∈A X , Fα (α ∈ A) X T2 , X T2 . X = α∈A Fα , Fα T2 , {Fα }α∈A . , X T1 . X . x ∈ X F , x∈ / F, U (x) = X − ∪{Fα : x ∈ / Fα }, V (x) = ∪{Fα : x ∈ Fα } = n Fαi . i=1 {Fα }α∈A U (x) x ( 3.5.2), V (x) , U (x) ⊂ V (x). Fα ( 3.5.8), x Fαi (i n), i n, x X Ui (x) Ui (x) ∩ Fαi ∩ F = ∅. W (x) = U (x) ∩ n Ui (x) , i=1 W (x) x , W (x) ∩ F ⊂ V (x) ∩ n i=1 n Ui (x) ∩ F ⊂ (Fαi ∩ Ui (x)) ∩ F = ∅. i=1 X . U X . U () Fα Fα Uα . Fα , Fα Vα Uα . Fα , Vα X ( 5.3.1 ). V = α∈A Vα , V X, U . {Fα }α∈A , V . X , 5.1.1 X , . P [194] (locally ﬁnite closed sum theorem), X {Fα }α∈A Fα (α ∈ A) P, 5.5 · 145 · X P. (closed sum theorem) (sum theorem). 5.5.2 “T2 ”. (generality theorem for sum theorems), 5.5.2 . 5.5.3 [26] P (i) P (ii) P ; , P . {Fα }α∈A X , Fα (α ∈ A) P. α ∈ A, Fα Fα Fα , {Fα }α∈A , fα : Fα → Fα . X ∗ = α∈A Fα , X ∗ X f ( ) x ∈ X ∗ , f (x) = fα (x), x ∈ Fα . (i) X ∗ P. ( x Fα ), f . f . E ∗ X ∗ . E∗ = α∈A α ∈ A, (E ∗ ∩ Fα ), f (E ∗ ) = E ∗ ∩ Fα Fα , fα Fα fα (E ∗ ∩ Fα ). α∈A Fα , fα (E ∗ ∩ Fα ) Fα , Fα X , fα (E ∗ ∩ Fα ) X . {Fα }α∈A , {fα (E ∗ ∩ Fα )}α∈A , f (E ∗ ) . , α∈A fα (E ∗ ∩ Fα ) ( 5.1.2), f X ∗ X P . . . (ii), X P, 5.5.1 5.2.3, 5.5.3 (i) (ii), (). 5.5.4 [26] {Fα }α∈A X , Fα (α ∈ A) X , X . 5.5.3 , , fα (E ∗ ∩ Fα ) ⊂ f (E ∗ ) ∩ Fα . , α∈A fα (E ∗ ∩ Fα ) , {fα (E ∗ ∩ Fα )}α∈A . {Fα }α∈A , {fα (E ∗ ∩ Fα )}α∈A ? , {Fα }α∈A , α ∈ A, Hα ⊂ Fα , {Hα }α∈A , 5.5.1. · 146 · 5 5.5.1 ( , ) [0, 1], [0, 1/2], ···, [0, 1/n], ···. , {[0, 1/n]}n∈N . {1} ⊂ [0, 1], {1/2} ⊂ [0, 1/2], ···, {1/n} ⊂ [0, 1/n], ···, {{1/n}}n∈N . 5.5.3 {Fα }α∈A “” “ ” ( 5.3.1), {fα (E ∗ ∩ Fα )}α∈A , f . . 5.5.5 [356] P (i) P ; (ii) P , P (hereditarily closure-preserving closed sum theorem). Singal Arya[356] “ ”. , “ ”. , T2 , 5.2.1, T2 , 5.5.5 . 5.5.6 [356] {Fα }α∈A X , Fα (α ∈ A) X T2 , X T2 . 5.5.6 “T2 ”. 5.5.3 5.5.5 . , . X P (point countable), X P . “” ( (Σ )) “ ” ( (Σ )) 5.5.3 5.5.5. 5.5.5 , “” (pointcountable and hereditarily closure-preserving closed sum theorem, (Σ ∗ ))[138] . ( , 5.6). X F (regular closed set), F = IntF . F F . , “”[133] (locally ﬁnite regular closed sum theorem, (Σ ◦ )). 5.5 · 147 · X, Y , f : X → Y (countable-to-one), −1 y ∈ Y , f (y) ; , X U , Intf (U ) ( 4.22). 5.5.3 , . 5.5.7 [138] P (i) P ; (ii) P , P . 5.5.8 [133, 138] P (i) P ; (ii) P P . , 5.5.3. f : X ∗ → X , f , 5.5.3 Fα , Uα , Fα = U α . , f , Fα E (⊂ Fα ) f (E) . fα : Fα → U α , fα (E) U α , G fα (E) = G ∩ U α . x ∈ fα (E) ⊂ G, x V (x) ⊂ G. x ∈ fα (E) ⊂ U α , V (x) ∩ Uα = ∅. V (x) ∩ Uα ⊂ fα (E), Intfα (E) = ∅. E ⊂ Fα , f (E) = fα (E). f . (ii), X P, P . . 4 , ( (i) , ) (Σ ) ⇒ (Σ ∗ ) ⇒ (Σ ) ⇒ (Σ ◦ ); ⇒ ⇒ ⇒ 5.5.7 Lindelöf 5.5.8 . ; . 5.5.3 X T1 , 5.5.9. , . 5.5.1 [327] {Fα }α∈A T1 X , X K, K Fα . , Fα K. K ∩ Fα1 = ∅, x1 ∈ K ∩ Fα1 . K − Fα1 , Fα2 (K − Fα1 ) ∩ Fα2 = ∅, x2 ∈ (K − Fα1 ) ∩ Fα2 . xn ∈ (K − i 0 y x} ( y = x ). y > x, {(x, y)} . (x, x) ∈ X, Vx = {(x, y) : (x, y) ∈ X y > x} ∪ {(y, x) : (y, x) ∈ X y < x} ( (x, x) ). (5.5.1) (x, x) {{(x, x)} ∪ (Vx − F ) : F }. ∆ = {(x, x) : x > 0}. (x, y) ∈ X − ∆, F (x, y) = {(x, x), (x, y), (y, y)}, (5.5.2) 5.6 · 149 · F (x, y) . F = {F (x, y) : (x, y) ∈ X − ∆} “ ” X . F . F ⊂ F . (x, y) ∈ (∪F )− . (x, y) ∈ X − ∆, (x, y) , (x, y) ∈ ∪F . (x, y) = (x, x) ∈ ∆, (x, x) {(x, x)} ∪ (Vx − F ) ∪F , F (a, b) ∈ F ({(x, x)} ∪ (Vx − F )) ∩ F (a, b) = ∅. (5.5.2), F (a, b) = {(a, a), (a, b), (b, b)}. (5.5.1) (x, x) = (a, a) (x, x) = (b, b). (x, x) ∈ F (a, b) ∈ F , (x, x) ∈ ∪F . F . X T2 , , ( X 2.2.3 2.3.4 ). 1973 , Potoczny[340] ( ), ( 6.1.2), ( 6.1.8 ). 1975 , Potoczny Junnila , ( 5.5.2)[341] . 5.5.3 ( [372] , ) X , p ∈ X. X (excluded point topol/ U . , X T0 ogy)U ⊂ X X U = X, p ∈ , X T1 . P X , P = P ∪ {p}. {Fα }α∈A X , α∈A Fα = (Fα ∪ {p}) = α∈A Fα . α∈A , X . , X {{p, x} : x ∈ X} . , 5.5.1 T1 T0 . P “”, P 5.5.9 ( X , {{p, x} : P). x ∈ X} X , {p, x} (x ∈ X) P, X P. 5.5.9 T1 T0 . 5.6 , ( 3.5.1) . T2 ( 3.5.8), , T2 [ 3.5.1 (i)]. T2 , . · 150 · 5 5.6.1 [110, 228] X (countably paracompact), X . , , . 5.6.1 [202] (i) X ; (ii) X {Ui }i∈N , X {Vi }i∈N Vi ⊂ Ui (i ∈ N); X {Wi }i∈N , X {Fi }i∈N Fi ⊂ (iii) Wi (i ∈ N) i∈N IntFi = X; (iv) X {Fi }i∈N i∈N Fi = ∅, X {Wi }i∈N Fi ⊂ Wi (i ∈ N) i∈N W i = ∅. (i) ⇒ (ii). U = {Ui }i∈N . (i) i(V ) V ⊂ Ui(V ) . V U . V ∈ V Vi = ∪{V : V ∈ V , i(V ) = i}. {Vi }i∈N , U Vi ⊂ Ui (i ∈ N). (ii) ⇒ (i). . (ii) ⇒ (iii). {Wi }i∈N X . (ii) X {Vi }i∈N Vi ⊂ Wi (i ∈ N). Fi = X − j>i Vj , Fi Fi ⊂ ji Vj . ji Vj ⊂ ji Wj = Wi , Fi ⊂ Wi (i ∈ N). {Vi }i∈N , x ∈ X x U (x) U (x) Vj . Vj i, U (x) ⊂ Fi , i∈N IntFi = X. (iii) ⇔ (iv). de Morgan . (iii) ⇒ (ii). {Ui }i∈N X . Wi = ji Uj , {Wi }i∈N . (iii) {Fi }i∈N Fi ⊂ Wi (i ∈ N) i∈N IntFi = X. Vi = Ui − j j x IntFj Vi . {Vi }i∈N Vi ⊂ Ui (i ∈ N). . 5.6.1 (iii) {Fi }i∈N ( ji Fj Fi ), (iv) {Wi }i∈N ( ji Wj Wi ). [110] 5.6.1 X X {Fi }i∈N i∈N Fi = ∅, X {Wi }i∈N Fi ⊂ Wi (i ∈ N) i∈N Wi = ∅. 5.6.1 {Ui }i∈N X , (i) {Ui }i∈N ; (ii) {Ui }i∈N ; 5.6 · 151 · (iii) {Ui }i∈N {Vi }i∈N V i ⊂ Ui (i ∈ N); (iv) {Ui }i∈N {Fi }i∈N Fi ⊂ Ui (i ∈ N); (v) {Ui }i∈N Fσ {Ai }i∈N Ai ⊂ Ui (i ∈ N); (vi) {Ui }i∈N {Fj }j∈N . (i) ⇒ (ii), . (ii) ⇒ (iii), 4.4.2 . (iii) ⇒ (iv), . (iv) ⇒ (v), 2.14 . (v) ⇒ (vi), Ai = j∈N Fi,j (i ∈ N), {Fi,j }i,j∈N . (vi) ⇒ (iv). U = {Ui }i∈N X , {Fj }j∈N X , U. U = {Ui }i∈N Fj Uj , U = {Uj }j∈N . U , , . U = {Uj }j∈N X , {Fj }j∈N U Fj ⊂ Uj (j ∈ N). (iv) ⇒ (iii), . (iii) ⇒ (i), Wi = Ui − j l + m, F (l, m) ⊂ F (l, n − l), n > l + m x Wn . . 6.1.6 θ . 6.1.8 θ . . 6.1.1 U X . n ∈ N, Fn = {x ∈ X : ord(x, U ) n}; x ∈ ∪U , Wx = ∩{U ∈ U : x ∈ U }, (i) Fn ; 6.1 · 163 · (ii) {Wx ∩ (Fn − Fn−1 ) : x ∈ Fn − Fn−1 } Fn − Fn−1 , Fn − Fn−1 , F0 = ∅; (iii) V X Fn−1 ⊂ V , {Wx ∩ (Fn − V ) : x ∈ Fn − V } X , Fn − V . (i) x ∈ Fn , Ui ∈ U x ∈ Ui (i = 1, 2, · · · , n + 1). n+1 U = i=1 Ui , U X . , x ∈ U U ∩ Fn = ∅ ( x∈U ord(x, U ) n + 1). (ii) x ∈ Fn − Fn−1 , x ∈ Wx ∩ (Fn − Fn−1 ), U n x, Ui ∈ U x ∈ Ui (i = 1, 2, · · · , n). y ∈ Fn − Fn−1 , Wy ∩ (Fn − Fn−1 ) = Wx ∩ (Fn − Fn−1 ), U ∈ U y ∈ U x ∈ U , U ∩ Wx ∩ (Fn − Fn−1 ) = ∅. , z ∈ U ∩ Wx ∩ (Fn − Fn−1 ), Ui (i = 1, 2, · · · , n) U z, z ∈ Fn − Fn−1 . , {Wx ∩ (Fn − Fn−1 ) : x ∈ Fn − Fn−1 } Fn − Fn−1 . (iii) {Wx ∩(Fn −V ) : x ∈ Fn −V } Fn −V . Fn −V ⊂ Fn −Fn−1 , x ∈ Fn − V , (ii) Wx = ∅. {Wx ∩ (Fn − V ) : x ∈ Fn − V } Fn − V . {Wx ∩ (Fn − V ) : x ∈ Fn − Fn−1 } X . x ∈ Fn − Fn−1 , (ii), Wx ∩ (Fn − V ) = Wx ∩ (Fn − Fn−1 ) ∩ (Fn − V ) Fn − V , (i), Fn − V X , Wx ∩ (Fn − V ) X . (ii) 5.5, {Wx ∩ (Fn − V ) : x ∈ Fn − Fn−1 } X , 5.6, X . . 6.1.1 (iii) {Wx ∩ (Fn − V ) : x ∈ Fn − Fn−1 } X . 6.1.2 X , U X , A, A , X σ A, U . n ∈ N, Fn = {x ∈ X : ord(x, U ) n} x ∈ ∪U , Wx = ∩{U ∈ U : x ∈ U }. 6.1.1, {Wx ∩ F1 : x ∈ F1 } X , F1 . , V1 F1 , U . . Fn nk=1 Vk U , Vk (1 k n) , Gn = ∪{V : V ∈ Vk , 1 k n}, Fn ⊂ Gn . 6.1.1 , {Wx ∩ (Fn+1 − Gn ) : x ∈ Fn+1 − Fn } , Fn+1 − Gn . , Vn+1 Fn+1 − Gn , U . n+1 k=1 Vk Fn+1 . , k ∈ N, Vk k∈N Vk n∈N Fn ⊃ A, U . . · 164 · 6 6.1.8 [416] θ . U θ X , {Vn }n∈N U θ . k ∈ N, An,k = {x ∈ X : ord(x, Vn ) k}. An,k , Vn An,k . 6.1.2, σ Wn,k An,k , Vn . ∪{Wn,k : n, k ∈ N} U σ . 5.1.2 . . 6.1.8 θ θ , , [78] [69]. Smith[368] , 6.1.8. 6.1.9 [368] . X θ . 6.1.10 [368] θ . θ 6.1.4 6.1.9 . . θ + ⇒ ( 6.1.6); θ + ⇒ ( 6.1.8); θ + ⇒ ( 6.1.10). ( θ + meta ) ⇒ . [49] 6.1.5 X (pointwisely collectionwise normal space), {Fα : α ∈ A}, {Gα : α ∈ A} Fα ⊂ Gα (α ∈ A), α = β, Fα ∩ Gβ = ∅. 6.1.11 [49] . X X θ , ( 6.2). . ) ( , 6.1.6 [119] X ortho (orthocompact space), X . ortho . ( 5.1.6), ( ( ) . meta ) ⇒ , 6.1 6.1 · 165 · 6.1 , [69] . ( meta )θ θ θ , . 6.1.7 X Lindelöf (para-Lindelöf space[55] ), X ; meta-Lindelöf [11, 55] , X ; δθ (weakly δθ-reﬁnable space[410] ), X V = n∈N Vn , x ∈ X, n ∈ N, 1 ord(x, Vn ) ω; Vn , δθ (δθ-reﬁnable space[30] ); δθ {∪Vn : n ∈ N} , δθ (weakly δθ-reﬁnable space[367] ). 6.2 . 6.2 [143, 367] δθ δθ ? 6.2 , Burke[67] Lindelöf . 6.1.12 [67] X Lindelöf X U V x ∈ X, x ∈ Int(st(x, V )). . . U X , V U . W W ∈ W , W V . , P W , x ∈ X, x ∈ Int(st(x, P)). V ∈ V , U (V ) ∈ U , V ⊂ U (V ). G(V ) = Int(st(V, P)) ∩ U (V ). G(V ) , x ∈ V, x ∈ Int(st(x, P)), V ⊂ Int(st(V, P)). G = {G(V ) : V ∈ V } X , U . G . · 166 · x ∈ X, 6 P , x N (x) P . P ∈ P W ∈ W , W V , P V . st(V, P) , P ∈ P {st(V, P) : V ∈ V } st(V, P) . N (x) G , G . . Z. Balogh[40] meta-Lindelöf , . Lindelöf [40] . —— . . , , , Fσ . 6.1.8 A (star-ﬁnite), A ∈ A , {B ∈ A : B ∩ A = ∅} ; (star-countable), A ∈ A , {B ∈ A : B ∩ A = ∅} . X (strongly paracompact space[109] ), X . , , ⇒ . ( ), 6.2.3. 6.1.1 . ( 6.1.13) , . 6.1.3 [184] A A = ∪{Bα : α ∈ Λ}, Bα , α = β , (∪Bα ) ∩ (∪Bβ ) = ∅. A, B ∈ A , A {C1 , C2 , · · · , Cn } A B , A = C1 , B = Cn Ci ∩ Ci+1 = ∅(1 i < n). A ∈ A , B(A) = {B ∈ A : A A B }. B(A) , A1 , A2 ∈ A , (∪B(A1 )) ∩ (∪B(A2 )) = ∅ B(A1 ) = B(A2 ). . 6.1.4 U = {Uk }k∈N X , U {Fk }k∈N Fk ⊂ Uk (k ∈ N), U . k ∈ N, Fk ⊂ Uk , {G(k, n)}n∈N Fk ⊂ G(k, 1) ⊂ G(k, 1) ⊂ G(k, 2) ⊂ · · · ⊂ G(k, n) ⊂ G(k, n) ⊂ · · · ⊂ Uk . n ∈ N, Vn = ∪{G(i, n) : 1 i n}. H(1, 1) = V2 ∩ G(1, 2); H(k, n) = (Vn+1 − Vn−1 ) ∩ G(k, n + 1) (n > 1; 1 k n). {H(k, n) : k, n ∈ N, 1 k n} , U . . 6.1 · 167 · 6.1.4 5.6.2 , . 6.1.13 [365] , (i) X ; (ii) X . (i) ⇒ (ii) . (ii) ⇒ (i). U X , U V . 6.1.3, V V = ∪{Bα : α ∈ Λ}, Bα α = β , (∪Bα ) ∩ (∪Bβ ) = ∅. (ii) , Bα (α ∈ Λ) Bα = {Bα,n : n ∈ N}, n ∈ N, {Bα,n : α ∈ Λ} , n∈N {Bα,n : α ∈ Λ} U σ ( 5.1.2 X ). α ∈ Λ, Zα = ∪Bα , Zα X , . Bα = {Bα,n : n ∈ N} Zα , {Fα,n }n∈N Fα,n ⊂ Bα,n , n ∈ N ( 5.6.1). 6.1.4, Wα Zα , Bα . α∈Λ Wα X , U . . 6.1.1 [301] Lindelöf . 6.1.13 (ii) . . 6.1.1 5.1.1. 1977 , [265] . 6.1.9 [265] X P X ⊂ X , x ∈ X X P . P = n∈N Pn , n−1 Pn X . P1 , n 2, Pn X − i=1 Pi∗ (Pi∗ = ∪{P : P ∈ Pi }), P σ (σ-relatively discrete); ∗ P1 n 2, Pn = {P − n−1 i=1 Pi : P ∈ Pn } n−1 ∗ X − i=1 Pi , P σ (σ-relatively discrete and relatively closed). X (quasi-paracompact) ( (strongly quasi-paracompact)), X σ (σ ) . . . n , P = n∈N Pn σ , n ∈ N, i=1 Pi∗ X . ( 6.1.1), . . 6.1.14 [265] . V X , V U = A m Am . {Uα }α∈A . m ∈ N, · 168 · 6 ξ ∈ Am , ξ = (α1 , α2 , · · · , αm ). Uξ = Uα1 ∩ Uα2 ∩ · · · ∩ Uαm , Um = {Uξ : ξ ∈ Am }. U m , Um∗ ⊃ Em . Em X Pm = {Uξ ∩ Em : ξ ∈ Am }. ∗ , Pm U , Pm = Em . U , X = i∈N Ei , P = i∈N Pi X. Am ξ, η, Uξ ∩ Uη ∩ Em = ∅, Uη ∗ Pm , Pm Um . X− m−1 Pi∗ = X − i=1 Pm X − m−1 i=1 m−1 Ei ⊂ Um∗ , i=1 Pi∗ , m−1 Uξ ∩ Em = X − Pi∗ ∩ (X − ∪{Uη : η ∈ Am , η = ξ}), i=1 Uξ ∩ Em (= Uξ ∩ Em − m−1 i=1 Pi∗ ) X − m−1 i=1 Pi∗ . P U , V σ . X . . 6.1.14 , . 6.1.15 [269, 446] θ . V θ X , {Un }n∈N V , x ∈ X, n ∈ N ord(x, Un ) < ω. 6.1.14 , Un (n ∈ N), σ Pn = m∈N Pn,m , Xn = {x ∈ X : ord(x, Un ) < ω}. n∈N Pn X. , W1 = P1,1 , W2 = P1,2 , W3 = P2,1 , W4 = P1,3 , W5 = P2,2 , · · · . , W = n∈N Wn V σ , X. X . . 6.1.16 [269, 446] θ . U X , σ P = P ∈ P1 x ∈ P n∈N Pn U . n = 1 , P1 X , x V (x) P1 . x ∈ P, V (x) P . V (P ) = x∈P V (x) P . U (P ) ∈ U P ⊂ U (P ). B(P ) = V (P ) ∩ U (P ), B1 = {B(P ) : P ∈ P1 }, 6.1 · 169 · , B1 U , B1∗ ⊃ P1∗ . x ∈ P1∗ , x ∈ P0 ∈ P1 . P = P0 , B(P ) ∩ P0 = ∅. x ∈ / B(P ). x ∈ P1∗ , ord(x, B1 ) = 1. ∗ n 2, Pn X − n−1 i=1 Pi . 6.1.9, x ∈ X − n−1 ∗ Pn . i=1 Pi , x X V (x) n−1 ∗ ∗ ∗ ∗ Bn U , Bn ⊃ Pn − n−1 i=1 Pi x ∈ X − i=1 Pi ⊃ ∗ Pn∗ − n−1 i=1 Pi , ord(x, Bn ) = 1. B = n∈N Bn . B X , U . x ∈ X, n0 x ∈ Pn∗ , ord(x, Bn0 ) = 1. X θ . . 6.1.17 [269, 446] θ . U X , σ P = n∈N Pn U . 6.1.16 , θ B = n∈N Bn ∗ U , B1∗ ⊃ P1∗ , Bn∗ ⊃ Pn∗ − n−1 i=1 Pi (n 2). P σ n , 6.1.9 , n ∈ N, i=1 Pi∗ X . B1 = B1 ; Bn = n−1 B− Pi∗ : B ∈ Bn (n 2). i=1 B = ∗ n∈N Bn θ , U . {Bn }n∈N . x ∈ X, n0 ∈ N x ∈ Pn∗0 , n > n0 , x ∈ / Bn∗ . X θ . . , 6.3 . 6.3 [446], [269], [158]). , θ ( [265], + ⇒ ? 6.1.10 “ θ + ⇒ ” , “ θ ” “ θ ” ( 6.1.8 ). [265] “” “ θ ” 6.3 , . 6.1.18 [265] . . · 170 · 6 6.1.11 “θ + ⇒ ”. [446] [269] “ ” “θ ”, Boone[49] . 6.1.19 [269, 446] X X . . 5.6 (countably subparacompact[195] ) (countably weakly paracompact countably metacompact[202] ) meso (countably mesocompact[418] ) θ (countably θreﬁnable[161] ) , meso θ “” “” . 6.1.6 6.1 , . 6.1.20 [195] X X U = {Un }n∈N {Fn,j }n,j∈N Fn,j ⊂ Un (n, j ∈ N). . U = {Un }n∈N X , F = j∈N Fj X σ , U . Fj Un Fn,j , Fn,j ⊂ Un (n, j ∈ N) . . Fn,j Fn,j = {Fn,j } (n, j ∈ N) . . 6.1.2 [195] . Fσ , 6.1.20 . . 6.1.21 [195] . U = {Un }n∈N X . {Fn,j }n,j∈N Fn,j ⊂ Un (n, j ∈ N). V1 = U1 , Vn = Un − ∪{Fk,j : k < n, j < n} (n 2), V = {Vn }n∈N , U . . , . , 6.1.1. , ( 5.6.1) . 6.1.22 [202] (i) X ; (ii) X {Un }n∈N , X {Vn }n∈N Vn ⊂ Un (n ∈ N); (iii) X {Wn }n∈N , X {Fn }n∈N Fn ⊂ Wn (n ∈ N); 6.1 · 171 · (iv) X {Fn }n∈N n∈N Fn = ∅, X {Wn }n∈N Fn ⊂ Wn (n ∈ N) n∈N Wn = ∅. 5.6.1 , . 6.1.23 [161] (i) X ; (ii) X θ ; (iii) X {Fn }n∈N n∈N Fn = ∅, X Gδ {Gn }n∈N Fn ⊂ Gn (n ∈ N) n∈N Gn = ∅. (i) ⇒ (ii), . (ii) ⇒ (iii). X θ , {Fn }n∈N X n∈N Fn = ∅. U = {X − Fn : n ∈ N}, U X , θ {Vn }n∈N . Gn,j = st(Fn , Vj ), Gn = Gn,j , j∈N Gn Gδ Fn ⊂ Gn (n ∈ N). n∈N Gn = ∅. , x ∈ n∈N Gn . j0 ∈ N ord(x, Vj0 ) < ω. ord(x, Vj ) = k, V1 , V2 , · · · , Vk ∈ Vj0 x ∈ Vi (i = 1, 2, · · · , k), V ∈ Vj0 V = Vi (i = 1, 2, · · · , k), x ∈ / V . i k, ni ∈ N Vi ⊂ X − Fni . n = max{n1 , n2 , · · · , nk }, i k, Vi ⊂ X − Fn . x ∈ Gn,j0 = st(Fn , Vj0 ), V ∈ Vj0 x ∈ V V ∩ Fn = ∅. x ∈ V ∈ Vj0 , V V1 , V2 , · · · , Vk , V ⊂ X − Fn . V ∩ Fn = ∅ . (iii) ⇒ (i). (iii) ⇒ 6.1.22 (iv), 6.1.22 . {Fn }n∈N X , n∈N Fn = ∅. (iii) X Gδ {Gn }n∈N Fn ⊂ Gn (n ∈ N) n∈N Gn = ∅. n ∈ N, Gn = j∈N Gn,j , Gn,j ; Un = ∩{Gi,j : 1 i, j n}. , Un , Fn ⊂ Un (n ∈ N) n∈N Un = ∅. 6.1.22 (iv). . , 6.4 . 6.4 · 172 · 6 6.1.1 [235] [0, ω1 ) × [0, ω1 ] . X = [0, ω1 ) × [0, ω1 ], H = {(α, ω1 ) ∈ X : α < ω1 }, K = {(α, α) ∈ X : α < ω1 }, H, K X . X , 6.1.20, X {X − H, X − K} F . {F ∈ F : F ∩ H = ∅} = {Hn }n∈N , {F ∈ F : F ∩ H = ∅} = {Gn }n∈N . n ∈ N α < ω1 , (α, ω1 ) ∈ / Gn , γn,α < ω1 α < γn,α ({α} × (γn,α , ω1 ]) ∩ Gn = ∅. γα = sup{γn,α : n ∈ N}, α < γα < ω1 , Gn ∩ {(α, γ) ∈ X : γα < γ ω1 } = ∅. , m ∈ N α < ω1 , β, γ < ω1 α β < γ (β, γ) ∈ Hm . , , n ∈ N, αn < ω1 αn β < γ < ω1 , (β, γ) ∈ / Hn . α0 = sup{αn : n ∈ N}, α0 < ω1 , γ0 ∈ (γα0 , ω1 ), (α0 , γ0 ) ∈ / n∈N (Hn ∪ Gn ) = X, . Y = ([0, ω1 ) × [0, ω1 )) ∩ Hm . Hm , n ∈ N, (βn , γn ) ∈ Y γn βn+1 < γn+1 . β = sup{βn : n ∈ N}, βn < γn βn+1 , [0, ω1 ) {βn } {γn } β. (β, β) ∈ Hm ∩ K, K ∩ Hm = ∅ . [0, ω1 ) × [0, ω1 ] ( 3.23), 6.1.1 . 5.6.1, 5.6.2, 6.1.22, 6.1.23, θ . 6.2 6.1 . . Micheal[280] ( 5.1.4), T2 ( 5.2.1) , . θ . 6.2.1 (Burke [60, 61] ) . 6.1.1 (iii), 5.2.1 . . [414] 6.2.2 (Worrell ) . 6.1.2 (ii), 5.2.1 . . 6.2 · 173 · 6.2.3 (Junnila [217] ) θ θ . 6.1.4 (ii), 5.2.1 . . meso , 6.1.3 (ii) , 6.1.1 (iii) 6.1.2 (ii) 6.1.4 (ii) , . 6.2.4 [147] Meso meso . X meso , f : X → Y ( 5.5.1). V = {Vβ }β∈B Y , U = {f −1 (Vβ )}β∈B X . 6.1.3 (ii), U F = {Fα }α∈A F X . f , {f (Fα )}α∈A Y f , {f (Fα )}α∈A Y , V . . 6.1.3 (ii), Y meso . . 6.2.5 [147] Meso meso . ( 5.5.1). . meso , 5.2.2 “ () ” , , , . 6.2.1 , 6.2.2 meso . 6.2.1 [244] T1 . X = (N ∪ {0}) × (N ∪ {0}) − {(0, 0)}, N . m, n, k ∈ N, V (n, m) = {(n, k) : k m}. X (i) (n, m) ∈ N × N {{(n, m)}}; (ii) (n, 0) {V (n, m) ∪ {(n, 0)}}m∈N ; (iii) (0, n) {V (n, m) ∪ {(0, n)}}mn . X T1 . X . X {0} × N 0∗ Y = (N × (N ∪ {0})) ∪ {0∗ }. {0} × N X , . T1 6.2.2 [244] f . Y . T0 meso . 6.2.1 X, , (i), (ii) ; (iii) {V (n, n) ∪ {(0, n)}} . (0, n) T0 , . (0, n) (n, n), · 174 · 6 , {0} × N 0∗ Y , f . ∗ Y . , Y 0 {{(n, m) : m n > 0} ∪ {0∗ }} . Y . ( 6.11), Y meso . 6.2.2 T0 meso . 6.2.2 meso , meso . T2 , meso ? [147] 6.2.6 meso meso . 6.6.9 . θ ( 6.1.5) , . θ ( , ). , θ . Burke , , [70], . 6.2.7 [70] θ θ . [66, 369] θ ? θ , . , θ ( 6.1.5) ( 6.1.5 (ii)). [369] θ ? ortho , Gruenhage[164] ortho . [65] Burke ortho . [66] ortho ? , . Ponomarev[336] “ ”, ( 6.2.3). 6.2.3 ( ) X F1 , F2 , X [ [114] 5.3.F(e)]. F = F1 ⊕ F2 (), f : F → X . , ( 6.14). , 5.5.3 X . , . 6.2 ( 6.1.7 ) , Burke Lindelöf . 6.2.8 [67] Lindelöf Lindelöf Lindelöf . 6.2 · 175 · X Lindelöf , f : X → Y Lindelöf . V −1 −1 Y , f (V ) = {f (V ) : V ∈ V } X . X U f −1 (V ). f (U ) = {f (U ) : U ∈ U }, f (U ) Y , V . f (U ) . y ∈ Y, f −1 (y) Lindelöf , U , f −1 (y) U ( 5.4). Uy , f −1 (y) ⊂ Uy . f , 1.5.1, Gy f −1 (y) ⊂ Gy ⊂ Uy , Gy = f −1 (f (Gy )) f (Gy ) Y . Gy U , f (Gy ) f (U ) . f (U ) . , y ∈ Y , y ∈ f (Gy ) ⊂ ∪{f (U ) : U ∈ U , y ∈ f (U )} = st(y, f (U )). f (Gy ) , y ∈ Int(st(y, f (U ))). 6.1.12 Y Lindelöf . . Burke[67] ( ) Lindelöf Lindelöf . , . 6.1 6.2 ( meso ortho ) Lindelöf ( “” , 5.2.3 ). 6.2.9 f : X → Y X Y Lindelöf ,Y θ θ , X θ θ . 5.2.3 () , 6.1.1 ( ) (ii) ; 6.1.13 () . ( {Wy,n }n∈N ). U = {Uα }α∈A X . V {V : V ∈ V } U . y ∈ Y, f −1 (y) Lindelöf , f −1 (y) V {Vy,i }i∈N . Uy,i ∈ U V y,i ⊂ Uy,i (i ∈ N). Wy,1 = Uy,1 ∩ Vy,i , i∈N Wy,n = Uy,n − i 1; n ∈ N), i∈N · 176 · 6 {Wy,n }n∈N X , f −1 (y). Wy = n∈N Wy,n f −1 (y) , f , 1.5.1, Wy f −1 (y) ⊂ Wy ⊂ Wy , Wy = f −1 (f (Wy )) f (Wy ) Y . {f (Wy )}y∈Y Y . Y , , , {Hy }y∈Y , y ∈ Y, Hy ⊂ f (Wy ) . f −1 (Hy ) ⊂ Wy ⊂ Wy = n∈N Wy,n . W = {Wy,n ∩ f −1 (Hy ) : n ∈ N, y ∈ Y }. , W X , U . W . x ∈ X. f (x) ∈ Hy x ∈ f −1 (Hy ), x f −1 (Hy ) (y ∈ Y ), f −1 (Hy1 ), · · · , f −1 (Hyk ). l k, {Wyl ,n }n∈N , x {Wyl ,n ∩ f −1 (Hyl )}n∈N , x W . W . . Meso Lindelöf meso ? [159] . , . [212] , . 6.2.10 Lindelöf ( [442] [156]). 6.2.10 f : X → Y X Y Lindelöf . Y θ Lindelöfmeta-Lindelöfδθ δθ δθ , X . 5.2.3 , , . 6.2.9 6.2.10 θ [156] δθ [156] D. K. Burke[69] . 6.2.11 f : X → Y X Y . Y [272] 6.2.10 , X 6.2.9, meso ( X T2 ). ⇒ Lindelöf , , 6.2.9 . , X T2 , [59] . . 2.1.1 X O, A ⊂ O, A O ClO (A). , (precise) ( 5.1.3). U , . 5.6.1 (i) ⇒ (ii). 6.2 · 177 · 6.2.1 T2 X U K, O K ⊂ O {U ∩ O : U ∈ U } O . U {Ui }in K. x ∈ K, i(x) n, x ∈ Ui(x) . Fx = K − Ui(x) , Fx X x ∈ / Fx . 3.1.4, Vx , Wx , x ∈ Vx , Fx ⊂ Wx Vx ∩ Wx = ∅, V x ∩ Wx = ∅. Gx = Wx ∪ Ui(x) , Gx K ⊂ Gx . K {xj }jm , K ⊂ jm Vxj . O= Gxj ∩ Vxj ∩ Ui , jm jm in O K ⊂ O. j m, Pj = ClO (Vxj ∩ O). Pj O , O = jm Pj , Pj − Ui(xj ) ⊂ (Gxj − Ui(xj ) ) ∩ ClX (Vxj ) ⊂ Wxj ∩ ClX (Vxj ) = ∅, Pj ⊂ Ui(xj ) . {Pj }jm {U ∩ O : U ∈ U } O . . 6.2.11 X T2 , Y , X . U X . y ∈ Y, f −1 (y) X , 6.2.1, X Oy , f −1 (y) ⊂ Oy Oy {U ∩ Oy : U ∈ U } . f , 1.5.4, Y Hy , y ∈ Hy −1 −1 f (Hy ) ⊂ Oy , f (Hy ) {U ∩ f −1 (Hy ) : U ∈ U } Fy . H = {Hy : y ∈ Y }. Y , H i∈N Wi , Wi = {Wiy : y ∈ Y } Wiy ⊂ Hy (i ∈ N; y ∈ Y ). i ∈ N, Pi = {f −1 (Wiy ) ∩ F : y ∈ Y, F ∈ Fy }. P = i∈N Pi U σ . , P U . i ∈ N, f −1 (Wi ) y ∈ Y , Fy , Pi σ . y ∈ Y F ∈ Fy , x ∈ X − (f −1 (Wiy ) ∩ F ), Lx = X − f −1 (Wiy ), x ∈ X − f −1 (Wiy ), f −1 (Hy ) − F, x ∈ f −1 (Wiy ) − F. Lx x X Lx ∩ (f −1 (Wiy ) ∩ F ) = ∅, f −1 (Wiy ) ∩ F . X . . 6.1.1 T1 X. f : , Y = X/X , X X → X/X, f · 178 · 6 . 6.2.11 T2 . 6.2.9 X . [156] 6.2.4 X = [0, 1), {[0, t) : 0 t 1} ∪ {{x} : x [0, 1) } X . X θ . Q [0,1) . Q . θ Q θ . ( 6.3.1), X U , , U ∗ = ∪{U : U ∈ U }. {rn } Q 1 , U = {[0, rn ) ∩ Q : n ∈ N} Q . V = n∈N Vn U θ . {Vn∗ : n ∈ N} [0, t) ∩ Q, t ∈ [0, 1). s1 ∈ Q, . , V n1 ∈ N, 0 < ord(s1 , Vn1 ) < ∞, ord(s1 , Vn1 ) = m1 . s2 ∈ Q, s2 > max{t : s1 ∈ [0, t) ∩ Q ∈ Vn1 }, n2 ∈ N, 0 < ord(s2 , Vn2 ) < ∞, ord(s2 , Vn2 ) = m2 . Vn2 Vn1 (n1 = n2 ), , s1 < s2 , ord(s1 , Vn1 ) = m1 + m2 , . {sk }k∈N ⊂ Q {nk }k∈N k ∈ N, 0 < ord(sk , Vnk ) < ∞, 0 s1 < s2 < · · · < sk < · · · < 1, k1 = k2 , nk1 = nk2 . , k ∈ N, s1 ∈ Vn∗k . {Vn∗ }n∈N . X θ . X X/Q Y . Y ( Y {Q} f : X → Y . 6.2.8 Y ), , 6.2.8 ( 5.2.3 ) . , Lindelöf [269] . X = N, T = {∅, X} ∪ {{1, 2, · · · , n} : n ∈ N}, (X, T ) . θ . g : X → X/X , . X/X X T0 . T2 . Bennett Lutzer[45] ( 6.4). X T2 , θ ( θ [159] ), θ . X T2 . f . 5.2.3 6.2.9 ( θ ) T2 . 6.2.5 [348] ortho ortho . p : [0, ω1 ) × [0, ω1 ] → [0, ω1 ). [0, ω1 ] , p . [0, ω1 ) ortho [0, ω1 ) × [0, ω1 ] ortho . , 6.2.2 pressing down lemma. 6.2.2 f : [0, ω1 ) → [0, ω1 ) x ∈ [0, ω1 ) 6.2 · 179 · f (x) < x, c ∈ (0, ω1 ) x ∈ [0, ω1 ), y x f (y) < c. , c ∈ (0, ω1 ), [0, ω1 ) c x y x f (y) c . x α f (x) < x. x1 α c , x1 x2 y x2 , f (y) x1 . x2 c , x2 x3 y x3 , f (y) x2 . , xn c , xn xn+1 . β {xn } . y xn+1 , f (y) xn . β xn+1 (n ∈ N), y β, f (y) xn (n ∈ N). β , y β , f (y) β. y = β, f (β) β. β x1 α, f (β) < β, . . 6.2.3 U = {Uα }α∈A [0, ω1 ) , c ∈ (0, ω1 ) st(c, U ) ⊃ [c, ω1 ). Uα , {0} (αx , x] , αx < x < ω1 . f (x) = αx , 0 < x < ω1 . 6.2.2 c ∈ (0, ω1 ) x ∈ [0, ω1 ), y x αy < c. c < x < ω1 ⇒ x, c ∈ (αy , y]. st(c, U ) ⊃ [c, ω1 ). . [0, ω1 ) ortho . U [0, ω1 ) (Uα ). 6.2.3, c ∈ (0, ω1 ) st(c, U ) ⊃ [c, ω1 ). [0, c] ( 3.1.1 ), U , U . V = U ∪ {Uα ∩ (c, ω1 ) : c ∈ Uα ∈ U }. V U . . [0, ω1 ) × [0, ω1 ] ortho . U = {[0, α] × (α, ω1 ] : α ∈ [0, ω1 )} ∪ {[0, ω1 ) × [0, ω1 )}. V U , 6.2.3 [0, ω1 ) × {ω1 } (V ), c ∈ (0, ω1 ) st((c, ω1 ), V ) ⊃ [c, ω1 ) × {ω1 }. ∩{V ∈ V : (c, ω1 ) ∈ V } ⊂ [0, ω1 ) × {ω1 }. V . [0, ω1 ) ortho , [0, ω1 )×[0, ω1 ] ortho . 6.2.5 . , 6.2.5 ortho ortho , ( 6.4.1). [0, ω1 ) , . [0, ω1 ) ( 3.11), ( 6.17), , [0, ω1 ) ( 3.5.10). 6.2.3 . [0, ω1 ) {x : x < α} (α ∈ [0, ω1 )) · 180 · 6 U. U V , 6.2.3 c ∈ (0, ω1 ) st(c, V ) ⊃ [c, ω1 ). st(c, V ) {x : x < α} , [0, ω1 ) ( 5.1.1). Stone ( 5.1.3) [0, ω1 ) . 6.1.10 6.1.18 [0, ω1 ) θ θ . [0, ω1 ) δθ ( 6.6.1 ). , [0, ω1 ) ( 3.10), [0, ω1 ) ( 3.10 ). , ( 5 ) . f : X → Y . X, Y , (open compact mapping[21] ), f y ∈ Y, f −1 (y) ( , 4.4.8 ). , . ( [16] , meso [147] [260] 6.10) , . , (meta )θ ortho meta-Lindelöf ( 6.2.12 6.2.13). 6.2.12 θ meta-Lindelöf θ meta-Lindelöf . meta-Lindelöf , . θ . . X θ , f : X → Y . V Y −1 , U = {f (V ) : V ∈ V } X . 6.1.4 (iii), {Wn }n∈N U , x ∈ X, n ∈ N U ⊂ U , x ∈ W ∈ Wn , W U . {Wn }n∈N . , {Wk }k∈N . Wk Wk = Wn1 ∧ Wn2 ∧ · · · ∧ Wnm , X , U . f , k ∈ N, Gk = {f (W ) : W ∈ Wk } Y , V . {Gk }k∈N Y . y ∈ Y , f , f −1 (y) = {x1 , x2 , · · · , xm }. xi (i = 1, 2, · · · , m), ni ∈ N Ui ⊂ U , xi ∈ W ∈ Wni , W Ui . xi (i = 1, 2, · · · , m), xi ∈ W ∈ Wk = Wn1 ∧ Wn2 ∧ · · · ∧ Wnm , W m i=1 Ui . y ∈ Y , k ∈ N V ⊂ V , 6.3 · 181 · V = {V ∈ V : f −1 (V ) ∈ m i=1 Ui }, y ∈ f (W ) ∈ Gk , f (W ) V . 6.1.4 (iii), Y θ . . , Junnila [219] θ [217] . Gittings[162] ortho ? Kofner[234] meta ( ortho ) ortho . , Scott[351] [435] . , . 6.2.13 [351, 435] Ortho ortho . X ortho , f : X → Y . V Y −1 , {f (V ) : V ∈ V } X . X ortho , −1 U = {Uα }α∈A {f (V ) : V ∈ V }. f , {f (Uα )}α∈A Y , V . {f (Uα )}α∈A . A ⊂ A, α∈A f (Uα ) = ∅, y ∈ α∈A f (Uα ), f −1 (y)∩ Uα = ∅ (α ∈ A ). f , f −1 (y) = {x1 , x2 , · · · , xk }. Ai = {α : α ∈ A , xi ∈ Uα }(i = 1, 2, · · · , k). , Ai . k A = i=1 Ai . xi (i = 1, 2, · · · , k), xi ∈ α∈A Uα , y ∈ f ( α∈A Uα ) ⊂ i i f (U ), i = 1, 2, · · · , k . α α∈A i k k y∈ f Uα ⊂ f (Uα ) = f (Uα ). i=1 i=1 α∈Ai α∈A α∈Ai k {Uα }α∈A , α∈A Uα X . f , i=1 i f ( α∈A Uα ) Y . y α∈A f (Uα ) . y i α∈A f (Uα ) . {f (Uα ) : α ∈ A} , Y ortho . . θ θ δθ δθ δθ ? . ( [162] [248]). [162] [248], . 6.3 ( 3.5.7 5.3.1) · 182 · 6 , . A ⊂ X ⊂ X, A X IntX A. 6.3.1 X 6.1 6.2 ( T2 , meso Lindelöf ), X Fσ ( X Fσ ). . . meso Lindelöf . X , X X Fσ . X X = n∈N Fn , Fn . U X , X . n ∈ N, Un = {U ∈ U : U ∩ Fn = ∅}, X Un ∪ {X − Fn } Gn . W1 = {G ∩ X : G ∈ G1 , G ∩ F1 = ∅}, Wn = G ∩ X − Fi : G ∈ Gn , G ∩ Fn = ∅ (n > 1). i n) . n∈N Wn X , U . Lindelöf . Gn , Wn . x ∈ X , x ∈ IntX Fn . IntX Fn x ( X ) Wm (m > n) . x n∈N Wn , n∈N Wn X , U . . 6.3.1 6.3.1 . 6.3.1 6.3.1 meso θ [160] δθ Burke[69] . 6.3.1 Fσ . N (A) ( 4.1.2), A , ( 6.3). N (A) [0, 1] N (A) × [0, 1] ( 6.4.1). N (A) × [0, 1] Fσ N (A) × (0, 1) ( 6.3). 6.3 · 183 · ( 5.3.2), . 6.3.2 [428] X , (i) X ; (ii) X G , G σ ; (iii) X G Fσ , G σ . (ii) ⇒ (iii) . (i) ⇒ (ii) (iii) ⇒ (i). (i) ⇒ (ii). X , G x, x U (x) U (x) ⊂ G. U = {U (x)}x∈G G . G G σ V = n∈N Vn , Vn G . V G V U (x) , U (x) ⊂ U (x) ⊂ G, V X . V (ii). (iii) ⇒ (i). G X . G . (iii), G G = i∈N ( αi ∈Ai Xαi ). Xαi X Fσ , i ∈ N, {Xαi }αi ∈Ai G . Xαi = j∈N Fαi ,j , j ∈ N, Fαi ,j X . U G . Uαi ,j = {U ∩ Fαi ,j : U ∈ U } (j ∈ N; αi ∈ Ai ). Uαi ,j Fαi ,j . Fαi ,j , ( 6.3.1). Uαi ,j Fαi ,j σ Vαi ,j = k∈N Vαi ,j,k . k ∈ N, Vαi ,j,k Fαi ,j . Fαi ,j , Vαi ,j,k X , G . Vα∗i ,j,k = ∪{V : V ∈ Vαi ,j,k } ⊂ Fαi ,j ⊂ Xαi . {Xαi }αi ∈Ai G , V = i,j,k∈N αi ∈Ai Vαi ,j,k G . Vαi ,j,k αi ∈Ai G σ , U . G . . , Junnila [224] 1986 (scattered partition) , . , [450] · 184 · 6 θ θ . , meso [450, 425] . 6.4 ( ). . ( 5.4.1). . 6.2.11 6.1 ( ortho ) 6.2 . . 6.4.1 X 6.1 (ortho , T2 ) 6.2 , Y , X × Y . 5.4.1 . 5.4.2: “T2 σ ”. ( 5.4.1) , T2 Fσ ( 5.3.1). 6.4.1 6.3.1 . 6.4.2 T2 X 6.1 ( ortho ) 6.2 (meso Lindelöf ), Y σ , X × Y . 5.4.2 . Ortho 6.4.1. 6.2.5 [0, ω1 ) ortho , [0, ω1 ) [0, ω1 ] [0, ω1 ) × [0, ω1 ] ortho , ortho 6.4.1, 6.4.2. 6.4.1, 6.4.2, 6.4.1. 6.4.1 6.3.1 N (A) ( 4.1.2), A , ( 6.3), N (A) σ (0, 1) N (A) × (0, 1) ( 6.3.1 6.3). 6.5 , . 6.1 6.2 . 6.5 · 185 · , (5.5 ) , . 6.5.1 [194] F = {Fα }α∈A X , Fα (α ∈ A) X , X . F A , F = {Fα : 0 α < η} ( α, η ). V = {Vσ : σ ∈ B} X . F , V Fα . V . α (0 α < η), Vα . V0 . V Vσ F0 , F0 {Vσ ∩ F0 : σ ∈ B}. F0 , ( F0 ) {Uσ : σ ∈ B} Uσ ⊂ Vσ ∩ F0 (σ ∈ B). Vσ0 = Vσ ∩ (X − (F0 − Uσ )), V0 = {Vσ0 : σ ∈ B}. V0 (i) V0 X ; (ii) Vσ0 ⊂ Vσ , σ ∈ B; (iii) x ∈ F0 , ord(x, V0 ) < ω; (iv) x ∈ Vσ − F0 , x ∈ Vσ0 . α (1 α < η). β < α Vβ = {Vσβ : σ ∈ B} (I) Vβ X ; (II) σ ∈ B, γ < β, Vσβ ⊂ Vσγ ; (III) x ∈ γ<β Fγ , ord(x, Vβ ) < ω; (IV) x ∈ γ<β Vσγ − Fβ , x ∈ Vσβ . Vα = {Vσα : σ ∈ B} (I)∼(IV). σ ∈ B, Wσα = Vσβ Wα = {Wσα : σ ∈ B}. β<α Wα X . x ∈ X, β = max{γ < α : x ∈ Fγ } ( F , . γ < α, x ∈ / Fγ , β = 0). (I), Vβ x ∈ / Fβ+1 , (IV), x ∈ Vσβ+1 , , x ∈ Vσβ . (II), x ∈ γβ Vσγ . x ∈ Vσβ (β > β). x ∈ Wσα , Wα X . Wσα . β = max{γ < α : Vσ ∩ Fγ = ∅} ( V Fα . γ < α, Vσ ∩ Fγ = ∅, β = 0). , Vσ0 ∩ F1 = ∅, Vσ0 = Vσ1 . , Vσ0 , Vσ1 , · · · , Vσβ , · · · . Wσα = β<α Vσβ . Wα . Wα Vα V V0 . . Vα (I)∼(IV). · 186 · 6 , Wσ = α<η Vσα W = {Wσ : σ ∈ B}. W Wα . , W V , (III), W , X . . “”. . 6.5.2 [356] θ . θ ( 6.2.1 6.2.2 6.2.3), 5.5.5 . . 6.5.1 6.5.2 , . Worrell . , θ Burke Junnila . . 6.5.3 Meso θ . meso θ ( 6.2.5 6.2.7), 5.5.3 . . 6.5.4 [250] T1 , meso . T1 , meso ( 6.2.4), 5.5.9 ( T1 ) . . 6.5.5 [138] Lindelöf . Lindelöf Lindelöf ( 6.2.8), 5.5.7 . . 6.5.2∼ 6.5.5 . , ( . ortho 6.2.7 ), , ortho Scott[349] . 6.5.1 , , Scott . 6.5.6 [349] Ortho . , ( 6.2.3), . 6.1 6.2 , θ meta-Lindelöf δθ δθ δθ . 6.6 Iso · 187 · 6.6 Iso iso . 6.6.1 [36] X iso (iso-compact space), . 6.1 6.1 6.2 6.3 ( ortho ) iso . iso Dieudonné[106] , Burke[60, 61] , Arens Dugundji[13] , meta-Lindelöf Aquaro[11] , θ Worrell Wicke[416] , δθ Aull[30] , θ δθ Wicke Worrell[410] , [265] . 6.1 6.2 6.3 , δθ iso . 6.6.1 X U S, M ⊂ S U M , S ⊂ ∪{U ∈ U : U ∩ M = ∅}. S . {xα }α<η xγ ∈ S − st({xα : α < γ}, U ). S {xα } x0 . α < γ, xα . Eγ = S − st({xα : α < γ}, U ). Eγ = ∅, ; , Eγ xγ . {xα }α<η . M = {xα : α < η}, M . . [410] 6.6.1 δθ iso . δθ , 6.6.1 δθ . X δθ , U X , V = n∈N Vn , x ∈ X, n ∈ N 0 < ord(x, Vn ) ω. Cn = {x ∈ X : 0 < ord(x, Vn ) ω}. , X = n∈N Cn . F (U ) X U . X ∈ F (U ). X . . X∈ / F (U ), Cm ∈ / F (U ). n0 m Cm ∈ / F (U ). W0 U Cm (m < n0 ), E0 = X −W0∗ . C1 ∈ / F (U ), E0 = X. E0 X , E0 ∩ Cm = ∅ (m < n0 ). k ∈ N, 0 j k, · 188 · 6 (i) Ej X ; (ii) nj m Ej ∩ Cm ∈ / F (U ); (iii) Ej+1 ⊂ Ej ( Ej+1 ); (iv) Ej ∩ Cm = ∅ (m < nj ). B = Ek − Vn∗k , B . B ∈ / F (U ). , U U1 B. ∅ = (Ek ∩ Cnk ) − U1∗ ⊂ Vn∗k , 6.6.1, M ⊂ (Ek ∩ Cnk ) − U1∗ , (Ek ∩ Cnk ) − U1∗ ⊂ ∪{V ∈ Vnk : V ∩ M = ∅} Vnk M . M ⊂ Ek − U1∗ ⊂ Vn∗k Ek − U1∗ , M ω , M . Ek ∩ Cnk ∈ F (U ), . / F (U ), Cm B ∩ Cm ∈ / F (U ). nk+1 m B∈ B ∩ Cm ∈ / F (U ). Wk+1 U , B ∩ Cm (m < nk+1 ). ∗ Ek+1 = B − Wk+1 , (i)∼(iv) j = k + 1 . , {Ej }j∈N , {nj }j∈N (i)∼(iv). (iv), j∈N Ej = ∅. X . X ∈ F (U ). . . Ortho iso . 6.2.3 [0, ω1 ) [0, ω1 ) ortho , [0, ω1 ) . 6.6.1 ortho iso . , 6.6.1 [0, ω1 ) δθ . , [0, ω1 ) δθ iso . iso Worrell Wicke[417] [ω1 , ∞)r (weakly [ω1 , ∞)r -reﬁnable space) δθ ; , Davis[102] θL (θL-property) , Arhangel’skiı̌[23] pure (pure space) δθ . Sakai[347] neat (neat space) . iso , . meta Lindelöf ( [372] 60). Scott[350] Watson[407] . Burke Davis[72] Lindelöf . , Watson[408] meta-Lindelöf . iso . 6.6.1 iso . iso Fσ . 6.6.2 [36] Iso Fσ iso . E iso X Fσ . E = n∈N Fn , Fn (n ∈ N) iso . M E . U X M . n ∈ N, M ∩ Fn Fn , , U 6.6 Iso · 189 · . M U . M , U , M . E iso . . f : X → Y k (quasi-k-mapping), Y K, f −1 (K) X . 5.5.1 k 5.2.1 . 6.6.3 f : X → Y iso X Y k , Y iso . K Y , f −1 (K) X , , K Y . . f : X → Y Y X , ⇒k ( 3.31), . [36] 6.6.1 iso . 6.6.4 [36] f : X → Y X iso Y ,X iso . M X , f (M ) Y , . k ( 3.30), f −1 (f (M )) . M f −1 (f (M )) , M . . 6.6.5 [293] f : X → Y X iso T1 Y , y ∈ Y, f −1 (y) iso , X iso . M X , f (M ) Y , f (M ) . f M f |M : M → f (M ), f |M . −1 −1 y ∈ f (M ), f |M (y) M , . , f |M (y) f −1 (y) , iso . f |−1 . f (M ) M (y) , f |M , M ( 3.30). X iso . . , (), ( 5.14 5.15). 6.6.6 [36] Iso iso . 6.6.4 3.3.1 . . 6.6.7 [36] iso iso . 6.6.2 . . 6.6.8 {Fα }α∈A T1 X , Fα (α ∈ A) iso , X iso . M T1 X . 5.5.1, M Fα . M ⊂ ni=1 Fαi . 6.6.7, ni=1 Fαi iso . M , X iso . . · 190 · 6 iso 6.6.2 ( 6.6.9 6.6.10). iso . iso 6.6.1, y ∈ Y , f −1 (y) ∂f −1 (y) () . “ ” ( 3.5.6) , 4.4.7 ( 4.4.7 ) . . 6.6.9 [140] iso X Y . K Y . X , f , Y −1 −1 T2 . K Y . f (K) X. f f (K) g = f |f −1 (K) . iso , g iso K . 6.6.2, g . 4.4.8, −1 . C ⊂ f (K) g C h = g|C C K C ( 3.3.3). C ⊂ f −1 (K) ⊂ X, f (C) = g(C) = h(C) = K. . 6.6.9 , 6.2.6 ( meso ( 6.2.4), meso ). meso iso ( 6.6.1), 6.2.6 . 6.6.2 [283] T2 X Y 6.6.3 [140] f X Y {y0 } y0 , f −1 (y0 ) ∩ n∈N f −1 (yn ) . K = f −1 (y0 ) ∩ n∈N f −1 (yn ). , f f . , {yn } Y − , f −1 (yn ) ⊃ {yn : n ∈ N} = {y0 , y1 , y2 , · · · , yn , · · ·}, n∈N K = ∅. . K , {xn } ⊂ K K . {xn : n ∈ N} , {{xn }}n∈N . 2.29 2.30, {Gn }n∈N xn ∈ Gn (n ∈ N). xn ∈ K ⊂ f −1 (y0 ) f −1 (y0 ) ∩ f −1 (yn ) = ∅, n∈N n ∈ N, ni=1 f −1 (yi ) , Gn ( n −1 (yi ) Gn ) i=1 f n Gn ∩ f −1 (yi ) = ∅ i=1 , Gn − (6.6.1). 6.6 Iso m ∈ N, xm ∈ · 191 · n∈N f −1 (y n ), Gm ∩ ( zm ∈ Gm ∩ n∈N f −1 (yn )) = ∅. f −1 (yn ) , n∈N i(m) ∈ N, zm ∈ f −1 (yi(m) ). (6.6.1), i(m) > m (m ∈ N). N {nj } {i(nj )} znj ∈ Gnj ∩ f −1 (yi(nj ) ) (j ∈ N). znj ∈ Gnj (j ∈ N), {{znj } : j ∈ N} , {znj : j ∈ N} . f , , {yi(nj ) : j ∈ N} . {yi(nj ) }j∈N {yn }n∈N y0 . K . . 2 Fréchet ( 2.3.1 ). X Fréchet , A ⊂ X, x ∈ A, A {xn } x; , A ⊂ X, A X A A. , ⇒ Fréchet ⇒ . X, Y . f : X → Y (irreducible[240] ), X F f (F ) = Y . 6.6.10 [140] f iso Y , X ⊂ X f |X X Y . Y y, xy ∈ f −1 (y). X0 = {xy : y Y } ∪ (∪{f −1 (y) : y Y }). (6.6.2) x ∈ / X0 , x ∈ f −1 (y), y Y . f −1 (y) , {xy } , x f −1 (y) − {xy } X0 , X0 . g = f |X0 X0 Y . g , . , X0 U0 g(X0 − U0 ) = Y . Zorn . U = {Uα : Uα X0 , g(X0 − Uα ) = Y }. U “<”Uα < Uα Uα Uα . U ⊂ U, U = {Uα : α < β}, β , U . (i) β , g|X0 −Uβ−1 , X0 U ⊂ X0 − Uβ−1 g((X0 − Uβ−1 ) − U ) = Y . Uβ = Uβ−1 ∪ U , Uβ−1 < Uβ g(X0 − Uβ ) = Y . Uβ U . (ii) β , Uβ = α<β Uα . g(X0 − Uβ ) = Y . , y0 ∈ Y − g(X0 − Uβ ) g −1 (y0 ) ⊂ Uβ , g −1 (y0 ) ⊂ Uα (α < β). (6.6.2) g , y0 Y ( y0 , g −1 (y0 ) = {xy0 }. · 192 · 6 ). {y0 } Y , Y , Y y0 {y0 }, {yn } ⊂ Y − {y0 } {yn } y0 . xy0 ∈ Uβ = α<β Uα ⇒ xy0 ∈ Uα , g(X0 − Uα ) = Y H= g −1 (yn ), K = g −1 (y0 ) ∩ H. n∈N iso , 6.6.3, K , X0 iso K , α < β K ⊂ Uα . K . K ⊂ g −1 (y0 ) ⊂ Uβ , H − Uα . Y T2 , {yn }n∈N ∪ {y0 } , g −1 (y0 ) ∪ H . g −1 (y0 ) ∩ H = ∅, K = g −1 (y0 ) ∩ (H − H). g −1 (y0 ) ∪ H , H ⊂ g −1 (y0 ) ∪ H, H − H ⊂ g −1 (y0 ). K = H − H, H − H ⊂ Uα . g , H − Uα = (H ∪ (H − H)) − Uα = H − Uα . H − Uα . g(H − Uα ) ⊂ {yn : n ∈ N}. {yn } y0 , g(H − Uα ) . m ∈ N, n m , g −1 (yn ) ⊂ Uα . g(X0 − Uα ) = Y . g(X0 − Uβ ) = Y . Uβ U . Zorn , U U . g|X0 −U . X = X0 − U . . . f |X X Y [140] Fréchet , . 1998 , G. Gruenhage[169] f : X → Y , X ⊂ X Lindelöf X , f |X X Y . 6.6.10 Y . 6.6.3 [240] f T2 X Fréchet Y X ⊂ X f |X X Y . , —— . 6.6.2 X U (minimal), U X. X (irreducible[51] ), X . Arens Dugundji[13] iso . 6.6.11 [13] . U = {Uα }α∈A X . U Uα U , Φ. Φ A U ∪ {∅} f (i) f (α) = Uα , f (α) = ∅; 6.6 Iso (ii) · 193 · α∈A f (α) = X. f ∈ Φ. Af = {α : α ∈ A, f (α) = ∅}. Φ “<”f < f Af ⊂ Af . Φ . Φ Φ0 , Φ0 . f0 (α) = f ∈Φ f (α) (α ∈ A). f0 . f0 ∈ Φ0 . f0 (i). x ∈ X, x Uα . Af0 = {α : α ∈ A, f0 (α) = ∅}. / Af0 , f0 (ii), Φ0 . α α ∈ α Af0 , . f ∈ Φ0 α Af , f (ii), f ∈ Φ0 . Zorn , Φ f ∗ . {Uα }α∈A−Af ∗ . . 6.6.4 [13] U = {Uα }α∈A . 6.6.2 6.6.11 . . 1965 , Worrell Wicke[416] θ , . [92, 93] . 1975 , Boone[51] θ ? 1977 , van θ , [107] Douwen Wicke θ , . θ . 1976 , Boone[52] Smith[367] θ . , . 1984 , Mashburn[277] T1 δθ T1 δθ . , . , , . , 1991 [449] T1 . T1 , T1 Lindelöf ( 6.22). [143, 145] . . 1972 , Christian 6.6.12 [51] X , X U = {Uα }α∈A {Fβ }β∈B , B ⊂ A, Fβ ⊂ Uβ (β ∈ B) {Uβ }β∈B X. . U = {Uα }α∈A X . {Vα }α∈B X , B ⊂ A, Vα ⊂ Uα (α ∈ B). β ∈ B, Fβ = X − ∪{Vα : α ∈ B, α = β}, Fβ = ∅. {Fβ }β∈B B ⊂ A, Fβ ⊂ Uβ (β ∈ B) {Uβ : β ∈ B} X. . U = {Uα }α∈A X . , · 194 · 6 {Fβ }β∈B , B ⊂ A, Fβ ⊂ Uβ (β ∈ B) {Uβ }β∈B X. β ∈ B, Vβ = Uβ − ∪{Fα : α ∈ B, α = β}, {Vβ }β∈B X , U . . 6.6.13 (i) ; (ii) ℵ1 ( ) Lindelöf . , ⇒ , Lindelö ⇒ ℵ1 ( 4.1.7 ). , U = {Uα }α∈A X . 6.6.12, {Fβ }β∈B 6.6.12 , (ℵ1 ) (), U () . . 1 (ℵ1 ) X (). , U = {Uα }α∈A (ℵ1 ) X , U (), A (). α ∈ A, xα ∈ Aα . F = ∪{{xα } : α ∈ A}, F ( 3.5.2), F . X (ℵ1 ) , F (), α, β ∈ A , {xα } = {xβ }, U xα (α ∈ A) . U (). 2 [0, ω1 ) , 6.6.13 (i) [0, ω1 ) . [0, ω1 ) ortho , ortho ( 6.6.1 ). 6.6.13 (i) iso ( 6.6.1) . iso , , . 6.6.14 [105] . X X × [0, ω]. X × [0, ω) , ϕ(X). , X ϕ(X) X × {ω}. U ϕ(X) . x ∈ X nx ∈ [0, ω) Ux X Ux × [nx , ω] U . Vx = (Ux × [nx + 1, ω]) ∪ {(x, nx )}. Vx U . n ∈ [0, ω), An = {x ∈ X : nx = n}. W0 = {Vx : x ∈ A0 }; n−1 / Wi∗ Wn = Vx : x ∈ An (x, ω) ∈ i=0 (n ∈ N); 6.6 Iso · 195 · Wω = {p} : p ∈ ϕ(X) − Wn∗ . n<ω ω W = n=0 Wn X , U . ϕ(X) . . 1976 , van Douwen “ ”, . 1979 Davis Smith[105] . (i) . X . X X ×{ω} ϕ(X) . (ii) iso . X iso , ϕ(X) iso . , iso , X iso . . , ( [143] [145]). 1975 , Boone[51] “ ? , Arhangel’skiı̌ MOBI ?” [449] . 6.6.15 [449] X , A ⊂ X, X/A . U X/A . U0 ∈ U A ∈ U0 . U = {U − {A} : U ∈ U } ∪ {U0 }. f X X/A . X/A , X/A {A} . U X/A , U . f −1 (U ) X . f −1 (U ) X V . V1 = {V : V ∈ V , V ∩ A = ∅}, V2 = V − V1 . f |X−A : X − A → X/A − {A} , f (V1 ) U ∗ ∗ ∗ X/A − f (V2 ) . A ⊂ V2 , f (V2 ) X/A A . V ∈ V2 , V ∩A = ∅, V f −1 (U0 ). V2∗ ⊂ f −1 (U0 ), f (V2∗ ) ⊂ U0 . f (V1 ) ∪ {f (V2∗ )} X/A , U . X/A . . 6.6.1 [449] . X , Z = X × {0, 1}. (x, 0) ∈ Z, {Vx × {0} : Vx x X } (x, 0) . (x, 1) ∈ Z, {Vx × {0} ∪ {(x, 1)} : Vx x X } (x, 1) . f : Z → X f ((x, i)) = x (i = 0, 1), f . Z . Z {Vx × {0} ∪ {(x, 1)} : x ∈ X} , . . · 196 · 6 6.6.15 6.6.1, [143] . ( ) ? 1966 , Arhangel’skiı̌[21] MOBI ( (i) (ii) ) ; . 1970 , Bennett[42] “ Y MOBI M φ1 , φ2 , · · · , φn , (φ1 ◦ φ2 ◦ · · · ◦ φn )(M ) = Y ”. [44] θ MOBI . 1990 , Bennett Chaber Boone[51] MOBI . 6.6.16 [145] . Arhangel’skiı̌ MOBI T1 , 7.6.2. , MOBI T1 . 6.1[20] ; k X X meso . 6.2 6.3 6 k X X U [49] ( 6.1.11). . Sorgenfrey , , Fσ , N (A) ( 4.1.2) A [114] . N (A) × (0, 1) A , (0,1) 6.4[45] . R {G(x, n)}n∈N , G(x, n) = {x} ∪ {y : y ; [316] . x |y − x| < 1/n}. X T2 θ θ . 6.5[164] X ortho (pointwise star-orthocompact), X U {Vx }x∈X x ∈ X, x ∈ Vx ⊂ st(x, U ). ortho [218] 6.6 . X ortho (discretely orthocompact), X {Fα }α∈A 6.7 {Uα }α∈A Fα ⊂ Uα , α ∈ A, X {Vα }α∈A Fα ⊂ Vα ⊂ Uα (α ∈ A). ortho [440] . X ortho (locally ﬁnitely orthocompact), {Fα }α∈A “” (i) ortho “”. ; (ii) 6 · 197 · ortho ⇒ ortho , ortho ⇒ ortho ⇒ ortho ⇒ ortho ; (iii) [0, ω1 ) × [0, ω1 ] [101] 6.8 ortho ortho . X *Lindelöf (*Lindelöf space), X U V {st(x, U )}x∈X . (i) X *Lindelöf X U {xn }n∈N n∈N st(xn , U ) = X, *Lindelöf Lindelöf ; (ii) *Lindelöf ; (iii) X Lindelöf X *Lindelöf meta-Lindelöf . 6.9[82] 6.10[147] meso [22] Arhangel’skiı̌ “ . . , . [132] X 6.12 . 6.13 () , X Lindelöf Lindelöf . A . α ∈ A, Iα 0 S(A). ρ(x, y) = ?” [48] 6.11 I = [0, 1]. α∈A Iα |x − y|, x, y ∈ Iα , α ∈ A, x + y, x ∈ Iα , y ∈ Iβ , α = β, . S(A) . S(A) 6.14 . 6.15 6.16[132] [336] . T2 [58] . f X Y , y ∈ Y, f −1 (y) , X . 6.17[274] [45] ( 1.2.3) [421] 6.19 iso 6.21[129] θ 6.20 . X X θ . 6.18 . . . iso iso [36] . iso Tychonoﬀ iso [113] . . iso . S(A) (hedgehog space), 1927 Urysohn [114] . , fα : Xα → Yα (α ∈ A). {fα }α∈A f = {fα }α∈A α∈A fα : {fα (xα )} ∈ α∈A Xα → α∈A Yα x = {xα } ∈ α∈A Yα . 7.2.5. α∈A Xα , f (x) = ( α∈A fα )({xα }) = · 198 · 6 6.22[269] T1 , . [13] 6.6.11 iso . [449] . , 6.23 6.24 . 6.25[449] T1 . T1 . 6.26[145] δθ δθ . T1 δθ . 6.27[433] X B (property B), {Fα }α∈A α∈A Fα = ∅, {Gα }α∈A Fα ⊂ Gα (α ∈ A) α∈A Gα = ∅. (i) ⇒ B ⇒ ; (ii) B B; (iii) B T2 ; (iv) B , , ℵ1 Lindelöf . 6.28[426] Lindelöf B. [393] [155] [369] ( 6.1.9) “” “” 6.29 6.30 6.31 Chaber ( 2.21) B Lindelöf . B. b1 (property b1 ). , ⇒ b1 . b1 ⇒ . 6.32[210] [154] 6.33 b1 . Lindelöf B. b1 . 7 () (generalized metric space generalized metrizable space) . 6 () , () , . . . , 4.1 ( 4.1.1) (M1) (ρ(x, y) = 0 x = y) (M1 ) (ρ(x, y) = 0 x = y), . (M3) ( ) , , . , . , Alexandroﬀ-Urysohn ( 4.5.10) , (i), ( 4.4.1). Moore . (i) (ii) M ( 7.2.2). Moore M . Bing-Nagata-Smirnov ( 4.3.6 4.3.7) , . 7.1 Moore , Gδ Alexandroﬀ-Urysohn ( 4.5.10) X X T0 {Un }n∈N (i) Un+1 Un (n ∈ N); (ii) {st(x, Un )}n∈N x ∈ X . {Un }n∈N (ii) ( 4.4.1), {Un }n∈N . Moore (Moore space[299] ). 7.1.1 [60] [416] . [ 6.1.1 (v)], . , F , F = n∈N st(F, Un ) ( 7.2). . · 200 · 7 () 4.5.6 , X × X ∆ = {(x, x) : x ∈ X} X × X . 7.1.1 [80] X Gδ (Gδ -diagonal), X × X ∆ X 2 Gδ . Gδ , . 7.1.2 [80] X Gδ X {Un }n∈N x, y ∈ X, x = y, n ∈ N, y ∈ / st(x, Un ) ( , x ∈ X, {x} = n∈N st(x, Un )). X Gδ ∆ = n∈N Vn , Vn X 2 . x ∈ X n ∈ N, Un (x) x Un (x) × Un (x) ⊂ Vn , Un = {Un (x) : x ∈ X}. {x, y} ⊂ n∈N st(x, Un ), x = y, n ∈ N, zn ∈ X {x, y} ⊂ Un (zn ), (x, y) ∈ Un (zn ) × Un (zn ) ⊂ Vn , (x, y) ∈ n∈N Vn . . {Un }n∈N , Vn = ∪{U × U : U ∈ Un }, ∆ ⊂ n∈N Vn . (x, y) ∈ n∈N Vn , n ∈ N, Un ∈ Un , (x, y) ∈ Un × Un , y ∈ n∈N st(x, Un ) = {x}, y = x. ∆ = n∈N Vn . . 7.1.2 X Gδ (Gδ -diagonal sequence[80] ). , T1 Gδ , Gδ T1 . 7.1.3 [370] Gδ T2 . X , 7.1.2, {Un }n∈N , x ∈ X, n∈N st(x, Un ) = {x}. X , {Vn }n∈N V n = {V : V ∈ Vn } Un , Vn+1 Vn . st(x, V n ) ⊂ st(x, Un ) = {x}. n∈N n∈N B = {X − (V 1 ∪ · · · ∪ V k ) : V1 , · · · , Vk ∈ Vn ; k, n ∈ N}, B , B X . x0 ∈ X x0 U . 3.1.3, n ∈ N, st(x0 , V n ) ⊂ U . Vn X , x ∈ X − U , Vx ∈ Vn x ∈ Vx . x0 ∈ / Vx ( , x ∈ U , ). X − U , Vx1 , · · · , Vxk X − U , X − (V x1 ∪ · · · ∪ V xk ) ∈ B, x0 ∈ X − (V x1 ∪ · · · ∪ V xk ) ⊂ U. B X , Urysohn ( 4.3.1) . . 7.1 Moore , Gδ · 201 · 7.1.4 (Bing [46] ) . 4.4.6 Nagata-Smirnov ( 4.3.6) T2 . 7.1.1 6.1.8 . . Bing[46] 1937 Jones[215] Moore ( Moore ) . ZFC (Zermelo-Fraenkel + ) . . 7.1.2 [41] X (quasi-developable space), {Un }n∈N , x ∈ X x U , n ∈ N x ∈ st(x, Un ) ⊂ U . X (quasi-development). 7.1.5 [43] X X . 7.1.2 7.1.1. . X Un∗ = ∪{U : U ∈ Un }, . {Un }n∈N 7.1.2, Un∗ , X , Un∗ = m∈N Fn,m , Fn,m . Vn,m = Un ∪ {X − Fn,m } (n, m ∈ N). , Vn,m . {Vn,m }n,m∈N X . x ∈ X x U , n ∈ N x ∈ st(x, Un ) ⊂ U , x ∈ Un∗ = m∈N Fn,m , x ∈ Fn,m ⇒ x ∈ / X − Fn,m , n, m ∈ N st(x, Vn,m ) = st(x, Un ) ⊂ U . . 7.1.6 [45] θ . , , θ . {Gn }n∈N X . U X , U , U = {Uα : α < Λ}, Λ . α < Λ, n ∈ N, P (α, n) = {x ∈ X : x ∈ Uα − ∪{Uβ : β < α} x ∈ st(x, Gn ) ⊂ Uα }, Pn = {P (α, n) : α < Λ}, P = ∪{Pn : n ∈ N} X U . X ∪Pn . x ∈ ∪Pn , α x ∈ P (α, n), st(x, Gn ) x Pn ( P (α, n)) , Pn ∪Pn , 6.1.5 (iii) X θ . . 7.1.3 [416] X θ (θ-base) B, B X , B = ∪{Bn : n ∈ N} X U x ∈ U , nx ∈ N ord(x, Bnx ) < ∞ B ∈ Bnx x ∈ B ⊂ U . , θ θ , θ ( 6.4 7.4 Bennett Lutzer ). · 202 · 7 () 7.1.7 [45] X X θ . X , {Gn }n∈N X . 7.1.6, X θ , X ∪Gn (n ∈ N) θ , Gn ( ∪Gn ) θ {Bn,m }m∈N , Bn,m X . {Bn,m }n,m∈N X θ . X U x ∈ U , {Gn }n∈N X , n ∈ N x ∈ st(x, Gn ) ⊂ U . {Bn,m }m∈N Gn θ , m ∈ N 1 ord(x, Bn,m ) < ∞, st(x, Bn,m ) ⊂ st(x, Gn ) ⊂ U . {Bn,m }n,m∈N X θ . X θ B = ∪{Bn : n ∈ N}. n, k ∈ N, X(n, k) = {x ∈ X : ord(x, Bn ) = k}, G(x, n, k) = ∩{B ∈ Bn : x ∈ B}, x ∈ X(n, k), Gn,k = {G(x, n, k) : x ∈ X(n, k)}, Gn,k G(x, n, k) x . {Gn,k }n,k∈N X . X U x ∈ U , B θ , n ∈ N ord(x, Bn ) < ∞, B ∈ Bn x ∈ B ⊂ U , ord(x, Bn ) = k, x ∈ X(n, k). , Gn,k G(x, n, k) x, st(x, Gn,k ) = G(x, n, k) ⊂ B ⊂ U , {Gn,k }n,k∈N X . . 7.1.5 . 7.1.1 [416] X X θ . 7.2 w∆ M p w∆ . 7.2.1 [55] X w∆ (w∆-space), X {Un }n∈N x ∈ X, xn ∈ st(x, Un ) (n ∈ N), {xn } . {Un }n∈N w∆ (w∆-sequence). x, x, {st(x, Un )}n∈N x ( 7.5). ( 4.4.1), w∆ . , w∆ . Morita[305] M . 7.2.2 [305] X M (M-space), X {Un }n∈N (i) Un+1 Un (n ∈ N); (ii) x ∈ X, xn ∈ st(x, Un ) (n ∈ N), {xn } . 7.2 w∆ M p · 203 · {Un }n∈N M (M-sequence). M ( Un = {X}). (ii) w∆ ( 7.2.1), M w∆ . M AlexandroﬀUrysohn , w∆ . M w∆ [198] (generalized countably compact space ). 7.2.1 (Frink [126] ) d : X × X → R+ ( ) ε > 0, d(x, y) < ε, d(y, z) < ε, d(x, z) < 2ε. (7.2.1) ρ : X × X → R+ x, y, z ∈ X, (i) ρ(x, z) ρ(x, y) + ρ(y, z); (ii) d(x, y)/4 ρ(x, y) d(x, y). d ( d(x, y) = d(y, x)), ρ . ρ ρ(a, b) = inf n−1 d(xi , xi+1 ) : n ∈ N, xi ∈ X, x0 = a, xn = b . i=0 , ρ (i) ρ(x, y) d(x, y). , a, b, x1 , · · · , xn ∈ X, (7.2.2) d(a, b) 2d(a, x1 ) + 4 n−1 d(xi , xi+1 ) + 2d(xn , b). (7.2.2) i=1 n=1 , max{d(a, x1 ), d(x1 , b)} = ε. (7.2.1), d(a, b) 2d(a, x1 ) d(a, b) 2d(x1 , b), (7.2.2) . . (7.2.2) n . a, b, x1 , · · · , xn ∈ X, (7.2.1) xi d(a, b) 2d(a, xi ) d(a, b) 2d(xi , b). k d(a, b) 2d(a, xk ). k = 1 , (7.2.2) . k > 1 , d(a, b) 2d(a, xk−1 ), d(a, b) 2d(xk−1 , b), d(a, b) d(a, xk ) + d(xk−1 , b). d(a, xk ) d(xk−1 , b), (7.2.2) . . 7.2.2 [69] X {Un }n∈N Un+1 Un (n ∈ N), X ρ · 204 · 7 (i) ρ(x, y) = 0 y ∈ () n∈N st(x, Un ); (ii) U ρ x ∈ U n ∈ N st(x, Un ) ⊂ U . X d : X × X → R+ , 0, y ∈ n∈N st(x, Un ), d(x, y) = 1/2n, n = min{i ∈ N : y ∈ / st(x, Ui )}. d(x, y) < 1/2n+1 , d(y, z) < 1/2n+1 , y ∈ st(x, Un+1 ), z ∈ st(y, Un+1 ). z ∈ st(st(x, Un+1 ), Un+1 ) ⊂ st(x, Un ), d(x, z) < 1/2n , d 7.2.1 (7.2.1). X ρ, d 7.2.1 (ii) (i). , 7.2.1 (ii), x ∈ X, st(x, Un+2 ) = S1/2 n+2 (x) ⊂ S1/2n+2 (x) ⊂ S1/2n (x) = st(x, Un ), (7.2.3) Sε (x) = {y ∈ X : ρ(x, y) < ε}, Sε (x) = {y ∈ X : d(x, y) < ε}. , ρ {st(x, Un )}n∈N x . (ii). . 7.2.3 {Un }n∈N X M , x ∈ X, C = n∈N st(x, Un ), C X , {st(x, Un )}n∈N C X ( U ⊃ C, n ∈ N U ⊃ st(x, Un )). st(x, Un+1 ) ⊂ st(st(x, Un+1 ), Un+1 ) ⊂ st(x, Un ), st(x, Un+1 ) ⊂ st(x, Un ), C = n∈N st(x, Un ) = n∈N st(x, Un ) X . 7.2.2 (ii), C . 3.5.2, C . , V ⊃ C xn ∈ st(x, Un ) − V (n ∈ N). {xn } x , x ∈ st(x, Un )(n ∈ N), x ∈ n∈N st(x, Un ) = n∈N st(x, Un ) ⊂ V , . {xn } , M . . , 7.2.2 ρ , n∈N st(x, Un ) . R n∈N st(x, Un ) [x], Y . , 7.2.3 Alexandroﬀ-Urysohn (ii), n∈N st(x, Un ) . 7.2.1 (Morita [305] ) X M Y X Y f . X M , {Un }n∈N M . 7.2.2 X ρ (i) ρ(x, y) = 0 y ∈ n∈N st(x, Un ); 7.2 w∆ M p · 205 · (ii) U ρ x ∈ U n ∈ N, st(x, Un ) ⊂ U . X R xRy ρ(x, y) = 0. Y = X/R f : X → Y , ρ : Y × Y → R+ , ρ ([x], [y]) = ρ(x, y), [x] x Y . , ρ ρ Y . Y TR T . X T . x ∈ X ε > 0, Sε (x) = {y ∈ X : ρ(x, y) < ε}, Sε ([x]) = {[y] ∈ Y : ρ ([x], [y]) < ε}, f −1 (Sε ([x])) = Sε (x) ∈ T , Sε ([x]) ∈ TR , T ⊂ TR . V ∈ TR , f −1 (V ) ∈ T , [x] ∈ V , X [x] ⊂ f −1 (V ), (i) 7.2.3, [x] = n∈N st(x, Un ) (X, T ) , n ∈ N st(x, Un ) ⊂ S1/2 f −1 (V ), S1/2n+2 (x) ⊂ f −1 (V ) [ 7.2.2 (7.2.3) ], n+2 ([x]) ⊂ V , V ∈ T . TR = T . f : X → Y . [x] ∈ Y , f −1 ([x]) = {y ∈ X : ρ(x, y) = 0} = [x] . f . H ⊂ X , f , −1 −1 −1 f (f (H)) . x ∈ / f (f (H)), f (f (x)) ∩ H = ∅, n∈N st(x, Un ) ∩ H = ∅. 7.2.3, m ∈ N st(x, Um ) ∩ H = ∅. st(x, Um+1 ) ∩ −1 −1 f (f (H)) = ∅, f (f (H)) . x ∈ st(x, Um+1 ), st(x , Um+1 ) ⊂ st(st(x, Um+1 ), Um+1 ) ⊂ st(x, Um ), st(x , Um+1 )∩H = ∅. n∈N st(x , Un ) x ∈ / f −1 (f (H)). st(x, Um+1 ) ∩ ∩ H = ∅, f −1 (f (x )) ∩ H = ∅. f −1 (f (H)) = ∅. f . , f X Y . Alexandroﬀ-Urysohn , Y {Un }n∈N , Un+1 Un (n ∈ N). Gn = −1 {f (U ) : U ∈ Un }, Gn+1 Gn . x ∈ X, xn ∈ st(x, Gn )(n ∈ N), {xn } , X M . xn ∈ st(x, Gn ) f (xn ) ∈ st(f (x), Un ), {Un }n∈N Y , Y f (xn ) → f (x). C = {f (x)} ∪ {f (xn ) : n ∈ N}, C Y , , f , f −1 (C) X ( , 3.31, 5.6.6 ). xn ∈ f −1 (C) (n ∈ N), {xn } . X M . . 7.2.1 X iso M Y X Y . · 206 · 7 () iso ( 6.6.1), iso ( 6.6.4), 7.2.1 . . 7.2.1 . 7.2.2 [305] Y . X M Y X iso ( 5.2.4). , iso M . . 7.2.1 7.2.2 M M M . 7.2.2, , 7.2.2 7.2.1 . 7.2.4 [127] {Xs }s∈S , As Xs , W s∈S Xs , W ⊃ s∈S As , Us ⊂ Xs , Us = Xs s , s∈S As ⊂ s∈S Us ⊂ W . 3.32 . A= As ⊂ W ⊂ s∈S Xs , s∈S A s∈S Xs . a ∈ A, s∈S Ws a ∈ s∈S Ws ⊂ W ( Ws s, Ws = Xs ). A , A . Ws1 ∪ A⊂ s∈S Ws2 ∪ · · · ∪ s∈S Wsk ⊂ W. s∈S S0 = {s1 , s2 , · · · , sl } ⊂ S Wsi = Xs (i = 1, 2, · · · , k) s ∈ S − S0 . k Wsi , W1 = i=1 W2 = s∈S0 Xs , s∈S−S0 As × s∈S0 As ⊂ W1 × W2 ⊂ W. s∈S−S0 3.32 (), Us ⊂ Xs (s ∈ S0 ) As ⊂ s∈S0 s ∈ S − S0 , Us = Xs , Us ⊂ W1 . s∈S0 s∈S As = s∈S Us ⊂ W . . 7.2 w∆ M p · 207 · {Xs }s∈S , {Ys }s∈S {fs }s∈S fs : Xs → Ys . f : s∈S Xs → s∈S Ys {fs }s∈S (product of mapping family, f = s∈S fs ), f s∈S Xs {xs }s∈S s∈S Ys {fs (xs )}s∈S . 7.2.5 [127] {fs }s∈S f = s∈S fs . fs : Xs → Ys (s ∈ S) . y = {ys }s∈S ∈ −1 (y) = s∈S fs−1 (ys ) . , s∈S Ys , f f . f . y = {ys }s∈S ∈ s∈S Ys s∈S Xs U ⊃ f −1 (y) = fs−1 (ys ), s∈S 7.2.4, Us ⊂ Xs Us = Xs S S0 s , U ⊃ s∈S Us ⊃ s∈S fs−1 (ys ). S0 = {s1 , s2 , · · · , sk }. i k, fsi , Usi ⊃ fs−1 (ysi ), 1.5.1, Us i i Usi ⊃ Us i ⊃ fs−1 (ysi ), i Us i = fs−1 (fsi (Us i )) fsi (Us i ) Ysi . i s∈S Us ⊃ s∈S Us ⊃ s∈S fs−1 (ys ), s ∈ S − S0 Us = Xs , U = s∈S Us , U , U ⊃ U ⊃ f −1 (y), 1.5.1 , f 7.2.2 M . [305] . U = f −1 (f (U )) f (U ) Y. . {Xi}i∈N M , i∈N Xi Xi (i ∈ N), 7.2.2, Yi Xi Yi , i∈N Yi fi . 7.2.5, i∈N fi : i∈N Xi → i∈N Yi ( 4.1.10), 7.2.2 . . ( 2.3.3 2.3.4), Isiwata[203] M M , 7.2.2 . . P ( ), P . . P 1960 , Frolı́k[127] P Čech (Čech-complete space ). X Čech , X Stone-Čech βX Gδ . Čech ( 7.9). 1963 , [79] · 208 · 7 Arhangel’skiı̌[18] Frolı́k () , P p , p ( 7.2.3). 7.2.3 [18] X p (p-space), βX {Un }n∈N (i) Un X; (ii) x ∈ X, n∈N st(x, Un ) ⊂ X; (iii) n∈N st(x, Un ) = n∈N st(x, Un ), X p (strict p-space). T2 X Stone-Čech βX ( 3.6.1), , T2 Čech , Čech p , Čech X = n∈N Un , Un βX, 7.2.3 Un {Un }. 1 7.2.3 Stone-Čech βX αX , {Un }n∈N 7.2.3 αX , 3.6.4, f : βX → αX, f |X . f −1 (Un ) = {f −1 (U ) : U ∈ Un } 7.2.3 βX . 7.2.3 . 2 “U ∧ V ” Un+1 Un (n ∈ N), U1 ∧ · · · ∧ Un Un ( U , V , U ∧ V = {U ∩ V : U ∈ U , V ∈ V }). 7.2.3 [18] {Xi }i∈N p , i∈N Xi p . Xi , {Ui,j }j∈N βXi , 7.2.3 (i), (ii). i∈N Xi i∈N βXi U1 = U1 × βXi : U1 ∈ U1,1 , i>1 U2 = U1 × U2 × βXi : U1 ∈ U1,2 , U2 ∈ U2,2 , i>2 Un = ······ in βXi : Ui ∈ Ui,n , i n . Ui × i>n 1, {Un }n∈N 7.2.3 (i), (ii). 7.2.4 . p . [77] 7.2.4 X p X {Gn }n∈N (i) x ∈ X, Px = n∈N st(x, Gn ) ; (ii) {st(x, Gn )}n∈N Px . 7.2 w∆ M p · 209 · X p , βX {Un }n∈N 7.2.3 (i)∼(iii), 2, Un+1 Un (n ∈ N). st(x, Un ) = st(x, Un ) Px = n∈N n∈N βX X , Px X , Gn = {U ∩ X : U ∈ Un }, , Px = n∈N st(x, Gn ). {st(x, Gn )}n∈N Px . X U ⊃ Px , xn ∈ st(x, Gn ) − U (n ∈ N), {xn } βX , x , x ∈ n∈N st(x, Un ) = Px ⊂ U , xn ∈ U , {st(x, Gn )}n∈N Px . . n ∈ N st(x, Gn ) ⊂ U , , {Gn }n∈N X , (i) (ii), Un = {U : U βX U ∩ X ∈ Gn }, {Un }n∈N βX . , {Un }n∈N 7.2.3 (i). p . 7.2.3 (ii), (iii), X Px = n∈N st(x, Gn ) , βX . y ∈ βX − X, Px ⊂ X, βX − {y} Px . βX , βX O Px ⊂ O ⊂ O ⊂ βX − {y}, (7.2.4) O O βX . (7.2.4), Px ⊂ O ∩ X. , (ii), m ∈ N, O ∩ X X Px ⊂ st(x, Gm ) ⊂ O ∩ X ⊂ O ∩ X, st(x, Gm ) = st(x, Um ) ∩ X, st(x, Um ) ∩ X ⊂ O ∩ X. st(x, Um ) X O , st(x, Um ) − O ⊂ βX − X. st(x, Um ) − O βX , βX X , st(x, Um ) − O = ∅. st(x, Um ) ⊂ O ⊂ βX − {y} ( (7.2.4)). y βX − X , n∈N st(x, Un ) ⊂ X, 7.2.3 (ii). , n ∈ N, (7.2.4) , βX O m ∈ N Px ⊂ st(x, Um ) ⊂ O ⊂ O ⊂ st(x, Un ), st(x, Um ) ⊂ st(x, Un ). (iii). . n∈N st(x, Un ) = (7.2.5) n∈N st(x, Un ), 7.2.3 · 210 · 7 () 7.2.3 (a) p [77] . (b) p w∆ [62] . (a) T1 , Px = {x}, 7.2.4 (i), (ii). (b) x, xn ∈ st(x, Gn ), Gn+1 Gn (n ∈ N) ( 7.2.3 n 2), k=1 st(x, Gn ) = st(x, Gn ) ⊃ {xn , xn+1 , · · ·}, Px = n∈N st(x, Gn ) , {xn } Px . . 7.2.6 [236] X , Y ⊂ X θ , {Un }n∈N X , Un Y . X {Vn }n∈N , Vn Y y ∈ Y , n∈N st(y, Vn ) = n∈N st(y, Vn ) ⊂ st(y, Un ). n∈N U X , U |Y = {U ∩ Y : U ∈ U }. Y θ X , m ∈ N, X {Vm,n }n∈N , Vm,n Y (i), (ii). U1 Y θ {V1,n |Y }n∈N (V1,n X Y ), V1,n (n ∈ N) U1 . U2 ∧ V1,1 Y θ {V2,n |Y }n∈N (V2,n X Y ), V2,n (n ∈ N) V1,1 U1 , U2 . U3 ∧ ( i,j<3 Vi,j ) Y θ {V3,n |Y }n∈N (V3,n X Y ), V3,n (n ∈ N) Vi,j (i, j < 3) Uk (k 3) . , (i) {Vm,n |Y }n∈N Vi,j |Y (i, j < m) Uk |Y (k m) θ ; (ii) V ∈ Vm,n , W ∈ Vi,j (i, j < m) V ⊂ W Uk ∈ Uk (k m) V ⊂ Uk . x ∈ i,j∈N st(y, Vi,j ), y ∈ Y , i, j, m > max{i, j}, n ∈ N y Vm,n , st(y, Vm,n ) = ∪{V : y ∈ V ∈ Vm,n }, (ii), ∪{V : y ∈ V ∈ Vm,n } ⊂ ∪{W : y ∈ W ∈ Vi,j } = st(y, Vi,j ). x ∈ i,j∈N st(y, Vi,j ) ⇒ x ∈ st(y, Vm,n ) ⊂ st(y, Vi,j ), i, j ∈ N, x ∈ st(y, Vi,j ), x ∈ i,j∈N st(y, Vi,j ). i,j∈N st(y, Vi,j ) = i,j∈N st(y, Vi,j ). (ii), i,j∈N st(y, Vi,j ) ⊂ n∈N st(y, Un ), {Vi,j }i,j∈N {Vn }n∈N , . . 7.2.5 [62] θ X, 7.2 w∆ M p (i) X · 211 · p ; (ii) X p ; (iii) X w∆ . (ii) ⇒ (i) (iii) ⇒ (i). (ii) ⇒ (i). X θ p , {Un }n∈N 7.2.3 (i), (ii). 7.2.6 X {Vn }n∈N x ∈ X, n∈N st(x, Vn ) = n∈N st(x, Vn ) ⊂ n∈N st(x, Un ), {Vn }n∈N 7.2.3 (i)∼(iii), X p . (iii) ⇒ (i). {Un }n∈N θ X w∆ , 7.2.6 {Vn }n∈N , 7.2.3 Px = n∈N st(x, Vn ) , {Vn }n∈N 7.2.4 (ii), Px , 7.2.4 (i), X 7.2.4 θ iso ( 6.6.1), p . . (i) X p ; (ii) X T2 w∆ ; (iii) X T2 M . (ii) ⇒ (iii), 7.2.5. {Un }n∈N T2 w∆ X w∆ , X , X {Vn }n∈N Vn+1 Vn ∧ Un (n ∈ N), {Vn }n∈N X M . . 7.2.2 . [18] 7.2.5 T2 X p X . 7.2.2 . 7.2.6 [18] . {Xi }i∈N p , i∈N Xi p 7.2.5 θ p p . p θ ? p (strict p-space problem), Chaber Junnila[88] , Burke[63] 1972 . p ( 7.2.6). , , [236] Kullman “ X X θ p Gδ ” Worrell ( [66] 1 [104] 1.8) “ θ p ”, 7.2.7 7.2.8. . , [104]. , Davis , [213] . 20 25 , · 212 · 7 () ( [168], [222] [71]). , , . 7.2.7 ( [213] ) p θ . 7.2.5, . 7.2.6 X p X θ p . Kullman[236] , . 7.2.7 Gδ . X X p Worrell , . 7.2.8 M p p . , p M [306] [306] [200, 306] [381] [18] [84] θ [66] θ M p , . Neimytzki ( 2.2.3) p , M . M p [166] 3.23. , T2 ( 7.2.4). ( 7.2.2 7.2.6). P . P . Morita P M , Arhangel’skiı̌ P p . p , M . , ( ) . 7.3 (σ Σ ), . 7.3 σ Σ 7.1 7.2 Alexandroﬀ-Urysohn . Bing-Nagata-Smirnov . , . , 7.3 σ Σ · 213 · Bing-Nagata-Smirnov ( 4.3.6 4.3.7) X X (i) σ ; (ii) σ . 3.1 ( 3.1.2). X A , U x ∈ U , A ∈ A x ∈ A ⊂ U . A σ σ σ , X σ σ . 7.3.1 [324] . σ X σ (σ-space), X σ Nagata-Smirnov “” “ ”. , . X , , (closed network). σ Fσ , . ( 6.1.1), 7.3.1 σ [60] , [324] . M p , σ . 7.3.2 [324] , 7.3.3 σ σ ( σ ). ( 7.14). [324] σ σ . X = n∈N Xn , Xn (n ∈ N) σ , σ n n m∈N Am , Am . , n, m ∈ N, n Amn ⊂ Am+1 , n An = Ai × Xi : Ai ∈ Ani , i n . i=1 i>n , An X , X σ . . 7.3.4 [324] , n∈N An X σ , σ σ . ( 7.14). ( 4.1.10), 7.3.2 7.3.3 , 7.3.4 , 4.1.4, . 7.3.1 σ . σ , 7.3.4 σ . . . · 214 · 7 () 7.3.5 (Siwiec-Nagata [362] ) X, (i) X σ ; (ii) X σ ( X σ (iii) X σ ); . (iii) ⇒ (ii) ⇒ (i), (i) ⇒ (iii). A = n∈N An X , An , X , A X , An . An = {Aα : α ∈ Γn }, Fα,m = ∪{A ∈ Am : A ∩ Aα = ∅} (α ∈ Γn ; m ∈ N). Fα,m . {Aα , Fα,m } Hn,m = Hn,m H (7.3.1) , , {{Aα , Fα,m } : α ∈ Γn }. H(Γ ) = (∩{Aα : α ∈ Γ }) ∩ (∩{Fα,m : α ∈ Γn − Γ }), Γ ⊂ Γn . , Hn,m , Hn,m , ( 5.6). (7.3.2) Hn,m {Γλ : λ ∈ Λ}, Γλ ⊂ Γn . x ∈ / ∪{H(Γλ ) : λ ∈ Λ}, λ ∈ Λ, x∈ / H(Γλ ). (7.3.2), x ∈ / ∩{Aα : α ∈ Γλ } x ∈ / ∩{Fα,m : α ∈ Γn − Γλ }. x ∈ / Aαλ , αλ ∈ Γλ ; x ∈ / Fαµ ,m , αµ ∈ Γn − Γλ . Λ = {λ ∈ Λ : x ∈ / ∩{Aα : α ∈ Γλ }}, Λ = {λ ∈ Λ : x ∈ / ∩{Fα,m : α ∈ Γn − Γλ }}. , Λ ∪ Λ = Λ, F1 = Aαλ , λ∈Λ F2 = Fαµ ,m , µ∈Λ F1 , F2 , V = X − (F1 ∪ F2 ), V x ∪{H(Γλ ) : λ ∈ Λ} , ∪{H(Γλ ) : λ ∈ Λ} . Hn,m , . H = n,m∈N Hn,m . x ∈ X x U , A , n ∈ N, α0 ∈ Γn x ∈ Aα0 ⊂ U . Γ = {α ∈ Γn : x ∈ Aα }, x ∈ ∩{Aα : α ∈ Γ } ⊂ Aα0 ⊂ U. (7.3.3) 7.3 σ Σ · 215 · ∪{A ∈ An : x ∈ / A} = ∪{Aα : α ∈ Γn − Γ } x , X − ∪{Aα : α ∈ Γn − Γ } x . A , m ∈ N, α1 ∈ Γm x ∈ Aα1 ⊂ X − ∪{Aα : α ∈ Γn − Γ }. Aα1 ∩ (∪{Aα : α ∈ Γn − Γ }) = ∅. (7.3.1), Aα1 ⊂ Fα,m α ∈ Γn − Γ . x ∈ Aα1 ⊂ ∩{Fα,m : α ∈ Γn − Γ }. (7.3.3) x ∈ (∩{Aα : α ∈ Γ }) ∩ (∩{Fα,m : α ∈ Γn − Γ }) ⊂ Aα0 ⊂ U. (7.3.2), x ∈ H(Γ ) ⊂ U . H X . . 7.3.5 , σ σ , σ . 7.3.2 [362] f σ X Y . X Y , Y σ . X , X σ A, f , f (A ) Y σ , 7.3.5 , Y σ . Y , A X σ , f , f (A ) Y σ , Y , 7.3.5, Y σ . . T2 σ T2 σ ( 7.3.6). . [324] 7.3.1 X, Y T2 σ , X × Y T2 σ , . 7.3.5, n∈N Vn σ X σ , Vn = {Vβ }β∈Bn . U = {Uα }α∈A X × Y . n ∈ N, β ∈ Bn , P (n, β, α) = ∪{P : P Y Vβ × P ⊂ Uα } (α ∈ A), P (n, β) = ∪{P (n, β, α) : α ∈ A}. {P (n, β, α)}α∈A P (n, β) . T2 σ Y , ( 5.3.1), P (n, β) Y Fσ , . Y ( 5.3.1 ) {Hα }α∈A P (n, β) {P (n, β, α)}α∈A . {Vβ }β∈Bn X , {Vβ × Hα }β∈Bn ,α∈A X × Y . H = {Vβ × Hα }β∈Bn ,n∈N,α∈A , H X × Y . · 216 · 7 () (x, y) ∈ X × Y , α ∈ A (x, y) ∈ Uα , X, Y V, U (x, y) ∈ V × U ⊂ Uα , n ∈ N, β ∈ Bn x ∈ Vβ ⊂ V , (x, y) ∈ Vβ × U ⊂ Uα , U ⊂ P (n, β, α), (x, y) ∈ Vβ × P (n, β, α) ⊂ Vβ × P (n, β), {Hα }α∈A P (n, β), α ∈ A (x, y) ∈ Vβ × Hα ∈ H , H X × Y . n ∈ N, Vn = {Vβ }β∈Bn . 5.1.2, X , {Wβ }β∈Bn β ∈ Bn , Wβ ⊃ Vβ . {Wβ × Hα }β∈Bn ,α∈A X × Y . W = {(Wβ × Hα ) ∩ Uα }β∈Bn ,n∈N,α∈A , W X × Y σ , {Uα }α∈A , X × Y ( 5.1.1). σ σ σ ( 7.3.3), X × Y T2 σ , . . K. Morita ( [324]). 7.3.2 n ∈ N, X1 ×· · ·×Xn T2 σ , X = ∞ i=1 Xi T2 σ . X σ ( 7.3.3), X . V X , n∈N Vn Vn in Vi × i>n Xi , Vi Xi . n ∈ N, X Vn in Xi Vn in Xi Gn . in Xi , Gn , Gn Wn Vn . Wn = Wn × Xi : Wn ∈ Wn , i>n , Gn × i>n Xi X , X ( 5.3.1 ). Fσ . W = n∈N Wn X σ , n∈N Vn , X ( 5.1.1). . 7.3.6 [324] {Xi }i∈N T2 σ , i∈N Xi T2 σ . 7.3.1 , n ∈ N, X1 × · · · × Xn T2 σ , . 7.3.2 i∈N Xi T2 σ . M 7.2.2 p 7.2.6 7.3.6 . σ M p . , M p , , σ . Wn Gn × i>n Xi 7.3 σ Σ · 217 · Z 3.1.2 Alexandroﬀ , Y Z C1 , f : Z → Y , f . Y , Y σ . Z T2 , Z σ ( 7.3.13). σ M “ ” ( M p ), “”, Nagami[312] M σ , σ M . , Alexandroﬀ σ , M σ ; 7.4 7.4.1 Michael , σ M ( 7.3.13). σ . σ ( 7.3.2) . σ 7.3.7 {Fα }α∈A X , Fα (α ∈ A) σ , X σ . 5.5.5 P σ 7.3.2 5.5.5 . . , 7.3.5 . 7.3.2 F X , X F (dominated[302] ), X K F ⊂ F , F K, F ∈ F , F ∩ K . X F = {Fα }α∈A , Fα (α ∈ A) P, X P, P (dominated closed sum theorem[357] ). , X () . Okuyama[326] , σ . Burke Lutzer “” , 1991 [253] . Okuyama , , , . [74] 7.3.8 [253] X {Fα }α∈A , Fα (α ∈ A) σ , X σ . , 7.1.3), “” “”. 7.3.3 [81] Šneider ( Gδ . X , {Gn }n∈N X Gδ . X , X U . x0 ∈ X, n0 ∈ N, U X − st(x0 , Gn0 ). , n ∈ N · 218 · 7 Un ⊂ U X − st(x0 , Gn ), (X − st(x0 , Gn )) = X − n∈N () st(x0 , Gn ) = X − {x0 }, n∈N n∈N Un X − {x0 } , U X. α ∈ ω1 , xα ∈ X nα ∈ N, (i) xα ∈ / β<α st(xβ , Gnβ ); (ii) U X − βα st(xβ , Gnβ ). , , α , γ < α, U X− βγ st(xβ , Gnβ ), U X− β<α st(xβ , Gnβ ). , {st(xβ , Gnβ ) : β < α} ∪ U X ( X ), st(xβ , Gnβ ) ( U X). st(xβ , Gnβ ) β γ, U X − βγ st(xβ , Gnβ ), . xα ∈ X − β<α st(xβ , Gnβ ) . , n ∈ N A ⊂ ω1 β ∈ A nβ = n, (i) Gn G {xβ : β ∈ A} , {xβ : β ∈ A} ω , X ( 3.5.2). X . . 7.3.9 (Chaber [81] ) Gδ T2 . 7.1.3 7.3.3 . . Gδ 7.1.2, Gδ G∗δ . 7.3.3 X G∗δ (G∗δ -diagonal[196] ), X {Un }n∈N x, y ∈ X, x = y, n ∈ N y ∈ / st(x, Un ) ( , x ∈ X, {x} = n∈N st(x, Un )). G∗δ (G∗δ -diagonal sequence). , G∗δ T2 . 7.2.6 Gδ θ G∗δ . , Gδ T2 , σ G∗δ [260] . 7.3.10 [196] G∗δ w∆ . {Hi }i∈N , {Ki }i∈N X w∆ G∗δ . n Gn = G : G = (Hi ∩ Ki ), Hi ∈ Hi , Ki ∈ Ki , i n (n ∈ N), i=1 {Gn }n∈N w∆ G∗δ , Gn+1 Gn (n ∈ N). 7.3 σ Σ · 219 · x ∈ X xn ∈ st(x, Gn ) (n ∈ N), {xn } p, {st(x, Gn )}n∈N , p ∈ n∈N st(x, Gn ). {Gn }n∈N G∗δ , n∈N st(x, Gn ) = {x}, p = x, X ( 7.5 7.2.1 ). . Gδ w∆ ?[196] w∆ (w∆-space problem). 1988 , Alster, Burke Davis[10] CH . 7.3.9 M . 7.3.11 [74] X X T2 , M Gδ . , . X T2 , M Gδ , 7.2.1, Y X Y f . y ∈ Y, f −1 (y) , Gδ , 7.3.3, f −1 (y) . f , 7.2.2, X . X T2 , X , 7.3.3 , X G∗δ . 7.3.10, X . X ( 7.1.4). . 7.3.12 σ [77] , σ G∗δ . X , {Gn }n∈N . , Gn σ Fn , F = n∈N Fn X σ , X σ . σ σ , , Gδ (T2 , 2.7). σ , G∗δ ( 7.3.3 ). . 7.3.4 [353] σ . X σ , ( 6.6.13 1), X , X Lindelöf , X . . 7.3.13 [353] X X T2 , M σ . . X T2 , M σ , 7.3.4 7.2.1 7.2.2, X , X . 7.3.11 7.3.12 . . 7.3.13, T2 σ . , Gδ X , T1 σ . , ( 2.3.1), X T2 , , X T1 , X Gδ ( 7.34). M σ “ ” , , “ ”. · 220 · 7 () M ( 7.2.2), , σ ( 7.3.6), σ M ( p ) “”, . Frolı́k[127] “ P , P f f −1 ({P}) ”. σ , f −1 ({σ})[312] , f −1 ({σ}) ; σ , , −1 f ({σ}) . M ( p ) , σ , M ( p ) f −1 ({σ}) , P f −1 ({σ}) , P Frolı́k, Morita, Arhangel’skiı̌, Okuyama . Nagami[312] ( 7.16). 7.3.4 [312] X Σ (Σ-space) ( Σ (strong Σ-space)), σ F () C X U C C ⊂ U , F ∈ F C ⊂ F ⊂ U . 7.3.4, σ Σ , f Σ . Σ f −1 ({σ}) . , Nagami[312] Σ σ , f −1 ({σ}) . Σ , σ Σ , iso ( 6.6.1) iso Σ Σ . , 7.3.14 [59] T2 X Σ X Σ . , iso ( 6.6.1). T2 Σ , . F = n∈N Fn Σ X σ , C X () , 7.3.4. U X , x ∈ X, Cx ∈ C x ∈ Cx . 6.2.11 6.2.1, X Ox , Cx ⊂ Ox U |Ox = {U ∩ Ox : U ∈ U } Ox Hx , Fx ∈ F , Cx ⊂ Fx ⊂ Ox . H = {Fx ∩ H : x ∈ X, H ∈ Hx }. H U σ . x ∈ X H ∈ Hx , z ∈ X − (Fx ∩ H), X − Fx , z ∈ X − Fx , Lz = Ox − H, z ∈ Fx − H. Lx z Lz ∩ Fx ∩ H = ∅, Fx ∩ H X . U σ . X . . 7.3 σ Σ · 221 · 7.3.15 [312] Σ ( Σ ) Σ ( Σ ). , . 7.3.5 [324] f : X → Y X Y , {Uα }α∈A X , {f (Uα )}α∈A Y . {Uα }α∈A X , {U α }α∈A , f , {f (U α )}α∈A Y . {f (U α )}α∈A . , y f (U α ), α ∈ A ⊂ A, A A , α ∈ A , xα ∈ f −1 (y) xα ∈ U α , xα ∈ f −1 (y) ∩ U α . {U α }α ∈A , , xα . {U α }α ∈A , {{xα } : α ∈ A } . {xα : α ∈ A } ⊂ f −1 (y), f −1 (y) (, 6.6.13 1). {f (U α )}α∈A , ( 5.6). , {f (Uα )}α∈A . . 7.3.16 [312] f : X → Y Σ ( Σ ) X Y () , Y Σ ( Σ ). X Σ ( Σ ). F = n∈N Fn X σ , Fn , C X () , Σ ( Σ ) ( 7.3.4). 7.3.5, n ∈ N, f (Fn ) = {f (F ) : F ∈ Fn } Y . , f (C ) = {f (C) : C ∈ C } Y () . , n∈N f (Fn ) f (C ) Σ ( Σ ) . . 7.3.3 [312] {Fα }α∈A X , Fα (α ∈ A) X Σ ( Σ ), X Σ ( Σ ). 5.5.3 Σ ( Σ ) , 7.3.16 . . 7.3.17 [312] f : X → Y X () , X Σ ( Σ ). Σ ( Σ ) Y Y Σ ( Σ ), F Y σ , C Y () , Σ ( Σ ) . f −1 (F ) = {f −1 (F ) : F ∈ F } X σ , f −1 (C ) = {f −1 (C) : C ∈ C } X () ( 3.31 3.30). C ∈ C X U ⊃ f −1 (C), f , 1.5.1, X V U ⊃ V ⊃ f −1 (C), f (V ) Y V = f −1 (f (V )). f (V ) ⊃ C, · 222 · 7 () F ∈ F f (V ) ⊃ F ⊃ C, U ⊃ V = f −1 (f (V )) ⊃ f −1 (F ) ⊃ f −1 (C), X Σ ( Σ ). 7.3.4 [312] f −1 (F ) ∈ f −1 (F ). . M Σ , M Σ . M , Σ , . . 7.3.5 [312] Σ Σ . Σ X Y X Σ . . ( 3.3.1), 7.3.18 [312] Σ Σ . X = i∈N Xi , Xi Σ , F i Xi σ , C i Xi , Σ . i ∈ N, i F i = n∈N Fni , Fni , Fni ⊂ Fn+1 (n ∈ N). Fn = Xi : F i ∈ Fni , i n , Fi × in i>n Fn X , F = n∈N Fn X σ , ( 7.2.4) F C = i∈N C i Σ , X Σ . . 7.3.19 [312] {Xi }i∈N T2 Σ , X = i∈N Xi T2 Σ . 7.3.14, Xi Σ , 7.3.18 . U X = i∈N Xi , Fni Xi , Xi T2 , Xi Vi,n , F i ∈ Fni V (F i ) ∈ Vi,n , F i ⊂ V (F i ) Vi,n ⊂ Vi,n+1 (n ∈ N) ( 4.4.1 ). Vn = Xi : V (F i ) ∈ Vi,n , i n . V (F i ) × in Vn X . i>n Fn A ( in F i × i>n Xi ) U U (A) ( X Σ ( 7.3.18), C () U , Σ A , 7.3 σ Σ · 223 · A ∈ F X), U (A) = {Uj (A) : j n(A)}, n(A) . Vn B ( in V (F i ) × i>n Xi ), B(A). Wj (A) = Uj (A) ∩ B(A), A ∈ Fn , j n(A); Wn,j = {Wj (A) : A ∈ Fn , A U }. Wn,j , W = n,j∈N Wn,j X σ , U . X , X . . X {Un }n∈N (reﬁned sequence), Un+1 Un (n ∈ N). 7.3.6 X G∗δ X . X , G∗δ {Un }n∈N , 7.3.3, {x} = n∈N st(x, Un ), x ∈ X . {st(x, Un )}n∈N x , ( 4.4.1). x ∈ X x U , {U } ∪ {X − st(x, Un ) : n ∈ N} X, X , U X − st(x, Un ) X, Un n0 , U ∪ (X − st(x, Un0 )) = X, st(x, Un0 ) ⊂ U . . 7.3.20 [74] . X σ X Gδ Σ . σ Σ Gδ ( 7.3.12). . X Σ , F = l∈N Fl X σ , Fl , C X , Σ . {Un }n∈N X Gδ , 7.3.3, C () , X Σ . T2 Σ ( 7.3.14), {Un }n∈N G∗δ ( 7.3.3 ), x ∈ X, {x} = n∈N st(x, Un ). X , n ∈ N, Hn = m∈N Hn,m X σ , Un , Hn,m . K (n, m, l) = {H ∩ F : H ∈ Hn,m , F ∈ Fl }, K (n, m, l) . K = ∪{K (n, m, l) : m, n, l ∈ N} σ , X σ . , K X x ∈ X x U , C ∈ C x ∈ C. 7.3.6 C ( ), n ∈ N C ⊂ U ∪ (X − st(x, Un )), (C − U ) ∩ st(x, Un ) = ∅. · 224 · 7 () y ∈ C − U , y Vy Vy ∩ st(x, Un ) = ∅. V = ∪{Vy : y ∈ C − U }, C ⊂ U ∪ V , F ∈ Fl C ⊂ F ⊂ U ∪ V . , Hn X , x H ∈ Hn,m . Hn Un , H Un , x ∈ H ⊂ st(x, Un ). x ∈ H ∩ F ∈ K (n, m, l). F ⊂ U ∪ V V ∩ st(x, Un ) = ∅, F ∩ st(x, Un ) ⊂ U ∩ st(x, Un ) ⊂ U , H ⊂ st(x, Un ), H ∩ F ⊂ st(x, Un ) ∩ F ⊂ U . K X . . Michael[286] Σ Σ Σ Σ . , 7.3.5 σ , σ Σ Σ , Michael 7.3.4 “” “” Σ (Σ -space) Σ (strong Σ -space), , 7.3.14[217] ( 7.18) 7.3.5[327] 7.3.18 7.3.19[331] 7.3.20[217] , Σ . 7.3.4 “” “ ” , , Okuyama[327] Σ∗ (Σ∗ -space) Σ∗ (strong Σ∗ -space), X Σ∗ Σ ( Michael[286] ), X Σ∗ , 7.3.5 , Σ∗ . Michael[286] , Σ Σ∗ . , σ . Lašnev[240] “ f : X → Y X Y , Y Y = Y0 ∪ ( n∈N Yn ), Yn ; y ∈ Y0 , f −1 (y) .” Lašnev (Lašnev’s decomposition theorem). σ . . x ∈ X, X N x X (network of a point in a space), x ∈ ∩N , X x U , N ∈ N N ⊂ U . 7.3.21 [86] f σ X Y , Y Y = Y0 ∪ ( n∈N Yn ), Yn (n ∈ N) ; y ∈ Y0 , f −1 (y) . n∈N Pn X σ , Pn Pn ⊂ Pn+1 (n ∈ N). n ∈ N, Fn = {f (P ) : P ∈ Pn }, Fn (y) = ∩{F ∈ Fn : y ∈ F }. f , Fn y ∈ Y , , {Fn (y) : y ∈ Y } . , A ⊂ Y , z∈ / ∪{Fn (y) : y ∈ A}, V = Y − ∪{F ∈ Fn : z ∈ / F }, V z 7.3 σ Σ · 225 · V ∩ Fn (y) = ∅, y ∈ A. Yn = {y ∈ Y : Fn (y) = {y}}, Yn {Fn (y) : y ∈ Y } , Yn . {y} = Fn (y) , , Y0 = Y − n∈N Yn , y ∈ Y0 , f −1 (y) . , f −1 (y) Lindelöf . y ∈ Y0 , (1) {Fn (y)}n∈N y Y . y ∈ Y0 , Fn (y) , Y T1 , Fn (y) . Y {yn } yn ∈ Fn (y) − {y}. (1), {yn } y. Fn (y) (2) P ∈ Pm P ∩ f −1 (y) = ∅ ⇒ n m, P ∩ f −1 (yn ) = ∅. (3) n ∈ N, {P ∈ Pn : P ∩ f −1 (y) = ∅} . (3) , m ∈ N Pm {Pn }n∈N Pn ∩f −1 (y) = ∅. (2), n m, Pn ∩ f −1 (yn ) = ∅, xn ∈ Pn ∩ f −1 (yn ). Pm , {xn : n ∈ N} , f , {yn : n ∈ N} . {yn } y . (3) . , f −1 (y) , f −1 (y) Lindelöf . f −1 (y) . K = f −1 (y). K X , (4) X {Uk }k∈N K K ∩ (Uk+1 − U k ) = ∅(k ∈ N). , K X ( 7.3.4), X V , V K, V K. x ∈ K, Vx ∈ V , X Wx , x ∈ Wx ⊂ W x ⊂ Vx . K Lindelöf , K {Wx }x∈K {Wi }i∈N , {W i }i∈N K. x1 ∈ K ∩ W1 , U1 = W1 . x2 ∈ K − U 1 , n1 ∈ N, x2 ∈ Wn1 . U2 = in1 Wi , U1 ⊂ U2 , x2 ∈ K ∩ (U2 − U 1 ). , X {Uk }k∈N (4). , xk ∈ K ∩ (Uk+1 − U k )(k ∈ N). xk ∈ X − U k , n∈N Pn , Pk ∈ Pnk xk ∈ Pk ⊂ X − U k , nk nk > nk−1 . xk ∈ Pk ∩ K (2), Pk ∩ f −1 (ynk ) = ∅, zk ∈ Pk ∩ f −1 (ynk ). {f (zk )} {yn } , y, {zk } z ∈ K, z ∈ Uk0 . , zk , zk ∈ Pk , Pk ∩ U k = ∅, zk ∈ / Uk ⊃ Uk0 , k > k0 . K = f −1 (y) . . Chaber[86] “ σ ” “ T2 , σ ” . · 226 · 7 7.4 Mi () Bing-Nagata-Smirnov ( 4.3.6 4.3.7) . “ X X (i) X σ ; (ii) X σ .” 1950 1951 , Michael 1953 ( 5.1.1 5.1.2) “ X X (i ) X σ ; (ii ) X σ .” , Michael , 1957 ( 5.1.4) ( 5.1.2), “ X X (iii ) X σ .” Bing-Nagata-Smirnov , “ σ ” “ σ ” “ σ ” ( 7.4.1), σ “” “ ” ( . Michael 1959 5.1.3), ( 5.1.6) “ T1 X X (iv ) X σ .” Michael ((iii ), (iv )) Ceder 1961 ( 7.4.1), M1 M3 ( 7.4.2) “ X M1 X σ ; T1 X M3 X σ .” . . Mi (i = 1, 2, 3) . 7.4.1 [80] X B X (quasi-base), x ∈ X x U , B ∈ B x ∈ B ◦ ⊂ B ⊂ U . (P1 , P2 ) P = {(P1 , P2 )}, P1 P1 ⊂ P2 , P X (pair-base), x ∈ X x U , (P1 , P2 ) ∈ P x ∈ P1 ⊂ P2 ⊂ U . 7.4 Mi · 227 · P (cushioned), P ⊂ P, ∪{P1 : (P1 , P2 ) ∈ P } ⊂ ∪{P2 : (P1 , P2 ) ∈ P }. , , (P1 , P2 ) (B ◦ , B). 7.4.2 [80] X M1 (M1 -space), X σ ; M2 (M2 -space), X σ . T1 X M3 (M3 -space), X σ . 7.4.1 [80] M1 ⇒ M2 ⇒ M3 ⇒ . , M1 ⇒ M2 . (1) M2 ⇒ M3 . n∈N Bn X σ , Bn . n ∈ N, Pn = {(B ◦ , B) : B ∈ Bn }, P ⊂ Pn , ∪{B ◦ : (B ◦ , B) ∈ P } ⊂ ∪{B : (B ◦ , B) ∈ P } = ∪{B : (B ◦ , B) ∈ P }. , n∈N Pn X σ . M3 ⇒ . X M3 . M3 ( 7.4.2) X , ( 5.1.4) M3 , ( 5.1.2). n∈N Pn X σ , Pn . G X , n ∈ N, (2) Fn = ∪{P1 : P2 ⊂ G, (P1 , P2 ) ∈ Pn }, Fn ⊂ ∪{P2 : P2 ⊂ G, (P1 , P2 ) ∈ Pn } ⊂ G. Fσ , X . . n∈N Pn , G = n∈N Fn M1 , M1 ( 7.4.1). 7.4.1 Michael [280] . N () Stone-Čech βN N ∪ {x} = X, x ∈ βN − N. N ∪ {x} Michael . Michael X , ( 3.6.2), . x , {n} (n ∈ N) X σ , X M1 . σ , , ? J. Nagata M1 , ( [80] 9.2). M3 . Sorgenfrey ( 2.3.3) ( 7.20). , · 228 · 7 () M2 ⇒ M1 M3 ⇒ M2 ( Mi ), Ceder[80] . Gruenhage[163] Junnila[216] M3 ⇒ M2 . . . M2 ⇒ M1 M3 g . 7.4.1 [319] T1 X M3 H U ⊂ X n ∈ N H(U, n), (i) U = n∈N H(U, n) = n∈N H(U, n)◦ ; (ii) U, V, U ⊂ V ⇒ H(U, n) ⊂ H(V, n)(n ∈ N). X M3 , P = n∈N Pn σ , Pn . U n ∈ N, H(U, n) = ∪{P1 : (P1 , P2 ) ∈ Pn , P2 ⊂ U }, (ii). Pn , H(U, n) ⊂ ∪{P2 : (P1 , P2 ) ∈ Pn , P2 ⊂ U } ⊂ U. x ∈ U , P , (P1 , P2 ) ∈ Pn x ∈ P1 ⊂ P2 ⊂ U , x ∈ , U = n∈N H(U, n) = P1 ⊂ H(U, n), P1 , x ∈ P1 ⊂ H(U, n)◦ . ◦ n∈N H(U, n) , (i). , X H, (i), (ii). n ∈ N, Pn = {(H(U, n)◦ , U ) : U ∈ τ }, τ X . U x ∈ U , (i), m ∈ N x ∈ H(U, m)◦ ⊂ U , P . Pn . τ ⊂ τ . V = ∪{U : U ∈ τ }, U ⊂ V ⇒ H(U, n) ⊂ H(V, n), ∪{H(U, n)◦ : U ∈ τ } ⊂ H(V, n) ⊂ V = ∪{U : U ∈ τ }. Pn . . 1 H X (stratiﬁcation[373] ), T1 (stratiﬁable space). M3 . H , (iii) H(U, n + 1) ⊃ H(U, n)(n ∈ N). H (U, n) = in H(U, i) H(U, n), (i), (ii). H, X U , Un = H(U, n)◦ , n ∈ N, U → {Un }. Borges[54] . , 7.4 Mi · 229 · 2 H [191] G F ⊂ X n ∈ N G(F, n), (i ) F = n∈N G(F, n) = n∈N G(F, n); (ii ) F, K, F ⊂ K ⇒ G(F, n) ⊂ G(K, n)(n ∈ N); (iii ) G(F, n + 1) ⊂ G(F, n)(n ∈ N). G. , , H U → {Un } ( 1), G ( 2). 3 H ( G) (i) ( (i )) U= H(U, n) n∈N F= G(F, n) , n∈N H ( G) (semi-stratiﬁcation), (semi-stratiﬁable space[98, 99] ). Henry[191] . (X, τ ) , g : N × X → τ N × X g [183] (g-function), n ∈ N x ∈ X, (i) x ∈ g(n, x); (ii) g(n + 1, x) ⊂ g(n, x). , g g . g(n, x) N × X g , A ⊂ X, g(n, A) = ∪{g(n, x) : x ∈ A}. 7.4.2 [186] T1 X M3 () N × X / F , n ∈ N y ∈ / g(n, F ). g y ∈ X , G ( 7.4.1 2). n ∈ N x ∈ X, g(n, x) = G({x}, n), g(n, x) g . y ∈ / F . (i ), n ∈ N y ∈ / G(F, n). g(n, x) G (ii ), G(F, n) ⊃ x∈F G({x}, n), y ∈ / g(n, F ). , g(n, x) N × X g . X F n ∈ N, G(F, n) = / F , x∈F g(n, x). , G(F, n) (ii ) n∈N G(F, n) ⊃ F . y ∈ n∈Ny∈ / g(n, F ) = G(F, n). F = n∈N G(F, n), (i ). G X , X . . . 7.4.3 [99] (i) X ; / F , n ∈ N y ∈ / g(n, F ); (ii) N × X g y ∈ · 230 · 7 () (iii) N × X g x ∈ X {xn }, x ∈ g(n, xn ), xn → x. 7.4.2 (i) ⇔ (ii), (ii) ⇔ (iii). (ii) ⇒ (iii). N × X g (ii). x ∈ X {xn }, x ∈ g(n, xn ), U x , x ∈ / X − U , n ∈ N x∈ / g(n, X − U ), k n , x ∈ / g(k, X − U ), xk ∈ / X − U , xk ∈ U . xn → x. (iii) ⇒ (ii). N × X g (iii). y ∈ / F , n ∈ N y ∈ g(n, F ), xn ∈ F y ∈ g(n, xn ), xn → y ∈ F , . 7.4.4 [318] . T1 X M2 N × X g (i) y ∈ / F , n ∈ N y ∈ / g(n, F ); (ii) y ∈ g(n, x) ⇒ g(n, y) ⊂ g(n, x). X M2 , B = n∈N Bn X σ , Bn . X , B . g(n, x) = X − ∪{B : B ∈ Bi , i n, x ∈ / B}, (7.4.1) g(n, x) N × X g . g(n, x) (ii). y ∈ / F , y ∈ X − F . ◦ B X , B ∈ Bn y ∈ B ⊂ B ⊂ X − F , B ∩ F = ∅. (7.4.1), B ◦ ∩ g(n, x) = ∅ x ∈ F . g(n, x) (i). , g (i), (ii). (i), X . Gn = {g(n, x) : x ∈ X}, Bn = {X − ∪Gn : Gn ⊂ Gn }, Bn . B= n∈N X U x ∈ U , x ∈ / X − U . (i), n ∈ N x ∈ / g(n, X − U ). Gn = {g(n, y) : y ∈ X − U } ⊂ Gn . x∈ / ∪Gn ⇒ x ∈ X − ∪Gn = (X − ∪Gn )◦ ⊂ X − ∪Gn ⊂ U, X − ∪Gn ∈ Bn ⊂ B, B . Bn . Bn , {∪Gn : Gn ⊂ Gn } . α ∈ A, Gn (α) ⊂ Gn , {∪Gn (α) : α ∈ A} α∈A (∪Gn (α)) . y ∈ α∈A (∪Gn (α)). α ∈ A, y ∈ ∪Gn (α) ⇒ y ∈ Gn (α) g(n, x). 7.4 Mi · 231 · (ii), g(n, y) ⊂ g(n, x) ⊂ ∪Gn (α). g(n, y) ⊂ . α∈A (∪Gn (α)), α∈A (∪Gn (α)) . σ 7.4.5 [187] ( 7.4.4 (i) ). X σ N × X g (i) y ∈ / F , n ∈ N y ∈ / g(n, F ); (ii) y ∈ g(n, x) ⇒ g(n, y) ⊂ g(n, x). σ σ 7.4.4. . ( 7.3.5), 7.4.2 7.4.4 M3 , M2 , “(ii) y ∈ g(n, x) ⇒ g(n, y) ⊂ g(n, x)”. M3 M2 , M3 g , (i) (ii). 7.4.4 {∪Gn : Gn ⊂ Gn } , Gn (). , g(n, x) “ ”, . (X, τ ) N : X → τ x ∈ X N (x). x N 2 (x) = ∪{N (y) : y ∈ N (x)}, N 3 (x) = ∪{N (z) : z ∈ N 2 (x)}. 7.4.2 [216] (X, τ ) , N : X → τ x ∈ X x N (x), X V , x ∈ X, ∩{V ∈ V : x ∈ V } ⊂ N 3 (x). X , 7.4.3, g x ∈ X {xn }, g(1, x) ⊂ N (x). k ∈ N, . x ∈ g(n, xn ) ⇒ xn → x. (7.4.2) Gk = {g(k, x) : x ∈ X}, Qk Gk Hk = {x ∈ X : x ∈ g(k, y) ⇒ y ∈ N (x)}. (7.4.3) Hk ⊂ Hk+1 , k ∈ N. (7.4.2) x ∈ X, x ∈ Hk ( , k ∈ N, yk ∈ X x ∈ g(k, yk ) yk ∈ / N (x), (7.4.1), yk → x, ). k(x) x ∈ H n n. Qn (x) = ∩{Q ∈ Qi : i n, x ∈ Q} (n ∈ N); V (x) = Qk(x) (x) − ∪{H i : i < k(x)}. · 232 · 7 () V = {V (x) : x ∈ X} X . x ∈ X, y, z ∈ X ∩{V ∈ V : x ∈ V } ⊂ N (y), y ∈ N (z) z ∈ N (x) . m = k(x). x ∈ H m , x g(m, x) ∩ Qm (x) Hm . z ∈ Hm ∩ g(m, x) ∩ Qm (x). z ∈ g(m, x) ⊂ N (x). Qm Gm , y ∈ X Qm (x) ⊂ g(m, y) ⊂ N (y), ∩{V ∈ V : x ∈ V } ⊂ V (x) ⊂ Qm (x) ⊂ N (y). , y ∈ N (z). z ∈ Qm (x) ⊂ g(m, y), z ∈ Hm , (7.4.3), y ∈ N (z). . 7.4.6 (Gruenhage-Junnila [163, 216] ) M3 M2 . X M3 (), 7.4.4 X M2 . 7.4.2, N × X g 7.4.4 (i), y ∈ / F , n∈Ny∈ / g(n, F ). n ∈ N, g 7.4.2 N (x), g 2 (n, x) = g(n, g(n, x)), g 3 (n, x) = g(n, g 2 (n, x)) = g(n, g(n, g(n, x))). g 3 (n, x) N × X g . g 3 (n, x) (i). g(n, x) (i), y ∈ / F , k ∈ N y ∈ / g(k, F ), m ∈ N, m k y ∈ / g(m, g(k, F )), n ∈ N, n m y∈ / g(n, g(m, g(k, F ))) ⊃ g(n, g(n, g(n, F ))) = g 3 (n, F ). g 3 (n, x) (i). 7.4.2, n ∈ N, X Vn ∩{V ∈ Vn : x ∈ V } ⊂ g 3 (n, x) (x ∈ X). (7.4.4) g (n, x) = ∩{V ∈ Vi : i n, x ∈ V }. (7.4.5) g (n, x) g . (7.4.4), g (n, x) ⊂ g 3 (n, x), g (n, x) (i). (7.4.5), y ∈ g (n, x) = ∩{V ∈ Vi : i n, x ∈ V }, ∩{V ∈ Vi : i n, y ∈ V } ⊂ ∩{V ∈ Vi : i n, x ∈ V }, g (n, y) ⊂ g (n, x). g (n, x) 7.4.4 (ii). X M2 . . 7.4 Mi · 233 · M3 ⇒ M2 , . [186] 7.4.7 M3 () σ . 7.4.4 7.4.5 7.4.6 . . M3 σ . Heath[186] 1969 , , Heath Hodel[187] σ (∗) “ X σ N × X g X x {xn }, {yn }, x ∈ g(n, xn ) xn ∈ g(n, yn ), yn → x.” (∗) M3 σ ( , 7.21). (∗) , , 7.5 ( 7.5.1), . Mi . , σ ( 7.3.3), . 7.4.8 [80] Mi Mi (i = 1, 2, 3) . ( 7.4.8) Mi (i = 1, 2, 3) . 7.4.9 7.4.10 7.4.1 M2 ( M3 ) . M2 , M3 . 7.4.9 [80] M2 ( M3 ) . M2 . X M2 , B X σ , B X . A ⊂ X, , B|A = {B ∩ A : B ∈ B} A σ , A M2 . . M3 () , . 7.4.3 [54] X . X (A, U ), A , U , UA ⊂ U (i) A, B, A ⊂ B; U, V, U ⊂ V ; UA , VB UA ⊂ VB ; (ii) A ∩ U ⊂ UA ⊂ U A ⊂ A ∪ U ; (ii ) A ⊂ U (A, U ), , A ⊂ UA ⊂ U A ⊂ U ((ii) ). X U → {Un } ( 7.4.1 UA = 1). {Un − (X − A)n }, n∈N (i). A ∩ U ⊂ UA . x ∈ A ∩ U , x ∈ U , k ∈ N x ∈ Uk , x ∈ A, x ∈ / X − A, x ∈ Uk − (X − A)k ⊂ UA ( , (X − A)k ⊂ X − A). UA ⊂ A ∪ U. x ∈ / A ∪ U, x ∈ / A, n ∈ N x ∈ (X − A)n , (X − A)n ∩ (X − U n ) x UA , (ii). (ii ) (ii) . . · 234 · 7 () 7.4.10 [54] . f : X → Y X Y , U → {Un } X . , Y T1 . Y V , ( 7.4.3 ) Tn = f −1 (V )n , Sn = f −1 (f (T n )), Qn = f −1 (V )Sn , Vn = f (Qn )◦ , (a) f (T n ) ⊂ Vn . 7.4.3 (ii ) , Qn Sn , f , 1.5.1, f (Qn ) f (T n ) , f (T n ) ⊂ Vn . (b) V n ⊂ V . V n ⊂ f (Qn ) = f (Qn ), 7.4.3 (ii ) , Qn ⊂ f −1 (V ), f (Qn ) ⊂ V , Vn ⊂V. (c) n∈N Vn = V . (b) (a), V ⊃ Vn ⊃ f (Tn ), V ⊃ Vn ⊃ n∈N Tn = V, f (Tn ) = f n∈N n∈N n∈N Vn = V . V, W X , V ⊂ W . X , Tn Sn , Vn , , Qn 7.4.3 (i) Vn ⊂ Wn (n ∈ N). , V → {Vn } Y , Y . . Ceder[80] M3 , 5.5.5 7.4.10, . 7.4.1 [356] . Borges[54] . 7.4.9 7.4.10 M1 ? M1 . 7.4.4 M1 ? M1 ? M3 ⇒ M1 ? M1 , , M1 M1 . , . 7.4.1 [204] M1 M1 . M1 X M1 X M1 ( M1 , ⇒ ). 7.4 Mi · 235 · D M1 X , D = X. X B B ∩ D = B ( 1.18). B = n∈N Bn M1 X σ , Bn . Bn |D = {B ∩ D : B ∈ Bn }. B ∩ D = B, Bn |D D . ∪{Bn |D : n ∈ N} D σ , D M1 . M1 X M1 , A X , A M1 , A M1 . . M1 , 7.4.11 7.4.9 ( 5.2.1, 5.5.7 ). 7.4.4 [133] f : X → Y , B X , C = {f (B)◦ : B ∈ B} Y . U U ∗ = ∪{U : U ∈ U }. B ⊂ B y ∈ ∪{f (B)◦ : B ∈ B }, f (B) ⊃ f (B)◦ , f (B ∗ ) ⊃ f (B ∗ ) ⊃ ∪{f (B)◦ : B ∈ B }. f , f (B ∗ ) , f (B ∗ ) ⊃ ∪{f (B)◦ : B ∈ B } y, f −1 (y)∩B∗ = ∅. B , B ∈ B f −1 (y)∩B = ∅. U y , f −1 (U )∩B = ∅. f , f −1 (U )∩B , f (f −1 (U ) ∩ B)◦ = (U ∩ f (B))◦ = U ∩ f (B)◦ , U ∩ f (B)◦ . y U f (B)◦ , y ∈ f (B)◦ . C . . 7.4.11 [133] M1 . f : X → Y M1 X Y . B = n∈N Bn M1 X σ , Bn . , U , U . Bn Bn . , C = {f (B)◦ : B ∈ B}, 7.4.4, C , Bn ⊂ Bn+1 , n ∈ N. Y σ . y ∈ Y , V y , B X , B ⊂ B f −1 (y) ⊂ B ∗ ⊂ f −1 (V ). n ∈ N, Bn = B ∩ Bn , B = Bn , n∈N f −1 (y) ⊂ Bn∗ ⊂ f −1 (V ). n∈N · 236 · 7 () Bn ⊂ Bn+1 (n ∈ N), {Bn∗ } . f , ∗ ◦ ∗ m y ∈ f (Bm ) ⊂ V . B ∈ Bm ⊂ B, B = Bm , f (B)◦ ∈ C y ∈ f (B)◦ ⊂ V . C Y σ . , Y , Y M1 . . 6.6 . f : X → Y , f X Y . X U y ∈ Y , y ∈ Y U ⊃ f −1 (y). k ∈ N, f : X → Y k (k-to-one), y ∈ Y , f −1 (y) X k . 7.4.2 M1 (i) [56] ; (ii) [133] ; (iii) k [162] . ⇒ ⇒ , (i). ( 7.23), 7.4.11 , 7.4.11 (ii). T2 k (iii). . ( 7.24), (ii) ( ) , . X , Y , f X × Y Y , f ( 3.16). , M1 ( 4.4.1). 7.4.3 [133, 204] M1 . 7.4.11 5.5.8 . . Heath-Junnila ( 7.4.12). M3 ⇒ M1 M1 . 7.4.5 X M1 X σ . B M1 X σ , B = {B : B ∈ B} X σ . , B X σ , B ◦ = {B ◦ : B ∈ B} X σ . . 7.4.6 [388] X σ . F X , X σ D, F ∈ F , F ∩ D F . 7.4.5 7.4.3, σ . X G ( 7.4.1 3) (i) F , F = n∈N G(F, n), G(F, n) ; 7.4 Mi · 237 · (ii) F, K, F ⊂ K ⇒ G(F, n) ⊂ G(K, n)(n ∈ N). Q(F ) = ∩F − ∪(F − F ), F ⊂ F, Q(∅) = X − ∪F . {Q(F ) : F ⊂ F } X . n ∈ N, Qn (F ) = ∩F − G(∪(F − F ), n), Qn (∅) = X − G(∪F , n). Q(F ) = X . , x ∈ X, (F )x = {F ∈ F : x ∈ F }, F ⊂ F, n∈N Qn (F ) {Qn (F ) : F ⊂ F } U = G({x}, n) − ∪(F − (F )x ), U x ∈ U . (F )x = F ⊂ F , F ∈ F − (F )x , Qn (F ) ⊂ F U ⊂ X − ∪(F − (F )x ) ⊂ X − F, Qn (F ) ∩ U = ∅; F ∈ (F )x − F , Qn (F ) ⊂ X − G(∪(F − F ), n) ⊂ X − G(F, n) ⊂ X − G({x}, n), Qn (F ) ∩ U = ∅. Q = {Qn (F ) : F ⊂ F , n ∈ N}, Q {Q(F ) : F ⊂ F } σ . H X σ H ∈ H , Q ∈ Q, H ∩ Q = ∅, x(H, Q) ∈ H ∩ Q. . D = {x(H, Q) : H ∈ H , Q ∈ Q, H ∩ Q = ∅}, D . , D X σ . F ∈ F , F W , X V , V ∩ F = W . y ∈ W . H ∈ H , Q ∈ Q, y ∈ H ∩Q ⊂ H ⊂ V. Q = Qn (F ), n ∈ N, F ⊂ F . Q ∩ F = ∅, F ∈ F , Q ⊂ F , x(H, Q) ∈ H ∩ Q ⊂ V ∩ F = W , F ∩ D F . , D X . . 7.4.12 (Heath-Junnila [188] ) M3 M1 . X M3 , S = {0} ∪ {1/n : n ∈ N} . Z = X × S, X × (S − {0}) Z ; X × {0} · 238 · 7 () X × S . , Z , X Z X × {0}. Z M1 . B ⊂ X, B[0] = B × {0}, B[n] = B × ({0} ∪ {1/k : k n}) (n ∈ N). B X , B[n] = B[n] − B[0], B[n] Z . 7.4.6, X B = n∈N Bn , Bn . Unm = {B[m] : B ∈ Bn } (n, m ∈ N), Vn = {{(x, 1/n)} : x ∈ X} (n ∈ N), Unm Z , Vn Z , Unm ∪ n,m∈N Vn n∈N Z σ . 7.4.5, Z M1 . Z Y , Y M1 Y X . 7.4.6, X D = n∈N Dn , Dn X Dn ⊂ Dn+1 , B ∈ B, B ∩ D B . Y = X[0] ∪ (∪{Dn × {1/n} : n ∈ N}). Y M1 . B ∈ B, (B[n] − B[0]) ∩ Y Y ClY ((B[n] − B[0]) ∩ Y ) = B[n] ∩ Y. (7.4.6) , , ClY ((B[n] − B[0]) ∩ Y ) ⊂ B[n] − B[0] ∩ Y = B[n] ∩ Y. x ∈ B, W x X , k n, B ∩ D B, m k x ∈ B ∩ W ∩ Dm , (x , 1/m) ∈ W [k] ∩ ((B[n] − B[0]) ∩ Y ), B[n] ∩ Y ⊂ ClY ((B[n] − B[0]) ∩ Y ). (7.4.6) . , B[n] ∩ Y Y , Unm |Y n,m∈N ∪ Vn |Y n∈N 7.4 Mi · 239 · Y σ . , Y X Y M1 . f . , f . f . Y F , x ∈ X − f (F ), (x, 0) ∈ / F , x V m ∈ N, V [m] ∩ F = ∅. n < m, x ∈ / f (F ∩(X × {1/n})) ⊂ Dn , x Vn , Vn ∩f (F ∩(X × {1/n})) = ∅. U = V ∩ ( n n, Em ⊂ En . , x ∈ D Ux . , x ∈ Bα ∩ Dn ( n ), Ux () Wn (x) ∩ (Bα )◦n ⊂ Wn (x) ∩ (Bα )n . x ∈ Bα ∩ Dn , ∪Ux ⊂ Wn (x) ∩ (Bα )n (n ∈ N). (7.4.7) ϕ Bα ∩ D x ∈ Bα ∩ D, ϕ(x) ∈ Ux (ϕ(x) x ). ϕ Φα . Bαϕ = Bα◦ ∪ (∪{ϕ(x) : x ∈ Bα ∩ D}) (ϕ ∈ Φα ), Bαϕ . (7.4.8) B # = {Bαϕ : α ∈ A, ϕ ∈ Φα }. B # F . G ⊃ F , B F , Bα ∈ B F ⊂ Bα◦ ⊂ Bα ⊂ G. x ∈ Bα ∩ D, x ∈ G, Ux ∈ Ux x ∈ Ux ⊂ G. ϕ ∈ Φα F ⊂ Bα◦ ⊂ Bαϕ ⊂ G. B # F . B ⊂ B# . x0 ∈ / ∪{Bαϕ : Bαϕ ∈ B }. A = {α ∈ A : ϕ ∈ Φα , Bαϕ ∈ B }, H = ∪{Bα : α ∈ A }, Φα = {ϕ ∈ Φα : Bαϕ ∈ B } (α ∈ A ). (7.4.9) · 242 · 7 () (7.4.8), Bα ∩ D ⊂ Bαϕ , Bα ∩ D ⊂ Bαϕ . Bα ∩ D Bα , Bα = Bα ∩ D. Bα ⊂ Bαϕ . (7.4.9), x0 ∈ / H, n ∈ N x0 ∈ / Hn . : Bαϕ ∈ B . C = ∪ Bα◦ ∪ ∪ ϕ(x) : x ∈ Bα ∩ Dk kn x ∈ Bα ∩ Dm , α ∈ A , m n. ϕ ∈ Φα , ϕ(x) ∈ Ux , (7.4.7), ∪Ux ⊂ (Bα )m , Bα ⊂ H, ϕ(x) ⊂ (Bα )m ⊂ (Bα )n ⊂ Hn . C ⊂ Hn . Hn , C ⊂ Hn , x0 ∈ / Hn , x0 ∈ / C. D = ϕ(x) : x ∈ Bα ∩ Dk , Bαϕ ∈ B . 1k n , x ∈ / Gm = Gm . D = {Uα : α ∈ D Uα ∈ ∪{Um |Gm : m > n}}. D = {Uα : α ∈ D }, D ⊂ D , Gm ⊃ ∪{Uα : α ∈ D }. x∈ / ∪{Uα : α ∈ D }. ∪{Uα : α ∈ D − D } ⊂ ∪{Ui |Gi : i n}, ∪{Ui |Gi : i n} , x∈ / U α (α ∈ D − D ) ⇒ x ∈ / ∪{Uα : α ∈ D − D }, x∈ / ∪{Uα : α ∈ D }, D . . 7.4.9 S X , T S , T X . S S X X . . 7.4.17 [204] X M1 , X M1 , X . M1 ⊂ P . 7.4.16, X . x ∈ X, x X , X H1 H2 (i) X = H1 ∪ H2 ; (ii) i = 1, 2, x ∈ Hi , Hi {Hi,n }n∈N {x} = ∩{Hi,n : n ∈ N}. X , X {Gn }n∈N X = G1 ⊃ G2 ⊃ G2 ⊃ G3 ⊃ · · · ; {x} = ∩{Gn : n ∈ N}. H1 = ∪{G2n−1 − G2n : n ∈ N}, H2 = ∪{G2n − G2n+1 : n ∈ N}. · 244 · 7 () Gk (k ∈ N) , , Gk − Gk+1 = Gk − Gk+1 , H1 = ∪{G2n−1 − G2n : n ∈ N} = ∪{G2n−1 − G2n : n ∈ N}, H2 = ∪{G2n − G2n+1 : n ∈ N} = ∪{G2n − G2n+1 : n ∈ N}. H1 , H2 , (i). n ∈ N, H1,n = H1 ∩ G2n−1 , H2,n = H2 ∩ G2n . H1,n = H1 ∩ G2n−2 , H2,n = H2 ∩ G2n−1 . H1,n , H2,n H1 , H2 . {x} = ∩{Hi,n : n ∈ N}, , H1 , H2 (ii). : x ∈ Hi , , H1 , H2 M1 . x ∈ Hi , 7.4.5, x Hi σ Ui = ∪{Ui,n : n ∈ N}, Ui Hi . (ii) 7.4.8, ∪{Ui |Hi,n : n ∈ N} x Hi , Ci . Ci Hi ( , 7.19). 7.4.9, Ci X X . x ∈ / Hi , Ci = ∅. C = {C1 ∪ C2 : Ci ∈ Ci , i = 1, 2}. (i) C x X , X ( , 7.19). C X , C X x . C ◦ = {C ◦ : C ∈ C } x . . Itō[204] 7.4.10 ( 7.4.18), [165] X . (G) Gruenhage . (G) M1 X M1 H, K, H ⊂ K, H K σ Gruenhage (∗), (G). , , Gruenhage . (G) M1 M1 , Gruenhage “ M1 M1 ?” 7.4.1 7.4.18 Gruenhage . 7.4 Mi · 245 · 7.4.18 [204] M1 M1 . f : X → Y M1 X Y . H, K X , H ⊂ K. K M1 , H K , 7.4.17, H K , X 7.4.10 (G), Y M1 . Y M1 . . 7.4.10 [204] 7.4.6 Nagata M1 . 7.4.5 7.4.9 , Nagata . . M1 , 7.4.18 Lašnev (Lašnev space). 7.4.7 [363] Lašnev M1 . 2004 , Mizokami[298] , Itō Mi . . 7.4.19 7.4.19 [298] M1 . 7.4.8 (i) X M1 ; (ii) X ; (iii) X ; (iv) X σ . 7.4.19, (i) ⇒ (ii); 7.4.16, (ii) ⇒ (iii); (iii) ⇒ (iv) ; 7.4.15, (iv) ⇒ (i). . Mizokami[297] P , P , M1 , M1 . M1 M1 . 7.4.9 M1 . f M1 X Y . 7.4.10, Y M3 . y ∈ Y , 7.4.19, f −1 (y) X B, 7.4.4, {f (B)◦ : B ∈ B} y . 7.4.8, Y M1 . . ( 7.23), , Tamano [388] . 7.4.10 M1 . 7.4.11 X M1 , A X , X/A M1 . f : X → X/A . f , X/A M3 . 7.4.8, X/A , X/A M1 . · 246 · 7 () . M3 ⇒ M1 Mi (). Heath Junnila[188] M0 (M0 -space, σ ), M0 M1 . Gruenhage[165] (G) ( 7.4.10) Fσ (Fσ -metrizable space, ) Fσ (G), M1 . , Mizokami[295] µ (stratiﬁable µ-space), Tamano[386] (regularly stratiﬁable space) (strongly regularly stratiﬁable space) Mizokami[297] M (stratiﬁable space with M-structures). Junnila Mizokami[223] . µ (µ-space[313] ). X µ , Fσ . , µ Fσ , µ σ . Mizokami[297] µ M1 , M0 , M0 µ . Lašnev Fσ Fσ ( [117] 2), [223] . Junnila Mizokami Fσ µ , µ Fσ . Tamono[389] Lindelöf σ µ . . X A Itō Tamano[206] x ∈ X (almost locally ﬁnite), x U B {A ∩ U : A ∈ A } ⊂ {B ∩ V : B ∈ B, V x }. A X , A X . σ , M1 , µ . . X A Ohta[322] x ∈ X (ﬁnitely closure-preserving), A ⊂ A , x U A B, U ∩ ∪{A : A ∈ A } = U ∩ ∪{B : B ∈ B}. A X , A X . Ohta σ , σ M0 , M1 . 7.1 [141] . 7.4 Mi · 247 · 7.1 7.1[141] . M3 M1 M1 . ( 7.1) “” M1 , M1 , M. Itō M1 (, T. Mizokami M1 M σ ). M3 ⇒ M1 . 7.1 , M1 M3 ( P ). · 248 · 7 () 1961 J. G. Ceder (Mizokami[298] , 7.4.19), “” . , [141], [165], [170], [388] . M3 ⇒ M1 Mary Rudin[346] , . , . G. Gruenhage[170] ZFC. , . 7.1 , M1 ? Fσ , , ? M0 µ σ σ M1 7.5 k , , σ Siwiec-Nagata ( 7.3.5). . g , σ ( 7.3.2). g σ . 7.4 g , M2 M3 , M2 = M3 ( 7.4.6), ( 7.3 ) g σ ( 7.4.5) M2 ( 7.4.4) ( 7.4.5) X σ N × X g (i), (ii) ((i) ⇔ (i ), 7.4.3) (i) y ∈ / F , n ∈ N y ∈ / g(n, F ); (i ) x ∈ X {xn }, x ∈ g(n, xn ), xn → x; (ii) y ∈ g(n, x) ⇒ g(n, y) ⊂ g(n, x). M2 σ ( 7.4.4 7.4.5), M3 ⇒ M2 , M3 ⇒ σ ( 7.4.7). Heath Hodel g σ ( 7.4.7 (∗)) . (∗) , , 7.3.5 7.4.5 7.5 k , , · 249 · , . ( 7.5.6). 7.5.1 (Heath-Hodel [187] ) X σ N×X g (∗) X x {xn }, {yn }, x ∈ g(n, xn ) xn ∈ g(n, yn ), yn → x. σ (∗), 7.4.5 (i ), (ii). {xn }, {yn} X , x ∈ g(n, xn ), xn ∈ g(n, yn ), 7.4.5 (ii) xn ∈ g(n, yn ), g(n, xn ) ⊂ g(n, yn ), x ∈ g(n, yn ). 7.4.5 (i ) yn → x. g (∗) σ . , (∗) X σ , 7.3.1 . , 7.4.5 (i ) (∗) ( (∗) yn xn ), 7.4.5 (i ) (i). X g (∗), X “ < ” , x ∈ X i, n ∈ N, H(x, i, n) = X − [(∪{g(i, y) : y < x}) ∪ (∪{g(n, y) : y ∈ / g(i, x)})]. (7.5.1) , H(x, i, n) ⊂ g(i, x). H (i, n) = {H(x, i, n) : x ∈ X}. H (i, n) . z ∈ X, y X z ∈ g(i, y) . , y < x , g(i, y) ∩ H(x, i, n) = ∅ [ (7.5.1) ]. x < y , y , z ∈ / g(i, x), g(n, z) ∩ H(x, i, n) = ∅ [ (7.5.1) ]. g(i, y) ∩ g(n, z) z H (i, n) H(y, i, n) , H (i, n) . m ∈ N, F (x, i, n, m) = {y ∈ H(x, i, n) : x ∈ g(m, y)}, F (i, n, m) = {F (x, i, n, m) : x ∈ X}. F (i, n, m) [ (7.5.2) ] (7.5.2) H (i, n) H(x, i, n) , H (i, n) , F (i, n, m) . F = {F (i, n, m) : i, n, m ∈ N} X . p ∈ U , U . i ∈ N, xi X p ∈ g(i, xi ) , p ∈ / X − g(i, xi ), 7.4.5 (i) ( ), n(i) ∈ N p∈ / ∪{g(n(i), y) : y ∈ / g(i, xi )}, · 250 · 7 () xi (7.5.1) , p ∈ H(xi , i, n(i)) ⊂ g(i, xi ). 7.4.5 (i ) ( ) xi → p. p g(m, p), i(m) m xi(m) ∈ g(m, p). (7.5.2) p ∈ H(xi , i, n(i)) i ∈ N , p ∈ F (xi(m) , i(m), n(i(m)), m). Fm . m Fm ⊂ U . p ∈ g(i(m), xi(m) ), , ym ∈ Fm − U , i(m) m, p ∈ g(m, xi(m) ). ym ∈ Fm , (7.5.2) , xi(m) ∈ g(m, ym ). (∗) ym → p. p ∈ U , ym ∈ / U . F X σ . . (∗) σ , . (∗) ⇒ σ , k ( 7.5.1) ⇒ σ ( 7.5.7). 7.4 ( 7.4.1 3). (Mi ), . . , ( H) U → {Un }, U , Un , U = n∈N Un , V ⊂ U Vn ⊂ Un (Vn ), n < m , Un ⊂ Um . 7.5.2∼ 7.5.4 Creede[99] . 7.5.2 , . , g ( 7.4.3). X . g(n, x) N × X g , 7.4.3 (iii). X A, N × A g g (n, x) = g(n, x) ∩ A (n ∈ N; x ∈ A), g (n, x) 7.4.3 (iii), A . , Xi (i ∈ N) , N × Xi g gi , 7.4.3 (iii). x = (xn ) ∈ n∈N Xn , g(n, x) = gi (n, xi ) × in Xn , i>n g N × ( n∈N Xn ) g , 7.4.3 (iii). . 7.5.3 . f : X → Y X Y , G Y , f −1 (G) X, X , f −1 (G) → {f −1 (G)n }. G → {f (f −1 (G)n )} Y , Y . . 7.5 k , , · 251 · 7.5.1 [356] . 5.5.5 . . 7.5.4 , . {Un }n∈N X . {Un }n∈N . N × X g g(n, x) = st(x, Un ) (n ∈ N, x ∈ X). , g 7.4.3 (iii). X . , U X , X U → {Un }. U “ < ” , U = {Oα : α ∈ A}, A , 0. n ∈ N, F0,n = (O0 )n , Fα,n = (Oα )n − ∪{Oβ : β ∈ A, β < α} (α > 0), Fn = {Fα,n : α ∈ A}, Fn . F = n∈N F . x ∈ X, U x Oα , n ∈ N x ∈ (Oα )n , , x ∈ Fα,n . Fn . , Fα,n . x ∈ X, U Fβ,n ∩ Oα = ∅. x Oα . β > α Fβ,n , Oα , β < α Oβ , , x ∈ / Oβ , Oβ ⊂ X − {x}, (Oβ )n ⊂ (X − {x})n , [X − (X − {x})n ] ∩ (Oβ )n = ∅, X − (X − {x})n x , Fn Fα,n β < α. x Oα ∩ (X − (X − {x})n ) . , F σ , U , 6.1.1, X . . , ( Gδ ). 7.5.2, X 2 , . T2 X ( ), X Gδ ( 2.7), T2 Gδ , 7.5.4, . G∗δ ( 7.3.3 ). 7.5.2 [201] T2 , M . X T2 , M , X Gδ . 7.3.11, X . . · 252 · 7 () 7.5.1 [270] X k (k-semistratiﬁable space), U → {Un }, K ⊂ U , n ∈ N K ⊂ Un . k (k-semistratiﬁcation). , k , k ( ). k . 8.2.3. 7.5.5 [270] k T1 , M1 . X k T1 . U → {Un } X k . x ∈ U , U X , {Wn (x)}n∈N x , U ⊃ W1 (x) ⊃ W2 (x) ⊃ · · ·. n ∈ N, Wn (x) ⊂ Un , yn ∈ Wn (x) − Un , {yn } x. K = {x} ∪ {yn : n ∈ N} , K ⊂ U . U → {Un } k , m ∈ N K ⊂ Um , ym ∈ Um , ym . U = n∈N (Un )◦ . 7.4.1, X Wn (x) ⊂ Un , x ∈ (Un )◦ . , 7.4.5, X M1 . . 7.5.6 [149, 243] T2 X k , N × X g x ∈ X X {xn }, {yn }, xn ∈ g(n, yn ), xn → x, yn → x. U → {Un } X k , N × X g g(n, x) = X − (X − {x})n . x ∈ X X {xn }, {yn }, xn ∈ g(n, yn ), xn → x, yn → x. U X , x ∈ U , xn → x, , {x} ∪ {xn : n ∈ N} ⊂ U , k , m ∈ N {x} ∪ {xn : n ∈ N} ⊂ Um . xn ∈ g(n, yn ) = X − (X − {yn })n , n m , xn ∈ Un ∩ (X − (X − {yn })n ) = Un − (X − {yn })n . (7.5.3) yn ∈ U , yn → x. , yn ∈ / U , U ⊂ X − {yn }, Un ⊂ (X − {yn })n , (7.5.3) . , g(n, x) N × X g , 7.4.3 (iii), X . X U , n ∈ N, Un = X − ∪{g(n, x) : x ∈ X − U }. U → {Un } k , K ⊂ U , n ∈ N K ⊂ Un . ( 7.4.3), (7.5.4) 7.5 k , , (7.5.4) ] · 253 · , n ∈ N, K ⊂ Un , K {xn : n ∈ N} [ xn ∈ K − Un = K ∩ (∪{g(n, x) : x ∈ X − U }). xn ∈ ∪{g(n, x) : x ∈ X − U }, xn ∈ g(n, yn ), yn ∈ X − U . {xn }, {yn }, {xn } ⊂ K, {yn } ⊂ X − U xn ∈ g(n, yn ). 7.5.2, K ( T2 ), ( 3.5.4), {xn } . , xn → x0 ∈ K ⊂ U , , yn → x0 . {yn } ⊂ X − U . . 7.5.7 [149, 243] k σ . X k . 7.5.6 ( ), N × X g g(n, x) x ∈ X X {xn }, {yn }, xn ∈ g(n, yn ), xn → x, yn → x. X x {xn }, {yn } x ∈ g(n, xn ), xn ∈ g(n, yn ). x ∈ g(n, xn ) xn → x, yn → x. Heath-Hodel ( 7.5.1 ) X σ . . , ⇒ k ⇒ σ ⇒ . . ℵ [ 8.1.1, k ( 8.2.1)], [328] . T2 cosmic ( 8.1.3, σ ), [185] , 7.5.5, k . σ ( [166] 9.10). T2 ( σ ), ( [261] 2.7.14 2.10.9). k , , , . Lutzer[270] k , [140] 6.6.9 k . 7.5.1 [258] f k X , Y T2 , f . Y . X X T2 , k , g(n, x) N × X g , 7.5.6 , x ∈ X, n∈N g(n, x) = {x}. 7.5.3, Y . K Y , 7.5.2, K Y . y ∈ K, xy ∈ f −1 (y). E = {xy : y ∈ K}. f (E) = f (E) = K. E ( 7.5.7 7.3.4), E , ( 3.5.2). {xn } ⊂ E, zn ∈ E ∩ g(n, xn )(n ∈ N). zn , z ∈ g(n, xn ), {xn } z, {xn } . zn · 254 · 7 () , f |E {f (zn )} ⊂ K , {zn } E . {zn } . x {zn } , X , X {Vi }i∈N x ∈ Vi+1 ⊂ V i ∩ g(i, x)(i ∈ N). zni ∈ Vi (i ∈ N). {zni } i∈N V i ⊂ i∈N g(i, x) = {x}, x {zni } . {zni } x, x U {zni } {znik } znik ∈ / U . {znik } {zn } , / U , . zni → x. g ( 7.5.6 x, x ∈ ), {xn } . E . . 7.5.8 f : X → Y , X, Y T2 . X k , Y k . X k U → {Un }, V Y , f −1 (V ) X, V → {f (f −1 (V )n )} Y ( 7.5.3). Y K, f ( 7.5.1), X C, f (C) = K. −1 K ⊂ V , C ⊂ f (V ), X k , n ∈ N, C ⊂ f −1 (V )n , K ⊂ f (f −1 (V )n ). V → {f (f −1 (V )n )} Y k . . , k [139] . 7.5.8 X T2 ?[427] —— ( 7.5.2). ( 7.5.9). ( 7.5.10), . , Borges ( 7.4.10) ( 5.1.4, ( 5.1.1) ), 7.5.11 . 7.5.9∼ 7.5.11 Heath, Lutzer Zenor[189] . , [95]. 7.5.2 T1 X (monotonically normal space), X F, K, D(F, K) (i) F ⊂ D(F, K) ⊂ D(F, K) ⊂ X − K; (ii) F ⊂ F , K ⊃ K , F , K , D(F, K) ⊂ D(F , K ). D X (monotone normal operator). D(F, K) ∩ D(K, F ) = ∅, , D (F, K) = D(F, K) ∩ (X − D(K, F )) D(F, K). ( 4.1.2) , . 7.5.9 X X . X , F → {Fn } ( G, 7.4.1 7.5 k , , 2), Fn F , F = Fn ⊃ Fn+1 , n ∈ N. F, K X D(F, K) = · 255 · n∈N F n , F ⊂ K ⇒ Fn ⊂ Kn , , (Fn − K n ). n∈N , D(F, K) ⊃ F . y ∈ K, y ∈ / F , m ∈ N, y ∈ / F m . (X − F m ) ∩ Km = Km − F m y D(F, K) . D(F, K) ⊂ X − K. D . , X , F → {Fn }, Fn F ( G, 7.4.1 3), D. Fn = D(F, X − Fn ), Fn F , F = n∈N (Fn )− . , F ⊂ D(F, X − Fn ) ⊂ D(F, X − Fn ) ⊂ Fn , F ⊂ (Fn )− = n∈N D(F, X − Fn ) ⊂ n∈N Fn = F. n∈N F → {Fn } X D , . . 7.5.10 . f : X → Y , DX X . F, K Y , f −1 (F ), f −1 (K) X , U DX (f −1 (F ), f −1 (K)) ( f ), U = {x ∈ X : f −1 (f (x)) ⊂ DX (f −1 (F ), f −1 (K))}, f (U ) ( 1.5.1), f (U ) ⊂ f (DX (f −1 (F ), f −1 (K))). DY (F, K) Y . , F ⊂ f (U ). f (U ) ⊂ Y − K. DX , f (U ) = f (DX (f −1 (F ), f −1 (K))) ⊂ f (X − f −1 (K)) = Y − K, Y − f (DX (f −1 (F ), f −1 (K))) ⊃ K, y ∈ K, Y − f (DX (f −1 (F ), f −1 (K))) y f (U ) , f (U ) ⊂ Y − K. · 256 · 7 () , DY 7.5.2 (ii), DY Y . . 7.5.9 7.5.10 7.5.3 ( 7.4.10), . 7.5.11 . F X , X D D(F, K) ∩ D(K, F ) = ∅. F ∈ F , F ∗ = ∪{F ∈ F : F = F }. UF = D(F, F ∗ ), F ⊂ UF . F0 , F1 ∈ F , F0 = F1 , UF0 ∩ UF1 = D(F0 , F0∗ ) ∩ D(F1 , F1∗ ) ⊂ D(F0 , F1 ) ∩ D(F1 , F0 ) = ∅. . ( 4.1.1) . 7.5.3 [7] X , x, y ∈ X d(x, y) (i) d(x, y) = 0 x = y; (ii) d(x, y) = d(y, x), d(x, y) X (symmetric). , , ε B(x, ε) = {y ∈ X : d(x, y) < ε} . , 4.1.1 “ y ∈ B(x, ε), δ > 0 B(y, δ) ⊂ B(x, ε)”, X , 7.5.4 . 7.5.4 [7, 21] X (symmetrizable), X d U ⊂ X x ∈ U , ε > 0 B(x, ε) ⊂ U . (X, d) (symmetric space). 7.5.4 , ε , 7.5.4 “ U U ε ”. 7.5.4 F ⊂ X x∈ / F , D(x, F ) > 0 ( 4.1.2), F x ∈ / F , ε > 0 B(x, ε) ∩ F = ∅. 7.5.5 [183, 412] X (semi-metrizable), X d 7.5.4 “ x ∈ X, ε > 0, x ∈ B(x, ε)◦ ”. (X, d) (semi-metric space). . “ x ∈ U , ε > 0, x ∈ B(x, ε)◦ ⊂ B(x, ε) ⊂ U ” ( 7.5.4 ), 7.5 k , , · 257 · {B(x, ε) : ε > 0} x ( ). “ ε ” “1/n”, {B(x, 1/n)}n∈N x , . , T1 . 7.5.12 [21] T2 X, (i) X ; (ii) X ; (iii) X Fréchet . 7.5.5 , (iii) ⇒ (i). X Fréchet (Fréchet 2.3.1 ), x ∈ X, ε > 0, x ∈ B(x, ε)◦ . , x ∈ X − B(x, ε)◦ = X − B(x, ε). X Fréchet , X − B(x, ε) {xn } xn → x. X T2 , ( 2.2.4), F = {xn : n ∈ N}, F = F ∪ {x} F . , y ∈ / F , y = x, δ > 0 B(y, δ) ∩ F = ∅, B(y, δ) ∩ F = ∅; y = x, B(y, ε) ∩ F = ∅. F , x ∈ B(x, ε)◦ . . [0, ω1 ) Fréchet , [179] . 7.5.13 [99] . . X X T1 . , X T1 , X F ⊂ X, G(F, n) = {y : D(y, F ) < 1/2n }◦ , x ∈ B(x, ε)◦ ( 7.5.5), F ⊂ G(F, n). F = G(F, n), F ⊂ K () ⇒ G(F, n) ⊂ G(K, n) (n ∈ N). n∈N F → {G(F, n)} X . . X T1 . x ∈ X, x {b(n, x)}n∈N , N × X g g(n, x) x ∈ g(n, yn ) ⇒ yn → x ( 7.4.3). h(n, x) = b(n, x) ∩ g(n, x), ⎧ ⎪ x = y, ⎪ ⎨ 0, n d(x, y) = / h(n, y) 1/2 , x = y, n ∈ N x ∈ ⎪ ⎪ ⎩ y∈ / h(n, x) , · 258 · 7 () d(x, y) = sup{1/2n : x ∈ / h(n, y) y ∈ / h(n, x)} ( “sup” 1/2n ). , d X ( T1 ). , y ∈ h(n, x) ⇒ d(x, y) < 1/2n , h(n, x) ⊂ B(x, 1/2n )◦ . {B(x, 1/2n ) : n ∈ N} x . , x U B(x, 1/2n ), n ∈ N, B(x, 1/2n )−U = ∅, yn ∈ B(x, 1/2n )− U . d(x, yn ) < 1/2n , yn ∈ h(n, x) ⊂ b(n, x) x ∈ h(n, yn ) ⊂ g(n, yn ) , yn ∈ b(n, x), yn → x, x ∈ g(n, yn ), yn → x, {yn } x. , yn U , U x. . 7.5.14 [183] . . T1 . ( 7.5.4) , 7.5.13 7.6 Nagata-Smirnov “ σ ” “ σ ”“ σ ”“”, “” (pointcountable base) . , Miščenko[294] “ T2 ” ( 7.6.1). Miščenko ( 7.6.1). “ T1 ”. Urysohn Miščenko . Miščenko , Miščenko , M. E. Rudin ( [97]), , “” “”. Miščenko , . 7.6.1 (Miščenko [294] ) A E . A , A E ( ). Vn (n ∈ N) A n E ( A n∈N Vn ). , n0 ∈ N Vn0 . k n0 A () A1 , A2 , · · · , Ak , SA1 A2 ···Ak Vn0 A1 , A2 , · · · , Ak . 7.6 p1 ∈ E, · 259 · Ap1 = {A ∈ A : p1 ∈ A} ( |Ap1 | ℵ0 ), Vn0 = SA . (7.6.1) A∈Ap1 (7.6.1), |Vn0 | > ℵ0 |Ap1 | ℵ0 , A1 ∈ Ap1 |SA1 | > ℵ0 . , E ⊂ A1 (, n0 = 1 Vn0 = Ap1 , |Vn0 | > ℵ0 |Ap1 | ℵ0 ). p2 ∈ E − A1 . Ap2 = {A ∈ A : p2 ∈ A} ( |Ap2 | ℵ0 ), SA1 = SA1 A . (7.6.2) A∈Ap2 (7.6.2), , A2 ∈ Ap2 |SA1 A2 | > ℵ0 , E ⊂ A1 ∪ A2 . , k < n0 , pk+1 ∈ E − ni=1 Ai Ak+1 ∈ Apk+1 |SA1 A2 ···Ak Ak+1 | > ℵ0 . , k = n0 − 1 , A A1 , A2 , · · · , An0 |SA1 A2 ···An0 | > ℵ0 . SA1 A2 ···An0 ⊂ Vn0 , SA1 A2 ···An0 ({A1 , A2 , · · · , An0 }) . . . 7.6.2 (Rudin) . X A , C X , A () C , A C A . . 7.6.3 (Rudin) T1 . X T1 B. {Cn }n∈N Cn ⊂ Cn+1 C = n∈N Cn X. x ∈ X, C1 = {x}. Cn , Bn = {B ∈ B : B ∩ Cn = ∅}. Bn F X − ∪F = ∅ , xF ∈ X − ∪F . Cn+1 Cn xF , Cn ⊂ Cn+1 . Cn , Bn ( B ), , Cn+1 . C = n∈N Cn , C , C X ( C = X). , x0 ∈ X − C, X T1 , X − {x0 } C , B X , U B () C x0 . U C , U () C , C , C , U ⊂ B , U . U C, U F0 . U C , Cn . F0 U , Cn ⊂ Cn+1 , n0 ∈ N F0 Cn0 , F0 ⊂ Bn0 . U x0 , X − ∪F0 = ∅. , Cn0 +1 X − ∪F0 xF0 , F0 Cn0 +1 (F0 C) . C X, X . . · 260 · 7 () T1 . X , ( 5.5.3, p ∈ X). X T0 , T1 . X − {p} X , X . p X, X . 7.6.2 7.6.1 [294] . T1 . Miščenko , “” “”. 7.6.1 (Miščenko [294] ) T2 . 7.6.1, T2 , Urysohn ( 4.3.1) . . Rudin . 7.6.1 X Y y ∈ Y, f −1 (y) . 7.6.2 [335] f : X → Y s (s-mapping), s . f : X → Y X Y s . B X , f , f (B) = {f (B) : B ∈ B} Y −1 . f (y) (y ∈ Y ) , f −1 (y) B ( 7.6.2), f (B) . . s . , 7.6.2, “ ”. 7.6.3 . 7.6.3 [335] s . s T0 4.4.4, Ponomarev “ T0 ” N (A) (), , , . U = {Uα }α∈A T0 X . N (A), N (A) A S = {(α1 , α2 , · · ·) : {Uαn }n∈N x ∈ X }. f : S → X f (α) = x, α = (α1 , α2 , · · ·), {Uαn }n∈N x . 4.4.4 f , f s , f −1 (x) (x ∈ X) . 7.6 · 261 · N (A) N (A) = n∈N An , An = A, A , , N (A) , S N (A) . U , x ∈ X U , f −1 (x) , ( 2.17). . Filippov[116] “ s ” 7.6.2, [75] , , , Burke Michael . 7.6.4 [75] Y Y P y ∈ Y y V , P P y ∈ (∪P )◦ , P ∈ P y ∈ P ⊂ V . . . Y P , Y . Φ = {F ⊂ P : F }. , Y , , y ∈ Y , {(∪F )◦ : F ∈ Φ, y ∈ (∪F )◦ , y ∈ ∩F } y . , {(∪F )◦ : F ∈ Φ} Y , (∪F )◦ . , F ∈ Φ, U (F ) = {A ⊂ (∪F )◦ : E F , A ⊂ (∪E )◦ }, V (F ) = (∪(U (F ) ∩ P))◦ . V = {V (F ) : F ∈ Φ} Y . V Y . y ∈ W, W Y . , F ∈ Φ ◦ E F , y ∈ / (∪E )◦ . , V (F ) ⊂ ∪U (F ) ⊂ (∪F )◦ ⊂ W . y ∈ V (F ). y ∈ (∪F )◦ , , S ⊂ P y ∈ (∪S )◦ , P ∈ S , y ∈ P ⊂ (∪F )◦ . E F , y ∈ P ⊂ (∪F )◦ y∈ / (∪E )◦ , P ⊂ (∪E )◦ . U (F ) , P ∈ U (F ). S ⊂ U (F ) ∩ P. (∪S )◦ ⊂ V (F ), y ∈ V (F ). y ∈ (∪F ) ⊂ W . V , y ∈ V (F ) F ∈ Φ . y ∈ V (F ), y ∈ A ∈ U (F ) ∩ P, P , y ∈ A A ∈ P . (∗) A ⊂ Y , A ∈ U (F ) F ∈ Φ . · 262 · 7 () n ∈ N, Φn = {F ∈ Φ : |F | = n}. A ∈ U (F ) F ∈ Φn . , A ∈ U (F ) Φn F , F Ψ , Ψ ⊂ Φn . R ⊂ P R ⊂ F , F ∈ Ψ . Ψ ∗ = {F ∈ Ψ : R F }, 0 |R| < n, F ∈ Ψ ∗ , A ∈ U (F ) R F . U (F ) , A ⊂ (∪R)◦ , y∈Ay ∈ / (∪R)◦ . E = Y − ∪R, y ∈ E. Y , y ∈ Z. F ∈ Ψ ∗ , y ∈ (∪F )◦ ( Z = {zn : n ∈ N} ⊂ E zn → y, y ∈ A ⊂ (∪F )◦ ). Z P ∈ F , P , Z P ∈ P . Z P0 ∈ P , P0 Ψ ∗ F . , P0 ∈ / R, P0 Z ∪R Z . R = R ∪ {P0 }, R R R ⊂ F F ∈ Ψ , R , (∗). V Y . 7.6.4 (Filippov [116] ) . . s f : X → Y X Y P = f (B), f s , 7.6.2 s , B X , P Y . y ∈ Y, W Y y , B B ⊂ f −1 (W ) B ∩ f −1 (y) = ∅ B , B ⊂ B, B f −1 (y). f , ◦ E ⊂ B , y ∈ (∪f (E )) . B ∈ E , f (B) ∈ f (E ), B ⊂ f −1 (W ) B ∩ f −1 (y) = ∅ y ∈ f (B) ⊂ W . 7.6.4 Y . . 7.6.5 T0 X, (i) X ; (ii) X s (iii) X (iv) X s [335] ; [116] s ; [287] . (i) ⇒ (ii), 7.6.3. (ii) ⇒ (iii) ⇒ (iv), ( 5.2.1). , 7.6.2, , 7.6.4 (iv) ⇒ (i). . s 6.6.1 Arhangel’skiı̌ MOBI . Y MOBI M φ1 , φ2 , · · · , φn , (φ1 ◦ φ2 ◦ · · · ◦ φn )(M ) = Y [42] . T1 , . 7.6 · 263 · 7.6.6 [145] f T1 X Y , y ∈ Y, f −1 (y) , Y . y ∈ Y , 7.6.3, f −1 (y) ; 7.6.1 3.5.9, f −1 (y) ( 6.6.1). f s . f . y ∈ Y X U f −1 (y), f , U U f −1 (y), f −1 (y) ⊂ ∪U , f , y ∈ f (∪U )◦ . f . 7.6.4, Y . . 7.6.6 . , ( 5.2.1) T1 . Filippov . 7.6.1 [115] T1 . “ T1 ( meta ) ”[21] , . MOBI MOBI1 MOBI , T1 . Chaber[87] T1 MOBI1 . , 7.6.6 MOBI1 . 7.6.2 MOBI1 . Miščenko ( 7.6.1) “” , ( T1 ) . 7.6.2 [286] X U T1 (T1 -separating), x, y ∈ X (x = y), U ∈ U x ∈ U y ∈ / U ( x ∈ U ⊂ X − {y}). ℵ1 ( 4.1.7 ). X ℵ1 , X ℵ1 , . 7.6.5 ℵ1 . U ℵ1 X . X {xα : α < κ} xα ∈ / β<α st(xβ , U ) X = α<κ st(xα , U ), {{xα } : α < κ} X . , x ∈ X, β < κ x ∈ st(xβ , U ), x st(x, U ) {{xα } : α < κ} . X ℵ1 , ( 6.6.13 1), |κ| < ℵ1 , U , {U ∈ U : α < κ xα ∈ U } U . . 7.6.7 [201] T1 T2 X . · 264 · 7 () X Gδ , 7.3.9 . 2 X Gδ X ∆ = {(x, x) : x ∈ X} X 2 Gδ . , X 2 T1 , U . V = {∪U : U ⊂ U ∆ }. Miščenko ( 7.6.1), V , ∆ = ∩V . , ∆ ⊂ ∩V ( , X = ∅ , V = ∅). ∆ ⊃ ∩V . p ∈ X 2 − ∆. p ∈ /V ∪U , p ∈ / ∩V . q ∈ ∆, U T1 , Uq ∈ U , q ∈ Uq ⊂ X 2 − {p}, {Uq : q ∈ ∆} ∆ p. ∆ X, ∆ , ℵ1 . 7.6.5, {Uq : q ∈ ∆} , / ∪U ∈ V , p∈ / ∩V . . U ( ∆ ). ∆ ⊂ ∪U , p ∈ [201] Ishii Shiraki T1 T2 , M . [76], [171], [259], [391] . 7.1[215] 7.2 7 ℵ1 Moore . , F = n∈N st(F, Un ), {Un }n∈N , . X T2 X 2 , X Gδ ; Gδ 7.3 . 7.4 Bennett Lutzer [45] ( 6.4) θ . 7.5 w∆ ( 7.2.1) , xn ∈ st(x, Un ) , {xn } x, {st(x, Un )}n∈N x , . 7.6 T2 θ w∆ . 7.7 M [317] 7.8 . T2 X M X T2 . 7.9[114] 7.10 Čech Čech . 7.2.3 (iii) n∈N st(x, Un ) = n∈N st(x, Un ) (iii ) n ∈ N n ∈ N st(x, Un ) ⊂ st(x, Un ). 7.11 A X Čech , A X Gδ . 7.12 Čech Gδ Čech . X Čech T2 Gδ . 7.13 M . T2 Gδ p . 7 · 265 · 7.14[324] (i) σ σ ; (ii) σ σ ; (iii) X Xn (n ∈ N) , Xn σ (σ ) , X σ (σ ) [62] 7.15 [287] 7.16 . , Moore = σ + w∆ . {An }n∈N X , A = n∈N An . (i) A , {An }n∈N A ; (ii) xn ∈ An (n ∈ N), {xn } A ω ; (iii) xn ∈ An (n ∈ N), {xn } X ω ; (iv) {Kn }n∈N , Kn ⊂ An (n ∈ N), n∈N K n = ∅. (i) ⇒ (ii) ⇒ (iii) ⇒ (iv). n∈N An = n∈N An , [312] Nagami . Σ X Σ (Σ-net) {Fi }i∈N K1 ⊃ K2 ⊃ · · · , x ∈ X, Ki ⊂ C(x, Fi ) = ∩{F : x ∈ F ∈ Fi } (i ∈ N), i∈N Ki = ∅. C(x) = i∈N C(x, Fi ), C(x) . X Σ , X Σ Nagami 7.3.14 Σ∗ [217] Σ θ . 7.18 7.19 U =U −◦ Michael 7.3.4. [59] 7.17 . . (regular open set) (regular closed set). U , . F , F = F ◦− . ◦ (i) F , F ; U , U − ; (ii) () () ; (iii) U, V , U ⊂ V U − ⊂ V − ; (iv) F, H , F ⊂ H F ◦ ⊂ H ◦ ; (v) , (vi) () (). Sorgenfrey ( 2.3.3) , M3 [197] . 7.20 7.21[187] 7.4.7 (∗) M3 ⇒ σ. [187] 7.4.5 (i) (ii) ⇒ 7.4.7 [133] 7.22 7.23 7.24 f X α ∈ A, Uα mapping). ; 7.25[133] Y . , X U = {Uα }α∈A , Y f (Uα ), f (locally homeomorphic f T2 X ; (ii) f (∗). Y k , (i) f . X , U = {Uα }α∈A X , Uα (α ∈ A) M1 , X M1 . · 266 · 7 7.26[445] X T1 . , f 7.27 () f : X → Y y ∈ Y, f −1 (y) . Heath-Junnlia ( 7.4.12) Z, Y , Z, Y ∈ P . 7.28[360] . [97] σ , ( 7.29 [97] 7.30 7.6.3). T2 . 7.31 Michael ( 5.4.1) . Čech [103] . 7.32[189] T1 X (p, C) H, C p ∈ X − C, H(p, C) (i) p ∈ H(p, C) ⊂ X − C; (ii) D p ∈ / C ⊃ D, H(p, C) ⊂ H(p, D); (iii) p, q ∈ X p = q, H(p, {q}) ∩ H(q, {p}) = ∅. 7.33 (i) g [183] n ∈ N, {p, xn } ⊂ g(n, yn ), {xn } p [197] w∆ n ∈ N, {p, xn } ⊂ g(n, yn ), {xn } (ii) 7.34 X N . ; . (i) X T1 ; (ii) X k ; (iii)[179] X d X {xn } xn → x, d(x, xn ) → 0. 7.35 {xn } X β (β-space[196] ), g n ∈ N, p ∈ g(n, xn ), . 7.33 w∆ ( 7.4.3) ⇒ β. w∆ ⇒ β. g [396] (i) X β ; (ii) X U , X {Fn (U )}n∈N (1) Fn (U ) ⊂ U , (2) V , U ⊂ V ⇒ Fn (U ) ⊂ Fn (V ), (3) {Un }n∈N X n∈N Un = X, n∈N Fn (Un ) = X; (iii) X F , X {Un (F )}n∈N (1) Un (F ) ⊃ F , (2) H , F ⊂ H ⇒ Un (F ) ⊂ Un (H), (3) {Fn }n∈N X n∈N Fn = ∅, n∈N Un (Fn ) = X. , β [197] ( 6.1.22). [396] β [242] β [197] Σ β . [196] , = β + G∗δ . 7.36 7.37 7.38 7.39 7.35∼ 7.39 , ! . . , β k ( 7.4.1). . β 7 7.40[197] · 267 · X γ (γ-space), g n ∈ N, yn ∈ g(n, p), xn ∈ g(n, yn ), {xn } p (i) γ ; (ii) β, γ T1 . . 8 () 8.1 ℵ0 7.3 σ ( 7.3.1) Nagata-Smirnov “” “”, “” , “” ( 3.1.2). k , “” “” ( 8.1.1), k k , “network” 1.4 “net” . 8.1.1 [328] X P X k ( k , k-network), X K U K ⊂ U , P P K ⊂ ∪P ⊂ U . X ℵ0 (ℵ0 -space [284] ) (ℵ (ℵ-space)), X k (σ k ). , k , k (closed k-network). P, , ℵ0 “ P ∈ P K ⊂ P ⊂ U ”. Michael [284] ℵ0 , P (pseudo-base), ℵ0 . 8.1.1 . ℵ0 , () ℵ0 . 8.1.1 [284] . ℵ0 , Lindelöf , , Gδ , . , ℵ0 G∗δ . 8.1.2 [284] ℵ0 , . . ℵ0 ℵ0 . X = n∈N Xn , Xn (n ∈ N) ℵ0 , k An . , X . m ∈ N, m Pm = An × Xn : An ∈ An , n m , n=1 Pm X . P = P X k . n>m m∈N Pm . , 8.1 ℵ0 · 269 · X K U K ⊂ U . x ∈ K, n∈N Un (x) x ∈ n∈N Un (x) ⊂ U , n ∈ N, Un (x) = Xn . K , K . K⊂ Un,1 ∪ n∈N Un,2 ∪ · · · ∪ n∈N Un,k ⊂ U. n∈N X T2 K , ( 3.7), X {Ki : i = 1, 2, · · · , k} K= k Ki ⊂ Ki , i=1 Un,i (i k), n∈N i k, n ∈ N, Un,i = Xn . i k. n ∈ N, X Ki n Xn Kn,i , pn (Ki ) = Kn,i , Kn,i Un,i . An Xn k , An An,i Kn,i ⊂ ∪An,i ⊂ Un,i . n > ni Un,i = Xn . n i Qi = An × Xn : An ∈ An,i , n>ni n=1 Qi Pni ⊂ P , Ki ⊂ Kn,i ⊂ ∪Qi ⊂ n∈N Un,i . n∈N k i=1 Qi P , K= k i=1 Ki ⊂ k (∪Qi ) ⊂ i=1 k i=1 Un,i ⊂ U. n∈N , P X k . . 8.1.1 [284] k . f : X → Y k P X Y , f (P) = {f (P ) : P ∈ P} Y k . C, U Y , C ⊂ U . f , X K f (K) = C, K ⊂ f −1 (U ), P P K ⊂ ∪P ⊂ f −1 (U ), C ⊂ f (∪P ) ⊂ U , f (∪P ) = ∪f (P ), f (P ) = {f (P ) : P ∈ P } f (P) . . · 270 · 8 () k . 8.1.3 [284] ℵ0 . f : X → Y ℵ0 X Y , 8.1.1, X Lindelöf , T2 , . f , Y , . 6.6.2, f , X k , 8.1.1, Y k . . ℵ0 Lindelöf , ℵ0 , 5.5.5 ℵ0 . 8.1.2 [284] X r (r-space), x ∈ X {Un (x)}n∈N , xn ∈ Un (x), {xn } . x r (r-sequence). 8.1.4 [284] ℵ0 X, r , X . X P, {P ◦ : P ∈ P} (P ◦ P ) X , X . , x ∈ X X U x ∈ U , P ∈ P x ∈ P ◦ ⊂ U , P U {P1 , P2 , · · · , Pn , · · ·}. X , x V , V ⊂ U , X r , x r {Un (x)}n∈N , Un (x) ⊂ V , n ∈ N. , Un (x) − Pn = ∅, xn ∈ Un (x) − Pn . A = {xn : n ∈ N}, r , A C , C , A ⊂ C, A , A ⊂ V ⊂ U . P , Pn A ⊂ Pn ⊂ U , xn ∈ / Pn . . r , () ℵ0 . 8.1.5 [284] X , (i) X ℵ0 k ; (ii) X . (i) ⇒ (ii). X ℵ0 k , F X k , . N = F ω , F , N , N {Fn } = (F1 , F2 , · · ·), Fn ∈ Fn (= F ). {Fn }, X x Fn , x ∈ n∈N Fn . {x} = n∈N Fn . {Fn }n∈N x ( 7.3.21 ). M , M ⊂ N. f : M → X, x {Fn }n∈N x ∈ X. , f ( 4.4.4). f M X 8.1 ℵ0 · 271 · . f , A ⊂ X , f −1 (A) M . X k , K K ∩ A . K σ ( k ), ( 7.3.13), x ∈ K − A an ∈ K ∩A an → x ( 2.3.1). m ∈ N, {x} ∪ {an : n m} , Zm . Zm F . F , F ∈ F F ⊃ Zm . F {Gn }n∈N . x Zm . F k , Gn , {Gn }n∈N x , {Gn } ∈ M f ({Gn }) = x ∈ A, {Gn } ∈ f −1 (A). , {Gn } M Bm = {{Fn } ∈ M : Fi = Gi , i m}, f (Bm ) = im Gi , m ∈ N [ 4.4.4 (4.4.11) ]. F , m ∈ N, im Gi ∈ F , Gi (i ∈ N) {an } , A ∩ ( im Gi ) = ∅, f −1 (A) ∩ Bm = ∅, {Gn } ∈ f −1 (A). f −1 (A) M . (ii) ⇒ (i). f : M → X M X . X k [ k , k ( 3.4.8)], X ℵ0 . B M . f (B) X k . , X K U K ⊂ U , K C = {C ∈ f (B) : C ⊂ U } , C {Cn }, xn ∈ K − in Ci . K , σ , , {xn } , , xn → x ∈ K x = xn . A = {xn : n ∈ N} X, f , f −1 (A) M . z ∈ f −1 (A)−f −1 (A), z ∈ f −1 (K) ⊂ f −1 (U ). B ∈ B z ∈ B ⊂ f −1 (U ), f (B) ∈ C . M z ∈ f −1 (A) − f −1 (A), f −1 (A) {zi } z. B , j ∈ N, i > j zi ∈ B, f (zi ) ∈ f (B), f (B) xn , xn . . 8.1.6 [284] X , (i) X ℵ0 ; (ii) X . (i) ⇒ (ii). ℵ0 X k F . 8.1.5 (i) ⇒ (ii) M f : M → X. f . K ⊂ X , K F , · 272 · 8 () {Fn }, L= {Fn } ∈ n∈N Fn F Fn : n∈N Fn ω () . ∩ K = ∅ . n∈N {Fn } ∈ L, ( n∈N Fn ) ∩ K = ∅, x ∈ ( n∈N Fn ) ∩ K. U x , K , K W x ∈ W ⊂ W ⊂ U ∩ K. F X k , F F F W ⊂ ∪F ⊂ U, K − W ⊂ ∪F ⊂ X − {x}. Fx = F ∪ F , Fx F , K ⊂ ∪Fx st(x, Fx ) ⊂ ∪F ⊂ U , Fx Fn , Fn ⊂ U . , {Fn }n∈N x , {Fn } ∈ N f ({Fn }) = x ∈ K. L ⊂ M , f (L) ⊂ K. , x ∈ K, {Fn } K , Fn ∈ Fn x ∈ Fn (n ∈ N). x ∈ ( n∈N Fn ) ∩ K, {Fn } ∈ L, , f ({Fn }) = x, K ⊂ f (L). f (L) = K. {Hn } ∈ n∈N Fn − L, ( n∈N Hn ) ∩ K = ∅, K Hn ∩ K (n ∈ N) , m ∈ N ( nm Hn ) ∩ K = ∅ ( 3.1.1). B= {Fn } ∈ Fn : Fi = Hi , i m , n∈N B {Hn } n∈N Fn , . n∈N Fn B ∩ L = ∅. L , f . (ii) ⇒ (i). 8.1.1 8.1.3 [284] . . cosmic (cosmic space). , ℵ0 cosmic , cosmic σ . , cosmic , . 8.1.7 [284] X , (i) X cosmic ; (ii) X . (i) ⇒ (ii). (X, τ ) F . F B, B , X τ B (X, τ ) . B τ , τ , (X, τ ) , B, (X, τ ) , (X, τ ) (X, τ ) ( τ ⊃ τ ). 8.1 ℵ0 · 273 · (ii) ⇒ (i). . . [284] cosmic 8.1.7, Continuous-images Of Separable Metrics. Michael ℵ0 , 2.1 , 2.1.4, γ∈Γ Xγ , . . 3.6 , 3.6.2 A , I = [0, 1] , I A |A| [0, 1] , A , |A| q ∈ I A q = {xα }α∈A , xα ∈ [0, 1], q A I A I , . , I A , . , X Y X, Y f W (x, U ) = {f ∈ Y X : f (x) ∈ U }, x ∈ X, U Y. (topology of pointwise convergence) Y X W (x, U ) , W (x, U ) . , “ ” (fine), [124] (compact-open topology), W (x, U ) x C ( k ). W (C, U ) = {f ∈ Y X : f (C) ⊂ U }, C X , U Y. W (C, U ) , C . W (C, U ) . {x} , , . X C (X, Y ). Y , ( 8.20∼ 8.24) , . 8.1.2 [12] C (X, Y ) Y . . Y C (X, Y ) ( 8.23). . Y X T1 , C (X, Y ) T1 . f ∈ C (X, Y ), f C (X, Y ) , C . f ∈ W (C, U ), C X , U Y , W (C, U ) f C (X, Y ) . f ∈ W (C, U ) ⇒ f (C) ⊂ U, f (C) Y . Y , Y · 274 · 8 () V f (C) ⊂ V ⊂ V ⊂ U ( 3.1.5). C f ∈ W (C, V ) ⊂ W (C, V ) ⊂ W (C, V ) ⊂ W (C, U ). C W (C, U ) ( 8.22, “ −C ” C ). . ( 3.6.4). 8.1.3 [12] C X T2 , C (X, Y ) × C → Y e : (f, x) → f (x) . x ∈ C, U Y (f, x) ∈ e−1 (U ), f ∈ C (X, Y ), f (x) ∈ U . C ( 3.4.3) f , x C N f (N ) ⊂ U , f ∈ W (N, U ), W (N, U ) f C (X, Y ) , W (N, U ) × N (f, x) C (X, Y ) × C e−1 (U ). e : (f, x) → f (x) . . F ⊂ C (X, Y ) A ⊂ X, W (C, V ) F (A) = {f (x) : f ∈ F x ∈ A}, F (A) = e(F, A). 8.1.4 X T2 . X xn → x K({x}) ⊂ U , n ∈ N, K({xn }) ⊂ U , K ⊂ C (X, Y ), U Y . , fn ∈ K, fn (xn ) ∈ / U . K , {fn } f ∈ K. C = {x} ∪ {xn : n ∈ N}, C . f (x) ∈ U , 8.1.3 e : (f, x) → f (x) . . 8.1.8 [284] X, Y ℵ0 , C (X, Y ) ℵ0 . 8.1.2 C (X, Y ) . C (X, Y ) k . 1. X . X ℵ0 , 8.1.6, M f : M → X. Φ : C (X, Y ) → C (M, Y ) Φ(g) = g ◦ f, g ∈ C (X, Y ). Φ (f Φ −1 ). , C (M, Y ) ℵ0 , ℵ0 ( 8.1.2), C (X, Y ) ℵ0 , 1. A ⊂X B ⊂ Y, W (A, B) = {f ∈ C (X, Y ) : f (A) ⊂ B}. 8.2 ℵ · 275 · B X , Q Y k , . P = {W (B, Q) : B ∈ B, Q ∈ Q}. 2. K ⊂ W (C, U ), K C (X, Y ) , C X U Y , P ∈ P, K ⊂ P ⊂ W (C, U ). 2, , B ∈ B 2 . X V ⊃ C Q ∈ Q K(V ) ⊂ Q ⊂ U . C ⊂ B ⊂ V K ⊂ W (B, Q) ⊂ W (C, U ). Q ∈ Q Q ⊂ U {Qn }. V Q , xn ∈ X D(xn , C) = inf{d(xn , x) : x ∈ C} < 1/2n , d X / in Qi (Q ). C , {xn } , fn ∈ K fn (xn ) ∈ . , xn → x ∈ C. K ⊂ W (C, U ), K({x}) ⊂ U , 8.1.4, n ∈ N, K({xn }) ⊂ U . A = {x} ∪ {xn : K({xn }) ⊂ U }, A , K(A) ⊂ U . 8.1.3, K(A) , K(A) ⊂ Q ⊂ U Q ∈ Q . Q Qm , K(A) ⊂ Q fn (xn ) ∈ / Q n m , 2. 3 8.1.8 . 3. P 2 , P P̂ C (X, Y ) k ( k ). W C (X, Y ) , W (C, U ) . 2 K ⊂ W ∈ W , K C (X, Y ) , P̂ ∈ P̂ K ⊂ P̂ ⊂ W . 3 ( , C (X, Y ) ). H ⊂ U, H, U C (X, Y ) , H ( ) , W ∈ W W ⊂ U ( 3.7). . 8.2 ℵ Michael [284] ℵ0 , O’Meara [328] ℵ k ( σ k ℵ ), σ k [245] ( σ ) T2 ( 8.10), ℵ k , ℵ0 . σ . k , k , ℵ , ℵ σ , G∗δ ( 7.3.1 7.3.12). · 276 · 8 8.2.1 [328] ℵ . () ℵ , ℵ ℵ . 8.1.2 ℵ0 , 7.3.3 σ ℵ , σ . ℵ 7.3.6 , ℵ σ . . 8.2.1 [121] X (F1 , F2 ) F = {(F1 , F2 )}, F1 F1 ⊂ F2 , k (pair-k-network), X K U ⊃ K, F () (F1(i) , F2(i) ), i n, K ⊂ ni=1 F1(i) ⊂ n (i) i=1 F2 ⊂ U . F ( 7.4.1), F ⊂ F , ∪{F1 : (F1 , F2 ) ∈ F } ⊂ ∪{F2 : (F1 , F2 ) ∈ F }. 8.2.2 [121, 139] k . 7.5.1). X k X T1 σ (X, T ) k , U → {Un } X k Fn = {(Un , U ) : U ∈ T }, F = ( Fn , n∈N Un . X K U ⊃ K, n ∈ N K ⊂ Un ⊂ U , (Un , U ) ∈ Fn . F k . n ∈ N, Fn ⊂ Fn , V = ∪{U : (Un , U ) ∈ Fn }. U ⊂ V ⇒ Un ⊂ Vn , ∪{Un : (Un , U ) ∈ Fn } ⊂ Vn , Vn . ∪{Un : (Un , U ) ∈ Fn } ⊂ Vn ⊂ V = ∪{U : (Un , U ) ∈ Fn }. Fn . F = n∈N Fn σ k . , (X, T ) σ k F = n∈N Fn , Fn = {(F1 , F2 )} , F1 , F1 ⊂ F2 . n ∈ N, U ∈ T , Un = ∪{F1 : (F1 , F2 ) ∈ Fn , F2 ⊂ U }, (8.2.1) 8.2 ℵ · 277 · Un . Fn , Un ⊂ ∪{F2 : (F1 , F2 ) ∈ Fn , F2 ⊂ U } ⊂ U. x ∈ U , {x} , k , m ∈ N (F1 , F2 ) ∈ Fm x ∈ F1 ⊂ F2 ⊂ U . (8.2.1) x ∈ Um , U = n∈N Un . , (8.2.1) U ⊂ V Un ⊂ Vn . U → {Un } X . k . , Fn ⊂ Fn+1 (n ∈ N), K ⊂ U . k , n0 ∈ N Fn 0 ⊂ Fn0 K ⊂ ∪{F1 : (F1 , F2 ) ∈ Fn 0 } ⊂ ∪{F2 : (F1 , F2 ) ∈ Fn 0 } ⊂ U. (8.2.1), K ⊂ Un0 . U → {Un } k , X k . . 8.2.1 [270] ℵ k . σ k ⇒ σ k ⇒ σ k , ℵ k . . k ( 8.2.2) Fréchet k T1 , , Lutzer “ k T1 ” ( 7.5.5 ), ⇒ Fréchet ( 2.3.1 ). 8.2.3 [148, 243] Fréchet k T1 . ( 7.5.9), k , Fréchet k T1 X . 8.2.2, k X σ k F = n∈N Fn , Fn . H, K X , ⎧ ⎫ ⎨ ⎬ Un = ∪ F1 : (F1 , F2 ) ∈ Fi , F2 ∩ K = ∅ − ⎩ ⎭ in Fi , F2 ∩ H = ∅} (8.2.2) ∪ {F1 : (F1 , F2 ) ∈ in D(H, K) = ( n∈N Un )◦ . D X ( 7.5.2). , H ⊂ H , K ⊃ K , H , K X , (8.2.2) D(H, K) ⊂ D(H , K ). H ⊂ D(H, K). , H ⊂ D(H, K), ◦ x ∈ H − (D(H, K) ∪ K) = H − Ui ∪K i∈N · 278 · ⊂ (X − K) ∩ X− Ui 8 () ∩H =X − K∪ i∈N Ui ∩ H. i∈N X Fréchet , {xn } ⊂ X − (K ∪ ( i∈N Ui )) xn → x. X {x} ∪ {xn : n ∈ N} ⊂ X − K (), F k , F (F1(i) , F2(i) ) (i m), (i) (i) {x} ∪ {xn : n ∈ N} ⊂ F1 ⊂ F2 ⊂ X − K. im im i0 m {xn } {xnj }, (i ) (i ) {xnj } ⊂ F1 0 ⊂ F2 0 ⊂ X − K. (8.2.3) F2(i0 ) ∩ K = ∅. , k ∈ N, (F1(i0 ) , F2(i0 ) ) ∈ Fk . x ∈ H, ⎧ ⎫ ⎨ ⎬ x ∈ X − ∪ F2 : (F1 , F2 ) ∈ Fi , F2 ∩ H = ∅ ⎩ ⎭ ik ⊂X −∪ ⎧ ⎨ ⎩ F1 : (F1 , F2 ) ∈ ik ⎫ ⎬ Fi , F2 ∩ H = ∅ . ⎭ n0 ∈ N, j n0 ⎧ ⎫ ⎨ ⎬ xnj ∈ X − ∪ F1 : (F1 , F2 ) ∈ Fi , F2 ∩ H = ∅ ⎩ ⎭ ik ⎫ ⎧ ⎬ ⎨ Fi , F2 ∩ H = ∅ . ⊂ X − ∪ F1 : (F1 , F2 ) ∈ ⎭ ⎩ ik , (8.2.3), (8.2.2) ⎧ ⎫ ⎨ ⎬ (i ) xnj ∈ F1 0 − ∪ F1 : (F1 , F2 ) ∈ Fi , F2 ∩ H = ∅ ⊂ Uk . ⎩ ⎭ ik Uk xn , {xn } . H ⊂ D(H, K). D(H, K) ⊂ X − K. , , D(H, K) ⊂ X − K, x ∈ D(H, K) ∩ K ∩ (X − H) ⊂ D(H, K) − H ∩ K. . 8.2 ℵ · 279 · X {xn } ⊂ ( n∈N Un )◦ − H xn → x. {xn } {xnj } (F1 , F2 ) ∈ Fm xnj ∈ F1 , F2 ∩ H = ∅, j ∈ N. Uk ⊂ X − F1 , k m. k m , xnj ∈ / Uk . xnj ∈ i n(1) fn(2) (P2 ) ⊂ R2 . , fn(3) n(3) > n(2) fn(3) (P3 ) ⊂ R1 fn(4) n(4) > n(3) fn(4) (P4 ) ⊂ R2 , · · ·. , Pi , Rj 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, · · · . R1 Rj ( Rj ), R1 . fi = fn(i) , xi ∈ Pi fi (xi ) Rj , Rj fn(i) Pi , {fi } F f0 , P (x) S x , {xi } x . · 288 · 8 () {fi (xi )} f0 (x) ( 8.1.3), {fi (xi )} U . n Rk ∈ R(x) fi (xi ) ∈ Rk i n , {fi } {xi } , m > n fm (xm ) ∈ / Rk . . x ∈ C, l(x), i(x) j(x), Fl(x) ⊂ (Pi(x) , Rj(x) ) ⊂ (x, U ). {Pi(x) : x ∈ C} C, {Pi(x1 ) , Pi(x2 ) , · · · , Pi(x }, r) m = max1tr {l(xt )}, Fm ⊂ 1tr (Pi(x , Rj(xt ) ) ⊂ (C, U ). t) [P, R] C (S, Y ) σ cs . Y , 8.1.2, C (S, Y ) , C (S, Y ) cs-σ . . Guthrie , Foged ( 8.3.2) , O’Meara [330] , Michael [289] ( 8.2 ). 8.3.4 [120] X ℵ0 , Y ℵ , C (X, Y ) ℵ . 8.4 σ k Lašnev Lašnev [240, 241] . . Foged [122] Lašnev . Lašnev Foged ( 8.4.1) , Burke, Engelking Lutzer [73] σ . 8.4.1 P X , P P ∩P . , P P {Pα : α ∈ A} Gα ⊂ ∩Pα (α ∈ A) α∈A Gα . X , {zn } ⊂ / {zn }. α∈A Gα zn → x ∈ X − α∈A Gα , α(n) ∈ A zn ∈ Gα(n) , x ∈ , α(n) . , m ∈ N, nm Pα(n) , {nm } P {Pm } [363] Pm ∈ Pα(n − {Pk : k < m}, m) Pm , znm ∈ Pm . P x∈ / {znm : m ∈ N}, . . 8.4 σ k Lašnev · 289 · 8.4.2 [122] P T1 X , {zn } X − {x} x, m ∈ N, {zn : n m} ∩ P = ∅ P ∈ P . , {zn } {znm } P {Pm : m ∈ N} znm ∈ Pm , m ∈ N. P X T1 {znm : m ∈ N} , . . 8.4.3 [122] X T2 , Fréchet , k P = n∈N Pn , Pn ⊂ Pn+1 . U , Z = {zn : n ∈ N} x ∈ U − Z, n ∈ N Z Int(∪{P ∈ Pn : P ⊂ U }). m ∈ N, Pm = {P ∈ Pm : P ⊂ U }. , Z {znm } znm ∈ U − Int(∪Pm ) ⊂ U − ∪Pm (m ∈ N). X Fréchet , znm , {znm ,k } ⊂ U −∪Pm znm ,k → znm . x ∈ {znm ,k : m, k ∈ N}. X Fréchet , x ∈ / Z T2 , x, Z Z {znm ,k : m, k ∈ N} Z Z = {znmi ,ki : i ∈ N} (mi < mi+1 ; i ∈ N). x∈ / Z ⊃ {znm : m ∈ N}, Z x. P k , m ∈ N Z , mj > m , znmj ,kj ∈ U − ∪Pm . , ∪Pm . 8.4.1 (Foged [122] ) X Lašnev X T2 , Fréchet σ k . . T2 , Fréchet X k P = n∈N Pn , Pn , 8.4.1, Pn . , Pn ⊂ Pn+1 (n ∈ N). Rn (P ) = P − Int(∪{Q ∈ Pn : P ⊂ Q}), P ∈ Pn ; (8.4.1) Rn = {Rn (P ) : P ∈ Pn }. (8.4.2) 4 . 1. Z = {zn : n ∈ N} x ∈ X − Z, Rn∗ = {R ∈ Rn : R ∩ Z }. (8.4.3) U x Z Int(∪{P ∈ Pn : P ⊂ U }), Z Int(∪Rn∗ ) ∪Rn∗ ⊂ U . · 290 · 8 () V = Int(∪Pn ) − ∪{Q ∈ Pn ∪ Rn : Q ∩ Z }. (8.4.4) 1 (8.4.4) Z ( 8.4.2), Z V , Z Int(∪Rn∗ ) ∪Rn∗ ⊂ U . V ⊂ ∪Rn∗ . Z Int(∪Rn∗ ). y ∈ V , Pn y . , y ∈ Q ∈ Pn , Q ∩ Z , 8.4.2, Pn Q , Pn y . P (y) = ∩{Q ∈ Pn : y ∈ Q}, . Pn , P (y) ∈ Pn , y∈ / ∪{Q ∈ Pn : P (y) ⊂ Q}. (8.4.1), (8.4.2), y ∈ Rn (P (y)) ∈ Rn , Rn (P (y)) ∩ Z . (8.4.3), Rn (P (y)) ∈ Rn∗ , y ∈ Rn (P (y)) ⊂ ∪Rn∗ . V ⊂ ∪Rn∗ , Z Int(∪Rn∗ ). ∪Rn∗ ⊂ U . Rn (P ) ∈ Rn∗ , Rn (P ) ⊂ P , {Q ∈ Pn : Q ⊂ U } Q P ⊂ Q, Rn (P ) ⊂ P ⊂ Q ⊂ U. ∪Rn∗ ⊂ U . , 1 (8.4.1) ,Z Int(∪{Q ∈ Pn : Q ⊂ U }) ⊂ Int(∪{Q ∈ Pn : P ⊂ Q}) ⊂ X − Rn (P ). Rn (P ) ∩ Z , (8.4.3) n ∈ N, . ∪Rn∗ ⊂ U . 1 . Rn = Rn ∪ {X − Int(∪Rn )} = {Rα : α ∈ In }. M = {σ = {σ(n)} ∈ n∈N In : {Rσ(n) }n∈N x ∈ X }. 8.4 σ k Lašnev · 291 · 7.3.21 . In (n ∈ N) , In (n ∈ N) . n∈N In , M . σ = {σ(n)} ∈ M , σ(n) ∈ In , Rσ(n) {Rα : α ∈ In } , {Rσ(n) }n∈N . f : M → X, f (σ) = x {Rσ(n) }n∈N x . 2. f (M ) = X. x , {x} ∈ Pn . (8.4.1) Rn ({x}) = {x}; x , Z X − {x} x, Rσ(n) ∈ Rn Rσ(n) ∩ Z , , Rσ(n) = X − Int(∪Rn ). , 8.4.2, x ∈ Rσ(n) , 8.4.3 1, {Rσ(n) }n∈N x . 2 . 3. f . U X , x ∈ U , σ = {σ(n)} ∈ M {Rσ(n) }n∈N x , n ∈ N Rσ(n) ⊂ U . M σ () B, B n σ n , f (B) ⊂ Rσ(n) ⊂ U . 3 . 4. f . F M , Z = {zn : n ∈ N} f (F ) x ∈ X − Z. n ∈ N, σn ∈ F ∩ f −1 (zn ), σn ( zn ) M . S0 = N. m ∈ N, Sm ⊂ Sm−1 τ (m) ∈ Im . 8.4.2, i ∈ N R ∗ = {R ∈ Rm : R ∩ Z } : R ∩ {zn : n i} = ∅} ⊂ {R ∈ Rm , n i, Rσn (m) ∈ R ∗ . 8.4.2, Rσn (m) . σn (m) , Sm ⊂ Sm−1 n ∈ Sm , σn (m) . σn (m) τ (m), τ (m) = σn (m). τ = {τ (m)} f −1 (x). m ∈ N, zn ∈ Rσn (m) = Rτ (m) n ∈ Sm , x ∈ m∈N Rτ (m) . U X , x ∈ U , 8.4.3 1, n ∈ N Rτ (m) ∈ Rn Rτ (m) ⊂ U . {Rτ (m) }m∈N x , f (τ ) = x. nm ∈ Sm m ∈ N nm < nm+1 , {σnm } τ . , m k, nm ∈ Sk , σnm (k) = τ (k). τ ∈ F, x ∈ f (F ), f (F ) . . · 292 · 8 () , . Fréchet , ( 8.3); , ; ( 6.6.9), k . . 6.6.9 7.5.1, T2 ℵ , σ k . σ k ℵ ? [168, 249] Foged . , Foged , k Fréchet Lašnev . 8.4.4 [247] P X , P = {P : P ∈ P} X . P = {Pα }α∈A . P X , α ∈ A, Hα ⊂ P α , α∈A H α . x ∈ α∈A Hα − α∈A H α . , α ∈ A, Vα , Uα , x ∈ Vα , H α ⊂ Uα Vα ∩ Uα = ∅, Hα ⊂ Uα ∩ P α ⊂ Uα ∩ Pα . x ∈ ∪{Uα ∩ Pα : α ∈ A} = ∪{Uα ∩ Pα : α ∈ A}. β ∈ A, x ∈ Uβ ∩ Pβ , Uβ ∩ Pβ ∩ Vβ = ∅, . P X . 8.4.1 [122] X X σ k . 8.4.4 Foged , σ k Lašnev , Morita-Hanai-Stone ( 4.4.2), Lašnev . . 8.4.2 [329] X X σ k . 8.4.3 (Burke-Engelking-Lutzer [73] ) X X σ . σ X . , σ , x ∈ X Gδ , {x} = n∈N Gn , Gn . P x . x X , P , X . P , P = {Pn : n ∈ N}. D1 = P1 ∩ G1 , Dn = Dn−1 ∩ Pn ∩ Gn (n 2). 8.4 σ k Lašnev · 293 · P1 ∩ G1 x , x D1 − {x} . x Dn − Dn+1 (n ∈ N) , P x ∪{Dn − Dn+1 : n ∈ N} = D1 − {x} , . P . . ℵ , Foged 8.3.2 ℵ ; 8.4.1 Lašnev —— . BurkeEngelking-Lutzer [73] σ ( 8.4.3), Foged , σ k ℵ Foged . Junnila [225] ( 8.4.3), , Foged . 4.1.5 , (fan space) Sω1 . R () S = {0} ∪ {1/n : n ∈ N}. α < ω1 , Sα S. α<ω1 Sα , () Sω1 . α<ω1 Sα , , Sω1 Lašnev . Foged ( 8.4.1) Sω1 σ , k , 8.4.2 Sω1 ℵ . 8.4.1 [150] X P X cs∗ (cs∗ -network), X Z = {z} ∪ {zn : n ∈ N} (zn → z) U ⊃ Z, P ∈ P Z Z = {z} ∪ {zni : i ∈ N}, Z ⊂ P ⊂ U . , cs cs∗ , k cs∗ ( 8.14). 8.4.2 [251] Sω1 cs∗ , ℵ . a Sω1 , α < ω1 , Yα = Sα − {a}. Sω1 cs∗ P, {P ∈ P : a ∈ P α < ω1 Yα ∩ P = ∅} (8.4.5) {Pn : n ∈ N}. Pn , yn ∈ Pn − {a}, yn Yα , {yn : n ∈ N} Sω1 . V = Sω1 − {yn : n ∈ N}, V . a ∈ P ⊂ V , P ∈ P, P ∩ {yn : n ∈ N} = ∅, P ∈ / {Pn : n ∈ N}. (8.4.5) , P Yα . H = ∪{P ∈ P : a ∈ P ⊂ V }. (8.4.6) H , H Yα , β < ω1 Yβ ∩ H = ∅. · 294 · 8 () , {xn : n ∈ N} = V ∩ Yβ , xn → a. P cs∗ , {xni } P ∈ P {a} ∪ {xni : i ∈ N} ⊂ P ⊂ V . (8.4.6) , Yβ ∩ H = ∅, . Sω1 cs∗ . ℵ σ k ⇒ k ⇒ cs∗ , Sω1 ℵ . . , Sω1 k . 8.4.2, Sω1 k cs∗ , k cs . , βN ( 3.6.2), P = {{x} : x ∈ βN} βN cs , P βN k . , βN , P βN, βN , . 8.4.3 (Junnila-Yun [225] ) σ k X ℵ X Sω1 . 8.4.2). , ℵ , Sω1 ℵ ( . 8.4.5 [225] X T1 , σ , , X X = n∈N Xn Xn . F = n∈N Fn X σ , Fn . n ∈ N, Xn = {x ∈ X : {x} ∈ Fn }, Xn . x ∈ X, F , {F ∈ F : x ∈ F } = {Fk : k ∈ N}, Fk = Fn (k ∈ N), nk {Fk }k∈N x , k ∈ N Fk = {x}. , Fk , {xk } xk ∈ Fk xk = x, k ∈ N, {xk : k ∈ N} ∪ {x} X , . Fk Fn , x ∈ Xn . . 8.4.6 [327] F T1 X , K , K A (K − A) ∩ F = ∅ F ∈ F . . . 8.4.6 8.4.2 5.5.1 , 8.4.2 F , D = {x ∈ X : |(F )x | ω}, D . D . , (F )x = {F ∈ F : x ∈ F }. , ( 5.5.3) 8.4.2 8.4.6 T1 T0 . 8.4.7 [225] T2 X Sω1 . F X , x ∈ X F ∈ F F − {x} 8.4 σ k Lašnev · 295 · x. , a ∈ X F {Fα : α < ω1 } α < ω1 , Fα − {a} Yα = {yα,n : n ∈ N} yα,n → a. 8.4.2, F yα,n . {F ∈ F : F ∩ Yα = ∅} , {Yα : α < ω1 } , σ ( 6.1.3). ω1 , {Yα : α < ω1 } {Yα(β) : β < ω1 }, {Yα : α < ω1 } . F , X {a} ∪ ( α<ω1 Yα ) Sω1 , . . 8.4.8 [225] σ k F X Sω1 , D X , X σ H X K d ∈ K ∩ D, {H ∈ H : d ∈ IntK (K ∩ H)} d X . 8.4.4, X σ k F = n∈N Fn , Fn , Fn ⊂ Fn+1 . x ∈ X, F (x) = {F ∈ F : K ⊂ F x ∈ K − {x}}. (8.4.7) X σ ( 7.3.5), X ( 7.3.13). 8.4.7, F (x) , {∪F : F F (x) } ∪ {{x}} = {Fk (x) : k ∈ N}. d ∈ D n ∈ N, Gn (d) d , Gn (d) ⊂ X − ∪{F ∈ Fn : d ∈ / F }, Sn (d) = ∪{F ∈ Fn : F ∩ D = {d}}. Fn , {Gn (d) ∩ Sn (d) : d ∈ D} . n, k ∈ N, Hn,k = {Fk (d) ∩ Gn (d) ∩ Sn (d) : d ∈ D} . σ H = n,k∈N Hn,k . K ⊂ X, d ∈ K ∩ D, O d X . V d , V ⊂ O − (D − {d}). F k , F F K ∩ V ⊂ ∪F ⊂ O − (D − {d}). n ∈ N F ⊂ Fn , k ∈ N Fk (d) = ∪(F ∩ F (d)) ∪ {d}, · 296 · 8 () F ∈ F ∩ F (d) ⇒ d ∈ F , d ∈ Fk (d) ∩ Gn (d) ∩ Sn (d) ⊂ ∪F ⊂ O. d ∈ IntK (K ∩ Fk (d) ∩ Gn (d) ∩ Sn (d)) (8.4.8) . d ∈ IntK (K ∩V ) ⊂ IntK (K ∩(∪F )) F K ∩V X , d ∈ IntK (K ∩(∪(F )d )). (∪F )∩(D−{d}) = ∅, ∪(F )d ⊂ Sn (d), (F )d = {F ∈ F : d ∈ F }. , d ∈ IntK (K ∩ Sn (d)). d ∈ IntX Gn (d), d ∈ IntK (K ∩ Gn (d)). d ∈ IntK (K ∩ Fk (d)), (8.4.8) . F ∈ F , EF = X − F ∩ K − {d}, E = ∩{EF : F ∈ F d ∈ EF }, E d . F ∈ F , d ∈ EF , E ⊂ EF , K ∩ F ∩ E ⊂ K ∩ F ∩ EF ⊂ K ∩ F ∩ (X − (F ∩ K − {d})) ⊂ {d}; d∈ / EF , d ∈ F ∩ K − {d}, (8.4.7) F ∈ F (d). K ∩ (∪F ) ∩ E ⊂ ∪(F ∩ F (d)) ∪ {d} = Fk (d). d ∈ IntK (K ∩ (∪F )) d ∈ IntX E, d ∈ IntK (K ∩ Fk (d)). . 8.4.3 F = n∈N Fn X k , Fn , Fn ⊂ Fn+1 , n ∈ N. X Sω1 . Dn = {x ∈ X : |(Fn )x | ω} (n ∈ N). 8.4.6 8.4.5, Dn Dn = k∈N Dn,k , k ∈ N, Dn,k X Dn,k ⊂ Dn,k+1 . , n, k ∈ N, Hn,k X σ , 8.4.8 ( H = Hn,k D = Dn,k ). n ∈ N, Fn , X −Dn , X −Dn . n, k ∈ N, Sn,k = ∪{F ∈ Fk : F ∩ Dn = ∅} Fn,k = {F ∩ Sn,k : F ∈ Fn }, Fn,k X . σ ⎞ ⎛ ⎞ ⎛ P=⎝ Hn,k ⎠ ∪ ⎝ Fn,k ⎠ n,k∈N n,k∈N 8.4 σ k Lašnev · 297 · X k . K ⊂ O, O X , K . F F K ⊂ ∪F ⊂ O, n ∈ N F ⊂ Fn . 8.4.6 , K ∩ Dn , k ∈ N K ∩ Dn ⊂ Dn,k . 8.4.8, d ∈ K ∩ Dn , Hd ∈ Hn,k d ∈ IntK (K ∩ Hd ) Hd ⊂ O. K = K − ∪{IntK (K ∩ Hd ) : d ∈ K ∩ Dn }, K K ⊂ X − Dn (). F F K ⊂ ∪F ⊂ X − Dn . l ∈ N F ⊂ Fl , ∪F ⊂ Sn,l K ⊂ Sn,l . P = {Hd : d ∈ K ∩ Dn } ∪ {F ∩ Sn,l : F ∈ F }, P P , K ⊂ ∪P ⊂ O. . Sω1 cs∗ ( 8.4.2), k . . 8.4.4 [152] X ℵ X σ k k . 8.4.5 [261] X ℵ X σ k cs∗ . 8.3.1 cs ⇒ k . , σ σ . ? . . 8.4.4 [320] P T2 X σ cs∗ , P X σ k . K T1 F , F F K ( 5.5.1). P = n∈N Pn X σ cs∗ , Pn . X K, U ⊃ K n ∈ N, Pn = {P ∈ Pn : P ⊂ U }, Fn = ∪Pn . K Fn . , K {xn } xn ∈ K − in Fi . 8.3.1 (ii) ⇒ (i) ( T2 ), K ⊂ in Fi . 5.5.1, in Pi P K ⊂ ∪P ⊂ U . . cs k ( 8.4.2 ). , , cs . Junnila-Yun ( 8.4.3), σ k · 298 · 8 () ℵ . [249] “ σ cs ”? , 8.4.5. 8.4.9 [256, 431] Sω1 σ cs . Sω1 σ cs F . Sω1 ℵ Sω1 = {(1/n, α) : α < ω1 , n ∈ N} ∪ {(0, 0)}. . α < ω1 , S (α) = {(1/n, α) : n ∈ N}. F σ , 8.4.2, {F ∈ F : |S (0) ∩ F | = ω} F . , {F ∈ F : |S (0) ∩ F | = ω} . , α, 0 < α < ω1 , Fα ∈ F zα ∈ S (α) ∩Fα , α = β , Fα = Fβ . α, 0 < α < ω1 , S (0) ∪ S (α) , S (0) , S (α) , (0, 0). F cs , F1 ∈ F S (0) ∪ S (1) F1 , z1 ∈ S (1) ∩ F1 . , |S (0) ∩ F1 | = ω. α < ω1 , β < α (β > 0), Fβ ∈ F zβ ∈ S (β) ∩ Fβ |S (0) ∩ Fβ | = ω. , Sω1 − {zβ : β < α} (0, 0) ∈ Sω1 − {zβ : β < α}. Fα ∈ F S (0) ∪ S (α) Fα Fα ⊂ Sω1 − {zβ : β < α}. , |S (0) ∩ Fα | = ω, zα ∈ S (α) ∩ Fα . Fα ⊂ Sω1 − {zβ : β < α}, β < α, Fα = Fβ . {F ∈ F : |S (0) ∩ F | = ω} . . 8.4.5 [254, 431] X ℵ X σ cs . Foged ( 8.3.2) . . X σ cs F . 8.4.4, F X σ k . , X Sω1 ( , F |Sω1 Sω1 σ cs . 8.4.9 ). 8.4.3 X ℵ . . ℵ , Sω1 Lašnev () ℵ , ℵ . , ℵ ( 8.5). Lindelöf ℵ . 8.4.10 [246] f : X → Y X Y Lindelöf , f . K Y , f −1 (K) X Lindelöf ( 3.3.3). X , f −1 (K) . g = f |f −1 (K) , g T2 f −1 (K) K . 6.6.2, g , f −1 (K) C g(C) = K, C X f (C) = g(C) = K. f . . (0, 0) Sω1 8.4 σ k Lašnev · 299 · 8.4.6 [246, 380] f : X → Y ℵ X Y Lindelöf , Y ℵ . ℵ X σ k P, Y , f : X → Y Lindelöf . f (P) = {f (P ) : P ∈ P}, 8.4.10, f , f (P) Y σ k . 8.3.1 , Y ℵ . . 8.4.11 [390] f : X → Y X Y , f Lindelöf ( y ∈ Y, ∂f −1 (y) Lindelöf ) Y Sω1 . f Lindelöf , M Lindelöf g : M → Y ( 4.4.8 ). 8.4.6, Y ℵ . 8.4.3, Y Sω1 . , f Lindelöf , y ∈ Y , ∂f −1 (y) X Lindelöf , ∂f −1 (y) {xα : α < ω1 }. X , X {Dα }α<ω1 , xα ∈ Dα . f , {f (Dα )}α<ω1 Y . α < ω1 , V y Y , f −1 (V ) ∩ (Dα − f −1 (y)) = ∅, V ∩ (f (Dα ) − {y}) = ∅, y ∈ f (Dα ) − {y}. Y Fréchet ( 8.3), f (Dα ) − {y} y. 8.4.7 . . 8.4.6 [152, 246] (i) X Fréchet ℵ ; (ii) X Lindelöf . (i) ⇒ (ii). X Fréchet ℵ . Foged 8.4.1, X Lašnev , M f : M → X. X ℵ , X Sω1 . 8.4.11, f Lindelöf . A Lindelöf g : A → X ( 4.4.8 ). X Lindelöf . (ii) ⇒ (i). 8.4.6 Fréchet . . 4.4.1 () ℵ . ℵ . 8.4.1 ℵ . X = {(x, y) ∈ R2 : y 0}. X V · 300 · 8 () (i) y > 0, (x, y) X ; (ii) y = 0, (x, y) V (x, n) = {(t, s) ∈ X : t = x ± s, 0 s 1/n}, (n ∈ N). X V , Heath V [183] . , X . R × {0} , ( 2.2.3), X . X k , X , 7.5.5, X , X . X k , X ℵ ( 8.2.1). r ∈ R, Xr = {(x, y) ∈ R2 : y = |x − r|}, Xr X ( σ ), {Xr : r ∈ R} X . M = r∈R Xr , f : M → X ( 3.4.9 ), M f . ℵ . ( 4.4.3). [249] “ℵ ?” [431] . . k . Guthrie [173] cs , Siwiec [359] , cs . Michael ( 5.5.1) 8.4.2 [359] f : X → Y (sequence-covering), Y {yn } y ∈ Y , xn ∈ f −1 (yn ) (n ∈ N) x ∈ f −1 (y) {xn } x. , cs . 4.4.9 , . 8.4.12 [144] X Gδ . f : X → Y X Y , Y f . f , Y T1 . y ∈ Y V y , f , x ∈ f −1 (y) x U, y ∈ Intf (U ). X , X U x ∈ U ⊂ U ⊂ f −1 (V ), y ∈ Intf (U ) ⊂ f (U ), y ∈ Intf (U ) ⊂ Intf (U ) ⊂ f (U ) ⊂ V. Y . {yn } Y y ∈ Y , {yn } y . f , x ∈ f −1 (y) x U, y ∈ Intf (U ). X Gδ , {Ui }i∈N x {x} = i∈N Ui 8.4 σ k Lašnev · 301 · U i+1 ⊂ Ui , i ∈ N. i ∈ N, m(i) ∈ N n m(i) , yn ∈ Intf (Ui ), −1 Ui ∩ f (yn ) = ∅. m(i + 1) > m(i). {xj } −1 j < m(1), xj ∈ f (yi ); m(i) j < m(i + 1) , xj ∈ Ui ∩ f −1 (yj ). f , {xj } ( , {xj : j ∈ N} , {yj : j ∈ N} , ). E {xj } , E ⊂ U i , i ∈ N. E⊂ Ui = i∈N Ui = {x}. i∈N {xj } x , {xj } x . {xj } x, f . . 8.4.7 [144] f : X → Y ℵ X Y , Y ℵ . ℵ 8.4.12 . 8.4.5 Y ℵ . . , , f , cs . 8.4.7 [431] f : X → Y ℵ X , Y ℵ . ℵ . Y 8.4.8 [250] {Fα }α∈A X ( ) , Fα (α ∈ A) ℵ , X ℵ . ℵ ( 8.4.6) 5.5.7 . . ℵ . Sω1 , Sω1 8.4.1 . α < ω1 , Sα ℵ . α<ω1 Sα Sω1 q, q . {Sα : α < ω1 } α<ω1 Sα , {q(Sα ) : α < ω1 } Sω1 , q(Sα ) Sα , Sω1 ℵ . Sω1 ℵ , Sω1 ℵ ( 8.4.2). Sω1 , α < ω1 , Sα = {0} ∪ {xα,n : n ∈ N}. n ∈ N, Xn = {xα,n : α < ω1 }, Xn Sω1 , ℵ . Sω1 = {a} ∪ ( n∈N Xn ), a Sω1 , ℵ . ( 5.2.1), 8.4.7 . M. Sakai . · 302 · 8 () 8.4.8 (M. Sakai, 2007) f : X → Y ℵ X Y , Y ℵ . X ℵ , P = n∈N Pn X σ k , Pn , Pn ⊂ Pn+1 , n ∈ N. 7.5.1, f , f (P) = {f (P ) : P ∈ P} Y σ k . Junnila-Yun ( 8.4.3), , Y Sω1 . , Y Sω1 , {∞} ∪ {yα,n : α < ω1 , n ∈ N}, {yα,n } α ∞ . P {Fα }α<ω1 (i) {nα }α<ω1 ⊂ N α < ω1 , ∪{f −1 (yα,n ) : n nα } ⊂ ∪Fα ; P ∈ Fα , P ∩ (∪{f −1 (yα,n ) : n nα }) = ∅; (iii) α < β < ω1 , Fα ∩ Fβ = ∅. (ii) γ < ω1 α < γ nα ∈ N P Fα . α < γ, (i), ∪{f −1 (yα,n ) : n nα } Fα Fα ∩ P = ∅ P ∈ Fα . F = ∪{Fα : α < γ}, F X . n ∈ N, Qn = {P ∈ Pn : P ∩ F = ∅ P ∩ f −1 (yγ,k ) = ∅ k ∈ N }. , Qn ⊂ Qn+1 . Qn , {Pi }i∈N ⊂ Qn i = j Pi = Pj . {xi } xi ∈ Pi {f (xi )} {yγ,n }n∈N . Qn ⊂ Pn , {xi : i ∈ N} , {f (xi ) : i ∈ N} , . Qn . nγ ∈ N ∪{f −1 (yγ,n ) : n nγ } ⊂ ∪Qnγ , Fγ = Qnγ . , n ∈ N, f −1 (yγ,k ) ⊂ ∪Qn k ∈ N . kn ∈ N xn ∈ f −1 (yγ,kn ) − ∪Qn . kn < kn+1 , n ∈ N. f , {xn : n ∈ N} X . {xn : n ∈ N} , f −1 (∞) . , x0 {xn } , {f (xn )} {yγ,n }n∈N , f (x0 ) = ∞, x0 ∈ f −1 (∞). X ℵ , {x0 } Gδ , {Gi }i∈N {x0 } = i∈N Gi , i ∈ N Gi+1 ⊂ Gi , Gi ∩ F = ∅. 8.5 · 303 · {xn } {zi} zi ∈ Gi (i ∈ N), {zi } x0 ( 8.4.12 ), . P X k {x0 } ∪ {zi : i ∈ N} ⊂ X − F , l ∈ N P ∈ Pl , P {xn } P ⊂ X − F , P ∈ Ql . n l, xn ∈ X − ∪Ql , ∪Ql {xn } , . . Fα , m ∈ N {αi : i ∈ N} ⊂ ω1 i ∈ N Fαi ⊂ Pm . Ei = ∪Fαi (i ∈ N). Pm , {X − ji Ej : i ∈ N} X , f −1 (∞). f , i0 ∈ N f (X − ji0 Ej ) ∞ Y . , (i), j i0 , n nαj , f −1 (yαj ,n ) ⊂ Ej , ⎞ ⎛ yαj ,n ∈ Y − f (X − Ej ) ⊂ Y − f ⎝X − Ej ⎠ . ji0 f (X − . ji0 Ej ) ∞ . Y Sω1 . 8.5 . , . . 8.5.1 [135] X q r , Y , p : X × Y → X ? 8.5.2 (§ 5.2) ( )? 8.5.3 [217, 230] (§ 6.1) 6.1.2 () (ii) 6.1.4 (θ ) (ii) “ ” “ ”? 8.5.4 [143, 367] (§ 6.1) δθ δθ ? 8.5.5 [40] (§ 6.2) Lindelöf ? 8.5.6 (§ 6.2) T2 , meso ? [66, 369] 8.5.7 (§ 6.2) θ ? 8.5.8 [369] (§ 6.2) θ ? [66] 8.5.9 (§ 6.2) ortho ? 8.5.10 (§ 6.2) Meso Lindelöf meso ? · 304 · 8 8.5.11 (§ 6.2) θ ? () Lindelöf T2 8.5.12 (§ 6.2) δθ δθ ? θ θ 8.5.13 (§ 6.5) θ Lindelöf δθ δθ δθ ? θ δθ meta- 8.5.14 [143] (§ 6.6) () ? 8.5.15 [143] 8.5.16 [135] (i) P B () ? P ; (ii) P ; (iii) P . θ ? X P , X 8.5.17 [135] (i) P P P ; (ii) P ; (iii) P ; (iv) P . X θ P , X P ? . 8.5.18 [134] Σ Σ wσ β ? 8.5.19 [80] (§ 7.4) M2 ⇒ M1 ? 8.5.20 [141, 204] M1 ? M1 ? M1 8.5.21 [141] (∗) ? 8.5.22 [141] M0 ? 8.5.23 [141] µ ? 8.5.24 [141] σ 8.5.25 [141] P ? ? ? [427] 8.5.26 (§ 7.5) f : X → Y , X, Y T2 . X k , Y k ? 8.5.27 [139, 264] (§ 8.2) σ k ? k σ 8 · 305 · 8.5.28 [168, 249] (§ 8.4) σ k ? 8.1 [291] ℵ 8 T2 X X . X q (q-space [283] ), x ∈ X, x 8.2 {Un (x)}n∈N xn ∈ Un (x) ⇒ {xn } . X (pointwise countable type [20] ), x ∈ X, K x ∈ K K X . , r ; r q . Gδ , r, q, [287] . X Fréchet (strongly Fréchet space [359] ), X 8.3 {An }n∈N x ∈ An (n ∈ N), xn ∈ An {xn } x ( Fréchet ). , Fréchet ⇒ Fréchet ⇒ An = A , ( 2.3.1 ). [125] Fréchet [359] . [125] [19, 125] 8.4 [137] Fréchet [19, 125] [359] . X () , (i) X M1 ; (ii) X σ 8.5 [358] 8.6 ℵ . σ k σ k . Lindelöf 8.7 k F , F ∈ F . [284] 8.8 ℵ . X T2 , k Y . , (i) X cosmic ; (ii) X Lindelöf σ ; (iii) X σ . 8.9 [284] (i) X Lindelöf ℵ0 ( ℵ0 ), X ℵ0 ; (ii) X Xn , Xn , X . 8.10 [245] T2 . σ ℵ0 . 8.11 [328] · 306 · 8 () (i) X ℵ0 ; (ii) X Lindelöf ℵ ; (iii) X ℵ . 8.12 . , . 8.13 K ⊂ C (X, Y ) . x ∈ X, φ(x) = {f (x) : f ∈ K} Y 8.14 k cs∗ . 8.15 P T2 , k X . , P = {P : P ∈ P} . X P [73] (weakly hereditarily closure- 8.16 preserving), x(P ) ∈ P ∈ P, {{x(P )} : P ∈ P} ; P (countably weakly hereditarily closure-preserving), P [227] 8.17 8.18 Chaber . Fréchet , σ [258] [83] . T1 . k σ ( σ σ ). 8.19 [263] X Lašnev X Fréchet σ k . 8.20 C (X, Y ) X Y W (C, U ) = {f ∈ C (X, Y ) : f (C) ⊂ U }, C X , U Y . . X , C (X, Y ) Y |X| = {Yx : x ∈ X}, |X| X , Yx = Y (x ∈ X). 8.21 n ∈ N, (i) (ii) (iii) 8.22 8.23 C (X, Y ) W (C, U ) . n n i=1 W (Ci , U ) = W ( i=1 Ci , U ); n n i=1 W (C, Ui ) = W (C, i=1 Ui ); n n n i=1 W (Ci , Ui ) = W ( i=1 Ci , i=1 Ui ). C −C W (C, U ) ⊂ W (C, U ), “ y0 ∈ Y , cy0 : X → Y y → cy Y → C (X, Y ) Y ” . , cy0 (x) = y0 , x ∈ X. C (X, Y ) . Y C (X, Y ) . 8.24 [12] 8.25 C (X, Y ) T2 f : X → Y (sequentially quotient mapping [53] Y T2 . [144] (weak sequence-covering mapping) ), Y {yn } 8 · 307 · y, {yn } {ynk } xk ∈ f −1 (ynk ) (k ∈ N), x ∈ f −1 (y) {xk } x. , ( 8.4.2) . (i) cs∗ (, k , cs ); (ii) Fréchet ( Gδ ) ; (iii) Y Fréchet Fréchet (iv) ( X) Y ; . [1] Alexander C C. Semi-developable spaces and quotient images of metric spaces. Paciﬁc J Math, 1971, 37: 277–293. [2] Alexandroﬀ P S. Sur les ensembles de la premiére classe et les ensembles abstraits. C R Acad Paris, 1924, 178: 185–187. [3] Alexandroﬀ P S. On bicompact extensions of topological spaces (in Russian). 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() ℵ1 85 α 156 B(x, ε) 256 βX 72 C[a, b] 124 β F2 45 βN F3 44 F4 42 c 4 Fσ ∩ 9 Fσ 246 73 54 1 χ(X) 37 Fσ 139 χ(x, X) Gδ 200 ∪ Gδ Gδ 200 266 1 δθ γ 9 37 267 G∗δ 218 G∗δ 218 ∈ 1 I 15 L2 [a, b] 124 Q 15 N (A) 79 R+ 11 Sω 1 293 P 240 Sω 89 µ 246 Sε (x) 8, 79 ∈ / 1 X/A 28 ω 5 X/R 28 ω ⇔ 1 ω1 5 ⇒ 1 ∂ 16 Σ Σ 220, 265 Σ # Σ ℵ ℵ0 ℵ1 5 σ σ 224 σ 268 σ 141 99 213 σ 268 130 131 224 5 ℵ0 63 σ 265 ∗ 165 99 σ 167 σ 167 · 329 · σ ∗ < 112 ⊂ 1 θL θ 188 139 (FB1)∼(FB2) (Fl1)∼(Fl3) 17 (G1)∼(G3) 113 (I1)∼(I4) 201 18 14 (M1)∼(M3) 76 θ 159 (N1)∼(N5) 8 θ 159 (NB1)∼(NB4) 10 ε 89 (O1)∼(O3) ε 79 (TG1)∼(TG2) 7 113 ∅ 1 (U1)∼(U4) 1 ∗ Lindelöf 197 ∧ 112 Čech 207 cs 112 280 cs-σ () 281 cs∗ 293 g 229 A Alexandroﬀ 69 k 252 Alexandroﬀ k 252 Alexandroﬀ-Urysohn k k k 61 268 k 268 k 148 l2 78 m n(X) 54 o(X) 5 p pγ 236 r 270 r 270 s 260 w(X) 36 Baire 96 Baire 96 104 Bing-Nagata-Smirnov 157 Birkhoﬀ-Kakutani Burke 172 C Cantor 94 Cantor-Bernstein 2 305 120 B Bing 208 q 55 CCC CH 49 5 Chaber Cl 218 12, 27 cosmic 272 D (B1)∼(B2) 9 (C1)∼(C4) 12 de Morgan (D1)∼(D4) 15 Dowker 154 (F1)∼(F3) 9 Dowker 155 1 4 126 104 · 330 · F Filippov Fr 262 16 Mi 228 meso 159 meta 159 Fréchet 37 meta-Lindelöf Frink 203 Miščenko G Gruenhage-Junnila Guthrie 232 286 H Hanai-Ponomarev Hausdorﬀ Hausdorﬀ 111 33 33 Heath-Hodel Hilbert 249 30 165 Miščenko 258 Michael 136 Michael 227 Michael 142 MOBI 196 MOBI1 263 Moore Moore 125 199 Morita-Hanai-Stone 14, 182 iso Nagata 187 neat 188 Niemytzki 173 Junnila-Yun 11 294 Niemytzki K O Kuratowski Kuratowski 242 Nagata-Smirnov J Junnila 12 58 ord(x, U ) ortho 224 Lašnev 245 Lindelöf 37 Lindelöf 138 164 P pressing down lemma pure 188 246 Siwiec-Nagata 40 203 Sorgenfrey 39 M0 246 st(A, U ) 101 M1 227 st(x, U ) 101 202 M M1 228 214 Smirnov Sorgenfrey M 178 S M M 11 158 L Lašnev 110 N I Int 260 Stone 11 99 M2 227 Stone-Čech 72 M3 227 Stone-Čech 72 103 · 331 · () T T0 T0 32 32 T1 T3 33 T5 34 35 35 142 Tietze 43, 65 totally 125 Tukey Tychonoﬀ Tychonoﬀ 43 Tychonoﬀ 30 68 46 Tychonoﬀ 35 U Urysohn 97 W 202 219 w∆ 202 Weil 126 Worrell 172 Z Zermelo 6 Zermelo 6 ZFC 201 Zorn 6 229 256 256 40 39 3 107 96 96 78 12 56 Tychonoﬀ w∆ 38 Tychonoﬀ w∆ 229 6, 56 Tychonoﬀ 34 Tamano 31 33 T5 273 B 33 T4 T4 33 T3 31 263 32 T2 T2 32 T1 T1 A 130, 239 145 9 34 4 4 213 138 22 27 k 268 16 16 110 1 1 5 192 191 · 332 · 1 37 37 82 258 C 228 228 µ 246 1 17 6 47 197 158 9 159 69 19 2 254 254 134 86 76 19 19 76 79 256 2 11 11 118 14 76 2 131 135 131 D 74, 106 7 256 256 117 124 2 49, 117 86 k 276 4 226 F 15 15 35 66 37 Lindelöf 37 196 131, 227 101 ortho 9 159 164 15 305 146 146 19 15 15 2 28 165 · 333 · 32 1 37, 51 14 1 5 127 82 71 133 G 35 35 19, 20 14 79 199 156 246 T2 66 222 Arens 41 5 45 4 273 51 110 46 22 7 60 4 30 207 17 59 107 48 265 144 30 131 176 4 1 159 46 148 68 9 12 48 J 212 5 45 1 H 110 66 203 Fσ 2 246 2 145 1 5 47 134 66, 239 19 146 3 ortho 196 · 334 · 14, 18, 19 meso 170 76 meta 170 76 117 76 107 K 1 9 7 180 217 217 42 8 L 79 124 4 4 49 8 8 138 22 77 27 ortho 126 126 91 46 47 124 256 97, 119 3 97 135 256 6 37 35 155 49 8 147 150 41 10 21 3 170 239 17 18 4 M 63 49 2 130 170 N 306 14 14 196 136 5 θ 170 135 167 226 · 335 · 113 R 125 201 75, 136 159 306 306 201 k 189 [ω1 , ∞)r 58 δθ 165 δθ 165 2 2 θ̄ O 162 θ̄ 162 77 θ 159 7, 76 θ 159 79 S P 149 3 3 293 28 28 28 8 80 Q 5 32 3 3 4 5 18 46 41 46 166 17, 18 19 15 220 1 Σ 224 # 224 Σ ∗ Σ 246 136 T Fréchet 305 71 37 71 76 4 7, 8 3 22 135 3 2 89 7 188 126 64 22 22 · 336 · 54 3 7 11 113 11 36 22 5 30 48 22 W 27 2 101 83 166 58 166 83 47 B 198 b1 198 92 126 35 43 19 3 12 19 281 89 306 54, 224 54 76 268 37 12 5 64 6 Y 76 p 76 15 5 X 78 7 145 2 30 167 3 3 3 211 126 112, 123 112, 123 112, 123 119 62 42 208 p 136 64 5 76 300 121, 123 122 146 · 337 · 36 36 2 139 197, 207 57 57 38 38 105 17 246 5 1 3 65 2 136 1 121 34 Moore 5 265 33 85 2 1 19, 281 18 37 1 27 28 18 1 3 4 5 192 4 5 201 28 107 Z 146, 265 6 3 2