Topologia 2015
111
Σηmειώσεις Γενικής Τοπολογίας Σηmειώσεις Μ. Γεραπετρίτη από τις παραδόσεις (διορθώσεις, 2015) Τmήmα Μαθηmατικών Πανεπιστήmιο Αθηνών Αθήνα, 2013
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Transcript of Topologia 2015
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. (, 2015)
, 2013
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ii
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1 31.1 , . . . . . . . . . . . . . . . . . . 31.2 , , . . . . . . . . . 8
2 152.1 . . . . . . . . . . . . . . . . . . . . . . 152.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 253.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 . . . . . . . . . . . . . . . . . . . . . . . . 33
4 374.1 . . . . . . . . . . . . . . . . . . . . . . 374.2 . . . . . . . 414.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 515.1 T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Hausdorff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.5 . . . . . . . . . . . . . . . . . . . . . . . . 66
6 696.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 . . . . . . . . . . 73
7 777.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.2 . . . . . . . . . . . . . . . . . 817.3 Tychonoff . . . . . . . . . . . . . . . . . . . . . . . . . . 84
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1
8 878.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878.2 . . . . . . . . . . . . . . . . . . . . . . . . 908.3 . . . . . . . . . . . . . . . . . . . . . . . 928.4 Lindelof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
9 999.1 Urysohn . . . . . . . . . . . . . . . 999.2 Nagata-Smirnov-Bing . . . . . . . . . . . . . . . . . . . 101
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2
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1
, , , , . , , , . , , , .
1.1 ,
1.1.1. X . T X X, :
(i) , X T,
(ii) T T. n N G1, G2, . . . , Gn T
ni=1Gi T
(iii) T T. I Gi T i I
iI Gi T.
(X,T) X T ( T (X,T))
1.1.2. () (ii)
(ii) G1, G2 T, G1 G2 T
() (iii) I = iGi = , T (
(i) ).
.
-
4
1.1.3. (X, ) .
T = {A X : x A > 0 : B(x, ) A}
X ( ) X .
1.1.4. (X,T) X, T = T. , T .
1.1.5. X . X X:() T1 = P(X), . (X,T1) -, T1 = T X.
() T2 = {, X}, . (X, ) , X \ {x} T, x X. X , (X,T2) , X \ {x} / T2 x X .
() T3 = {} {A X : x0 A} ( x0 X), (x0). , X , (X,T3) , X \{x0} / T3. X = {a, b} ( a 6= b), T = {, X, {a}} a X (X,T) Sierpinski.
() T4 = {X} {A X : x0 / A} ( x0 X), (x0). X , (X,T4) -, X \ {x} / T4, x X \ {x0}.
() T5 = {} {A X : Ac X \ A }, . T5 X:
(i) T5 X T5, X \X = , .
(ii) G1, G2 T5. , G1 = G2 = G1 G2 = T5. G1 6= G2 6= , Gc1, Gc2 , Gc1 Gc2 , (G1 G2)c . G1 G2 T5.
(iii) I 6= Gi T5 i I. Gi = i I, iI Gi = T5. i0 I : Gi0 6= . (Gi0)c ,
(
iI Gi)c (Gi0)c, (iI Gi)c , -
iI Gi T5.
X , (X,T5) = (X,T1), . X , (X,T5) . , , , ( x 6= y, B(x, ) B(y, ) = , = (x,y)2 > 0). (X,T5) : G1, G2 , Gc1, G
c2 , X = X \ (G1 G2) = Gc1 Gc2, ,
.
() T6 = {} {A X : Ac X \ A }, . , (X,T6) X .
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1.1. , 5
1.1.6. T X, T2 T T1.
X . - , X.
1.1.7. X T1,T2 X. T1T2 X. (Ti)iI X, iI Ti X.
. ( ) (i) , , X T1 T2.
(ii) G1, G2 T1 T2. G1, G2 T1 G1, G2 T2. T1,T2, G1 G2 T1 G1 G2 T2. G1 G2 T1 T2.
(iii) Gi T1 T2 i I, I . i I,Gi T1 Gi T2. T1,T2 ,
iI Gi T1
iI Gi T2.
iI Gi T1 T2.
1.1.8. , iI Ti Ti -
X . , T X T Ti, i I, T
iI Ti.
() . , :
1.1.9. X = {a, b, c}, a, b, c , T1 = {, X, {a}},T2 = {, X, {b}}, T1,T2 X, T1 T2 -, {a}, {b} T1 T2 {a} {b} = {a, b} / T1 T2.
T1 T2. C, X.
1.1.10. X C P(X). T X T C.
. = {S P(X) : S X C S}. 6= P(X) . T =
{S : S }. , 1.1.7 T
X C T. S X C S, S , T S ( T). T C
1.1.11. (X,T) B T. B T, B., G T (Bi)iI B, G =
iI Bi.
B .
,
T =
{G X : (Bi)iI B, G =
iI
Bi
}(1.1)
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6
1.1.12. X , B P(X) G X, G B x G B B x B G. : G =
iI Bi
(Bi)iI B, x G i0 I x Bi0
iI Bi = G. , x G Bx B
x Bx G, G =xGBx, (Bx)xG B.
1.1.13. (X,T) B T. :
(i) B T.
(ii) G T x G, B B x B B G.
, G X :
G T x G B B x B G.
. 1.1.12 (1.1).
1.1.14. () (X, ) . B ={B(x, ) : x X, > 0} B = {B
(x, 1n
): x X, n N}
T. R , B = {(a, b) : a, b R a < b} ( (a, b) = B
(a+b
2 ,|ab|
2
)).
() (X,T) B T, B B B T T. , T T.
() (X,T) , B = {{x} : x X} () T.
(1.1), . , , - X X. , X. , . :
1.1.15. X B P(X). B ( (1.1) ) X, :
() X =B
{B : b B}
() B1, B2 B x B1 B2, B3 B x B3 B1 B2(, 1.1.12, B1, B2 B, B1 B2 B).
. () B T. X T B1 B2 T B1, B2 B, (i) (ii).
() T = {G X : C B G =C}.1
T :
1 C x X : C C
x C
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1.1. , 7
(i) T ( C = ) X T ( C = B).
(ii) G1, G2 T. G1 =C1 G2 =
C2 C1, C2 B.
G1 G2 ={B C : B C1, C C2}. B C
B, (). G1 G2 B, G1 G2 T.
(iii) I Gi T i I. Ci B Gi =Ci i I.
C =iI Ci. C B
iI Gi =
C.
iI Gi T.
B T: B B B
C C = {B}, B
T. , T, B T.
1.1.16. () B = {(a, b] : a, b R, a b} ( (a, a] = ). B R, :
(i) R =B(=nN(n, n]
)
(ii) (a1, b1], (a2, b2] B (a1, b1] (a2, b2] B, , , (a1, b1](a2, b2] = (a, b] a = max{a1, a2} b = min{b1, b2}.
TS R B - . (R,TS) RS . TS T|| R, a, b R a < b (a, b) =
n=n0
(a, b 1n ], n0 N a b1n0
. (a, b) TS . , (a, b] / T|| ( a < b) (a, b] TS T|| ( TS .
() C = {(, a) : a R} {(b,+) : b R} R () 1.1.15. , a, b R, a < b, (, b) (a,+) = (a, b), C. , , B C : B,C C R.
.
1.1.17. (X,T) . C T T C T. ,
B =
{ni=1
Ci : n N, Ci C i = 1, 2, . . . , n
} {X}
T.
C T, T
G =iI
nij=1
Cij (1.2)
I , ni N i I Cij C.
1.1.18. C X. T X C.
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8
. B = {ni=1 Ci : n N, Ci C i = 1, 2, . . . , n} {X}.
X B B B. , 1.1.15, B T X. T C . (1.2).
1.1.19. C P(X) T X C, (1.2) T X C. 1.1.10, T.
1.1.20. () (X,T) . B T T. , B - B, B B T. B T., B T.
() C = {(, a) : a R} {(b,+) : b R} R. , B C. a, b R, a < b, (, a) (b,+) = (a, b). B {(a, b) : a, b R, a < b} . , B , C R.
() (), C = {(, a] : a R} {(b,+) : b R} TS .
1.2 , ,
, , - , . . - , .
1.2.1. (X,T) . F X X ( T), . F F c T .
1.2.2. (X,T) . :
(i) , X X.
(ii) .
(iii) ( ) .
. (i) c = X T = , Xc = T = X .
(ii) n N F1, F2, . . . , Fn X . F c1 , F c2 , . . . , F cn .
ni=1 F
ci = (
ni=1 Fi)
c .
ni=1 Fi
.
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1.2. , , 9
(iii) , .
1.2.3. () ( ).
() , X.
() (X,T) , X.
() (X,T) , X.
1.2.4. (X,T) A X. {G :
G A G } A A intTA.
1.2.5. A , A A.
1.2.6. (X,T) . A,B X:
(i) A A
(ii) A T A = A
(iii) (A) = A
(iv) A B, A B
(v) (A B) = A B
. (i), (ii) , 1.2.5.
(iii) (ii) A .
(iv) A B, A, B. , A B.
(v)A B AA B B
}(iv)= (A B)
A(A B) B
}= (A B) A B
(i) :
A AB B
}= A
B A BA B
}= A B (A B)
A B = (A B).
1.2.7. (X,T) A X. {F :
F A F } A A clTA.
1.2.8. A , A - A.
1.2.9. (X,T) . A,B X:
(i) A A
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10
(ii) A A = A
(iii) (A) = A
(iv) A B, A B
(v) (A B) = A B
. 1.2.6.
1.2.10. () (X,T) , A = A = A A X.
() (X,T) ,
A =
{, A 6= XX, A = X
, A =
{, A = X, A 6=
() (X,T) ,
A =
{A, Ac Ac , A =
{A, A X, A
1.2.11. () (iv) , n N A1, A2, . . . , An X, :
(A1A2. . .An) = A1A2. . .An (A1 A2 . . . An) = A1A2. . .An
. R :(
n=1
( 1n,
1
n
))= {0} =
n=1
( 1n,
1
n
)=
n=1
( 1n,
1
n
)= {0}
( n=1
( 1n,
1
n
))$n=1
( 1n,
1
n
).
( n=1
[0, 1 1
n
])= [0, 1) = [0, 1]
n=1
[0, 1 1
n
]=
n=1
[0, 1 1
n
]= [0, 1).
( n=1
[0, 1 1
n
])%n=1
[0, 1 1
n
].
() (A B) = A B, (A B) = A B. R, A = Q B = R \Q :
(A B) = R = R A B = = (A B) = = A B = R R = R
-
1.2. , , 11
1.2.12. (X,T) A X.
(i) X \A = (X \A) , A = X \ (X \A).
(ii) X \A = (X \A) , A = X \ (X \A).
. (i) :
X \A = X \{G X : G , G A}
={X \G X : G , G A}
={F X : F , X \A F}
= (X \A) .
(ii) (i), X \A A, :
X \ (X \A) = (X \ (X \A)) = A
(X \A) = X \A.
1.2.13. (X,T) , A X x X.
x A G A 6= G T : x G.
. 1.2.12(ii), :
x A x / (X \A) ={G X, G G X \A}
G T G X \A x / G G T x G G * X \A, G A 6= .
1.2.14. B T,
x A B A 6= B B : x B.
, B B x A, G T x A, B B x B G. x A 1.2.13. 1.2.13.
1.2.15. RS , (a, b) = (a, b], a, b R a < b. -, (a, b) (a, b) x R \ (a, b)
x a, B = (x 1, x] B B (a, b) = . x / (a, b).
x > b, B = (b, x] B B (a, b) = . x / (a, b).
x = b B B x B, B (a, b) 6= . x (a, b).
( B RS)
1.2.16. (X,T) D X. D X ( X) D = X .
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12
1.2.17. 1.2.13, D G D 6= G T, G 6= . , B T, D B D 6= B B, B 6= .
1.2.18. () (X,T) , D X .
() (X,T) , D X X D = X, {x} T x X.
() (X,T) x0, {x0} X.
() RS Q , Q (a, b] 6= a, b R a < b.
1.2.19. (X,T) , A X x X. x A G T x G A G \ {x} 6= . A A. A A. A , A.
1.2.20. x A x A \ {x}. , 1.2.13 x A \ {x} G x G (A \ {x}) G 6= , A G \ {x} 6= .
1.2.21. () (X,T) , A = A X ( G = {x} ). A .
() (X,T) x0, {x0} = X \ {x0}., x0 / {x0} {x0} \ {x0} = = , x0 / {x0} \ {x0}. , x X \ {x0}, x {x0} {x0} \ {x} = {x0} = X, x {x0} \ {x}.
1.2.22. (X,T) A X.
(i) A = A A
(ii) A A A.
. (i)
A A ( 1.2.9)A A ( 1.2.13)
}= A A A.
, A \ A A. x A \ A G T x G. x A, A G 6= ( 1.2.13). x / A, (A \ {x}) G = A G 6= . x A.
(ii) (i).
1.2.23. (X,T) A X. A (X \A) A BdA A.
1.2.24. , . BdA = BdAc A X, .
-
1.2. , , 13
1.2.25. (X,T) A X.
(i) BdA = A \A
(ii) BdA A =
(iii) A = BdA A
(iv) A, (X \A),BdA X.
. (i) BdA = A (X \A) = A (X \A) = A \A.
(ii) BdA A = BdA X \A = X \A.
(iii) BdA A = A (A (X \A)) = A (A (X \A)) = A.
(iv) (iii) BdA A = A
A BdA (X \A) = A (X \A) = A (X \A) = X.
, A (X \ A) A (X \ A) = , A BdA = ( (ii)) (X \A) BdA = (X \A) Bd(X \A) = ( (ii) X \A).
1.2.26. (X,T) A,B X.
(i) Bd() =
(ii) BdA = Bd(Ac)
(iii) Bd(BdA) BdA
(iv) A B Bd(A B) = A B (BdA BdB)
(v) A A BdA =
(vi) A BdA A
. .
-
14
-
2
2.1
, - , . , - . , - , .
2.1.1. (X,T) x X. U X x x U. x x Nx ( NTx ).
2.1.2. ()
U Nx G T x G U,
U ={G T : G U}.
() , U , . , a, b R, a < b, (a, b] R x (a, b) ( b), (a, b] R.
2.1.3. (X,T) . {Nx, x X} :
(i) U Nx, x U ( U 6= ).
(ii) U1, U2 Nx, U1 U2 Nx.
(iii) U Nx U V X, V Nx.
(iv) G X G Nx x G.
(v) U Nx G Nx G U G Ny y G.
-
16
. (i) U Nx, x U U . x U .
(ii) U1, U2 Nx, x U1 x U2 . x U1 U2 = (U1 U2). U1 U2 Nx.
(iii)U Nx x UU V U V
}= x V = V Nx.
(iv) G X. G G = G G G x G x G G Nx x G.
(v) U Nx. G T x G U ( 2.1.2 ()). , (iv), G Ny y G. , y = x G Nx.
(iv) . (i), (ii), (iii), (v) , , .
2.1.4. X x X Nx X :
() U Nx, x U .
() U1, U2 Nx, U1 U2 Nx.
() U Nx U V X, V Nx.
() U Nx, G Nx, G U G Ny y G.
T = {G X : G Nx x G} {}.
T X.
x X, T, Nx. , NTx = Nx x X.
. T X:
(i) T, T. , Nx 6= x X, Ux Nx, Ux X. , (), X Nx x X. X T.
(ii) G1, G2 T. G1G2 = , G1G2 T. , x G1 G2. x G1 x G2. G1 Nx G2 Nx. () G1 G2 Nx. x G1 G2 , G1 G2 T.
(iii) I Gi T i I. x iI Gi.
i0 I x Gi0 T. Gi0 Nx. (),
iI Gi Nx.
iI Gi T.
NTx = Nx x X:
x X U NTx , x T. x U, U T. U Nx. (), U Nx. NTx Nx x X. , U Nx. (), G Nx G U . (), x G. G NTx . , G U , U NTx . Nx NTx . NTx = Nx x X.
-
2.1. 17
2.1.5. (X,T) x X. Bx Nx x, U Nx B Bx B U . Bx x. Bx ( ) BTx .
2.1.6. x X Bx x, x
Nx = {U X : B Bx B U}.
X x x ( 2.1.3 (iii)).
2.1.7. () Nx x.
() Bx x, Bx = Nx B Bx U X B U U Bx. ,() 2.1.3 (iii).() U Nx. , , B Bx
B U . , U Bx. Nx Bx Nx = Bx.
() Bx = {G X : G x G} x: Bx Nx U Nx, U Bx U U .() (X, ), Bx = {B(x, ) : > 0} x ( x X). Bx Nx B(x, ) x. U x (U Nx), x U, > 0 B(x, ) U U , B(x, ) Bx. , {B(x, ) : > 0, Q} {B
(x, 1n
): n N}
x.
() , Bx = {{x}} x, Bx Nx U Nx {x} Bx {x} U .
2.1.8. (X,T) x X, Bx x.
(i) B Bx, x B ( B 6= ).(ii) B1, B2 Bx, B3 Bx B3 B1 B2.(iii) G X x G Bx Bx
Bx G.(iv) B Bx, G X x G B y G By By
By G.
. (i) B Bx, B Nx. 2.1.3 (i), x B.(ii) B1, B2 Bx, B1, B2 Nx. , 2.1.3 (ii), B1B2 Nx., Bx x, B3 Bx B3 B1 B2.(iii) G G Nx x G ( 2.1.3 (iv)) x G Bx Bx Bx G ( 2.1.6).(iv) B Bx. B Nx, G X , x G B. (iii), y G By By By G.
-
18
, - 2.1.4, .
2.1.9. X x X Bx X :
() B Bx, x B.
() B1, B2 Bx, B3 Bx B3 B1 B2.
() B Bx, G X, x G B y G By By By G.
T = {G X : x G B Bx B G} {}.
T X.
x X Bx x T.
. 2.1.4 - .
2.1.10 ( Hausdorff). X T1,T2 - X. x X B1x,B2x x T1 T2. :
() T1 T2
() x X B1 B1x, B2 B2x B2 B1.
. :
() N1x N2x x X
N1x,N2x x T1 T2 .
() () ().
()() x X U N1x. G T1 x G U . () G T2. U N2x. N1x N2x.
()() G T1. G N1x x G ( 2.1.3 (iv)). () G N2x x G. G T2. T1 T2.
()() x X B1 B1x. B1 N1x, (), B1 N2x. B2x x T2, B2 B2x B2 B1.
()() x X U N1x. B1x x T1, B1 B1x B1 U . (), B2 B2x B2 B1. B2 U U N2x. N1x N2x.
2.1.11. (X,T) B T. , B T x X Bx = {B B : x B} x.
-
2.2. 19
. T ( 1.1.13).
() x X. Bx Nx Bx ( ) x. U Nx. x U T. B T, 1.1.13, B B : x B U. B Bx B U . Bx x.() G T x G. G Nx. Bx x, B Bx, B G. B B x B G. , 1.1.13, B T.
, , , .
2.2 1 , , . - Stone-Cech , . Ramsey, . - .
2.2.1. X F X. F X :(i) / F(ii) A,B F , A B F .(iii) A F B A, B F .
, X. , L, X; :
X L / L. A L B A, B L.
, . :
A,B L, A B = . A L B / L B A, A \B L.
L: - - . A,B L , A B L. , AB / L, A\(AB) B\(AB) L), . , .
1 ( . . )
-
20
2.2.2. . , X X.
2.2.3. S X, F(S) = {A X : S A} . F(S) S = {x}, . F(x) F(S).
2.2.4. F ( ) x X F = F(x).
2.2.5. I X n N A1, A2, . . . , An I ni=1Ai 6= .
2.2.6. .
2.2.7. F X , F X F F ., .
.
= {I P(X) : F I I }
(,) I1 I2 I1 I2. C . I =
C . ,
F I A1, A2, . . . , An I, I0 A1, A2, . . . , An I0 ,
ni=1Ai 6= .
. , Zorn, F . F . ,
/ F F .
A F B A, B F , B F .
A,B F , A B F , .
, F . G F G, G F , G = F . F .
2.2.8. F G X.
(i) B X B A 6= A F , : B F .
(ii) A,B X BA F , F A,B.
(iii) F 6= G, A F B G A B = .
. (i) B = {AB : A F} , F . B F , F F (;). F = F B F .
-
2.2. 21
(ii) A,B / F , (i), C,D F AC = B D = . (A B) (C D) = C D F A B / F , .
(iii) G * F , B F\G. , (i), A F AB = .
, (ii) 2.2.8 :
2.2.9. F A F . A A =A1 A2 An, Ai F . Ai , Ai F .
.
2.2.10. X F X. F :(i) / F .(ii) A,B F , A B F .(iii) A X, A F Ac F ., G X A X, A G Ac G.
. F , -. , F (i), (ii), (iii). A F B A. B / F Bc F A Bc = F , . F . (iii).
- .
2.2.11. X . - X. , .
. X, C - X ( ) . , 2.2.7, C - F , (;). , 2.2.8 (ii).
.
2.2.12. F N ( 2.2.11).
= N {F} :
Bn = {{n}} n N BF = {A {F} : A F}
Bn, n N - 2.1.9. BF .
-
22
() F A {F} A F BF() A1{F}, A2{F} BF , (A1{F})(A2{F}) = (A1A2){F}
BF , (A1 A2) F ( F ).
() A{F} BF , G = A{F}. F G A{F}. y G. y = F , By = G By By G. y A, By = {y} By By G.
, 2.1.9, T x Bx .
n N , F . , N . F . , .
- . S n N [S]n {A S : |A| = n}, S n .
2.2.13 (Ramsey). X , n, r N [X]n = A1 A2 Ar. j {1, 2, . . . , r} S X, [S]n Aj
, . - Ai [X]n (;). , 1, 2, . . . , r - x [X]n i x Ai. , [X]n , S X, [H]n .
. X, X () . , X , N .
n = 1, . - , n = 2. .
c : [N]2 I = {1, 2, . . . , r} c({x, y}) = i {x, y} Ai
F N. i I x X
Ai(x) = {y N : c({x, y}) = i}.
x N Ai(x) X \ {x} F , F . i I Ai(x) F .
Bj = {x N : Aj(x) F}.
-
2.2. 23
Bj N . , j0 I Bj0 F .
. a1 Bj0 . a1, a2, . . . , am c({as, at}) = j0 s, t {1, . . . ,m} s 6= t,
S = Bj0
(ms=1
Aj0(as)
).
S F , S F . am+1 S \ {a1, a2, . . . , am}. , F ( 2.2.11).
-
24
-
3
3.1
, . - , ( ). - . , .
3.1.1. X , (xn)nN X x X. (xn) x ( xn x) U Nx n0 = n0(U) N xn U n n0.
3.1.2. (xn) x, (xkn) (xn) x.
3.1.3. () (N,T), T - , (xn) xn = n n N x N., x N U Nx. x U T, X \ U X \ U . n0 N xn / X \ U n n0, xn U . xn x.() (X,T), X T - , :
(xn) X x X xn x, (xn) x ( n0 N xn = x n n0)., , (xn) x. {n N : xn 6= x} . (xkn) (xn) xkn 6= x n N. U = X \ {xkn : n N}. {xkn : n N} x, x U U T, U Nx. xkn / U n N. (xkn) x, xn x.
X = X ( X ). , x X. U x U U X \{x} = U \{x} 6= , U (;). x X .
-
26
, x X x X \ {x}, (xn) X \ {x}, xn x.
( ), , ( 3.1.3()), , ( 3.1.3()). , . , , . , , , ( ).
3.1.4. ( a a a ) ( a, b, c a b b c, a c) 1 :
a, b c a c b c. a b, a b.
3.1.5. () . N,Z,Q,R .
() P(X) X :
A 1 B A B A 2 B A B.
() (X,T) x X. Nx
U V U V ()
, (Nx,) . U1, U2 Nx. U = U1 U2 Nx U1 U U2 U ., Bx x, Bx ().
3.1.6. () X . X p : X, (,) . p p() p (p) (p).
() (X,T) , (p) X x X. (p) x, U Nx 0 = 0(U) p U 0. p x lim p = x.
3.1.7. X - (N,). , .
1 , . a b b a a = b
-
3.1. 27
3.1.8. () (X,T) , (p) x X 0 p = x 0 ( {x} Nx).
() (X,T) , (p) X x X, Nx = {X}.
3.1.9. (X,T) , x X - (Bx,). U Bx 2 pU U . (pU )UBx . pU x., V Nx. U0 Bx U0 V , Bx x. U Bx U U0 pU U U0 V . pU x.
3.1.10. (X,T) , A X x X. x A (p) X p A p x.
. () x A. A U 6= U Nx. - pU U A U Nx. (pU )UNx X. pU A pU U U Nx, 3.1.9, pU x.
() U Nx. p x, 0 p U 0., p0 U A ( p A ) U A 6= . U A 6= U Nx x A.
3.1.11. (X,T) A X.
(i) (p) X x X p A p x, x A.
(ii) x X A (p) X, p A \ {x} p x.
. (i) A A = A A A. 3.1.10.
(ii) 3.1.10, x A x A \ {x}.
3.1.12. () (,) N . N ( ), N .
() (,), (M,) . : M 1, 2 M 1 2 (1) (2).
() p : (,) X X. q : (M,) X, (M,) , p : (M,) (,) (M) q = p. q (M, ). q (p())M (p()) .
2 U pU .
-
28
3.1.13. () M N M .
() (X,T) x X, B Nx x, (B,) (Nx,).
() (,) 0 , M = { : 0} . N .
() (xkn) (xn) ( k(N) ). (xn) .
() M - ( ). , 1, 2 M. , 1 2. M -, M . 1 2. ,
p : (,) X M , q = p|M p.
, : M () = (M ) , (M)(= M) q = p .
() (p) X x X,
p 9 x U Nx 0 0 p / U U Nx M p / U M.
(), (p)M (p) p / U M.
3.1.14. (X,T) , (p) X x X.:
p x (p) x.
. () q = p : M X (p) U Nx. p x, 0 p U 0. (M) , 0 M (0) 0. M 0 () (0) 0 p() U . p() x.
() , (p) (p).
3.1.15. (p) X x X. x (p) U Nx p U . x (p) U Nx { : p U} .
3.1.16. p x, x (p). -, U Nx. 0 p U 0, { : p U} { : 0} , 3.1.13(), { : p U} , x .
3.1.17. X , (p) X x X. x (p) (p) x.
-
3.2. 29
. () x . :
M = {(U, ) : U Nx, p U}
M :
(U1, 1) (U2, 2) U1 U2 1 2
M (;).
(M,) . , (U1, 1), (U2, 2) M, U3 = U1U2 0 0 1 0 2. x (p), 3 3 0 p3 U . (U3, 3) M (U3, 3) (U1, 1) (U3, 3) (U2, 2)
: M (U, ) = . ( ) (M) ( (X,) = ).
, p (p). p x. U0 Nx. x p, 0 p0 U0. (U0, 0) M (U, ) M (U, ) (U0, 0) p U ( M)
(p )(U, ) = p((U, )) = p() = p U U0.
, p x.
() q = (p())M (p) p() x. U Nx. 0 M p() U 0, :
{ : p U} {() : 0}.
, {() : 0} . 0 . M, 0 () 0.
(M) , 1 M (1) 0. , M , M 0 1. () (1) 0, () 0. {() : 0} .
3.2
. .
3.2.1. X,Y x0 X. f : X Y x0, V Nf(x0) U Nx0 f(U) V . f , X.
3.2.2. () f x0 V Nf(x0) f1(V ) Nx0 . ,
() V Nf(x0). U Nx0 f(U) V . U f1(V ) f1(V ) Nx0 .
-
30
() V Nf(x0). U = f1(V ) Nx0 f(U) V .() 3.2.1 f(x0) x0, Nf(x0) Nx0 . , .
() f , ( U = X).
() X , f (U = {x0})
3.2.3. X,Y f : X Y . :
(i) f .
(ii) G Y , f1(G) X.(iii) F Y , f1(F ) X.(iv) A X, f(A) f(A).(v) B Y , f1(B) f1(B).(vi) B Y , f1(B) (f1(B)).
. (i) (ii) G Y x f1(G). f(x) G G Nf(x0). f , f1(G) Nx ( 3.2.2()). f1(G) , f1(G) .
(ii) (iii) F Y . Y \F , (ii), f1(Y \ F ) = X \ f1(F ) . f1(F ) .(iii) (iv) A X. (iii) F = f(A) f1(f(A)) X. , A f1(f(A)) , A f1(f(A)). , , A f1(f(A)) f(A) f(A).(iv) (v) B Y . (iv) A = f1(B)
f(f1(B)) f(f1(B)).
, f(f1(B)) B , f(f1(B)) B. :
f(f1(B)) f(f1(B))f(f1(B)) B
} f(f1(B)) B f1(B) f1(B) .
(v) (vi) B Y . X \A = X \A,
X \ (f1(B)) = X \ f1(B)= f1(Y \B) f1(Y \B) ( (v))= f1(Y \B)= X \ f1(B)
f1(B) (f1(B)).(vi) (i) x X V Nf(x). f(x) V , x f1(V ). f1(V ) (f1(V )) ( (vi)) f1(V ) Nx. f x. x X , f .
-
3.2. 31
3.2.4. () (X,T), (Y,S) , B S f : X Y . f f1(B) T B B. :() , B S.() G S. G =
iI Bi (Bi)iI B.
f1(G) =iI f
1(Bi) T, . f .() B S, (;).
3.2.5. f : RS RS f(x) = x ., (0, 1] RS , f1 ((0, 1]) = [1, 0) RS (;). 3.2.6. X,Y , x0 X f : X Y . f x0 , (p) X p x0, f(p) f(x0).. () f x0. (p) X p x0. V Nf(x0). f x0, f1(V ) Nx0 . p x0, 0 p f1(V ) 0. f(p) V 0. f(p) f(x0).() (p) p x0, f(p) f(x0) , , f x0. V Nf(x0) U Nx0 f(U) * V . U Nx0 pU U , f(pU ) / V . , (pU )UNx0 pU x0. , f(pU ) / V V Nf(x0), f(pU ) 9 f(x0), .
3.2.7. X,Y f : X Y . f , x X (p) X p x, f(p) f(x). 3.2.8. 3.2.6 3.2.7 . , id : (R,T) R id(x) = x, x R, T . (xn) (R,T) x R, xn x, (xn) x id(xn) x = id(x) . , a, b R, a < b, (a, b) R, id1((a, b)) = (a, b) / T. id .
-. .
3.2.9. X, Y , Z , x0 X f : X Y , g : Y Z. f x0 g f(x0), g f : X Z x0. f, g , g f .. (p) X p x0. f x0, f(p) f(x0) 3.2.6). g f(x0), g(f(p)) g(f(x0)). (g f)(p) (g f)(x0). , 3.2.6, g f x0.
-
32
3.2.1 ,
3.2.10. X,Y . f : X Y , A X f(A) Y . f , A X f(A) Y .
3.2.11. () B X, f B B f(B) Y . ,
() , B B .
() A X , A =iI Bi (Bi)iI B.
f(A) =iI f(Bi) , . f
.
() f : X Y 1 1 , , f(A) = (f1)1(A) A X, :
f f1 : Y X f .
() T1 T2 X id : (X,T1) (X,T2) id(x) = x x X. :
T1 T2 id id1 id .
3.2.12. 1 : R2 R 1(x, y) = x , . ,
(x, y) R2 > 0, 1(B((x, y), )) = (x , x+ ) , 3.2.11(), 1 .
F ={(x, 1x
): x > 0
}, F R2 (;),
1(F ) = (0,+) R.
3.2.13. X,Y f : X Y . f f(A) (f(A)), A X.
. () A X. A X, f(A) Y . f(A) f(A), , f(A) (f(A)).
() A X ( A = A). , , f(A) = f(A) (f(A)). f(A) (f(A)) ( , f(A) = (f(A))). f(A) . f .
3.2.14. X,Y f : X Y . f f(A) f(A), A X.
. ( 3.2.13).
3.2.15. X,Y . 1 1 f : X Y , f f1 , X Y . f : X Y , X Y X Y .
-
3.3. 33
3.2.16. . . , , .
3.2.17. X,Y f : X Y 1 1 . :
(i) f .
(ii) f .
(iii) f .
(iv) A X f(A) = f(A).
. (i), (ii) (iii) 3.2.11(). (iii) (iv) 3.2.14 3.2.3.
3.2.18. () R (1, 1). , .. f : R (1, 1) f(x) = x1+|x| , 1 1, f
1 : (1, 1) R f(y) = y1|y| .
() id : RS R id(x) = x x RS 1 1 , ( a, b R a < b, id1((a, b)) =(a, b) TS). id1 : R RS , (a, b] TS , (id1)1((a, b]) = (a, b] R.
() X , x1, x2 X x1 6= x2 T1,T2 x1, x2 . f : (X,T1) (X,T2)
f(x) =
x2, x = x1x1, x = x2x, x 6= x1, x2
. , f 1 1 . G T2. G 6= , x2 G, x1 f1(G) f1(G) T1. f . , f1 .
() X,Y , f : X Y 1 1 ( ).
3.3 3 (xn) . (xn), lim
xn. :
: () limn xn , limn xn =lim xn.
3 ( . . )
-
34
: (yn) - c1 c2 R, lim(c1xn + c2yn) = c1 lim xn + c2 lim yn.
: xn 0 n N, lim xn 0.
, , - :
|xn| A n N, | lim xn| A.
, , () (xn) :
limnxn = l > 0, |N \ {n N : |xn l| < }| 0 , (xn) l ). , .
3.3.1 ( Banach). - (xn), l R > 0.
U(l, ) = U(l, )[(xn)] = {n N : |xn l| < }
F N,
limFxn = l > 0, U(l, ) F .
3.3.2. Banach .
3.3.3. Banach. Banach , ( ).
. (xn), (yn) c1, c2 R. -:
L = supnN |xn| I0 = [L,L]. I0 I01 = [L0, 0), I11 = [0, L]. {n N : xn I01}, {n N : xn I11} F . , I1 . I0, I1, . . . , k N Ik 2L/2k {n N : xn Ik} F . ,
kN Ik = {l}
l R , > 0, U(l, ) F .
l 6= l < |l l|/2, U(l, ) U(l, ) , F .
, (3.1).
-
3.3. 35
limF xn = l1, limF yn = l2 . c1, c2 6= 0. U (l1, /2c1) [(xn)],U (l2, /2c2) [(yn)] F , . n |xn l1| < /2c1 |yn l2| < /2c2. ,
|(c1xn + c2yn) (c1l1 + c2l2)| < .
U(c1l1 +c2l2, )[(c1xn+c2yn)] F . c1, c2 0, ( ).
xn 0 n N l < 0. , = |l|/2 > 0, U(l, ) F . limF xn = l.
2.2.12, Stone-Cech .
3.3.4. U N = N{U}. a : N R () (, ), aU : R (aU aU |N = a).
. a : N R . aU : R :
aU (x) =
{a(x) x N` = limU an x = U
aU a . U . (` , `+ ) `. , ` = limU an, U(`, ) U . , U(`, ) {U} U aU (U(`, ) {U}) (` , `+ ). aU .
3.3.5. Banach , Theorie des operations lineaires. ` ||(xn)|| = supnN |xn|. C, `. , f : C R, , 1. Hahn-Banach, f f : ` R, 1. , Banach (xn) ` f((xn)). .
-
36
-
4
, , ( 1.1.15, 1.1.18, 2.1.4, 2.1.9). , , . .
4.1
, , , - . , , (X,T) A X . X A, S = {G A : G T} A. (; A, S A). .
4.1.1. (X,T) A X. TA = {A G : G T} (;) A ( T). (A,TA) X TA A .
A TA. , F A A TA( A \F TA). , B A, clAB intAB clTAB intTAB . N
Ax N
TAx ,
x A. , TX = T.
4.1.2. (X,T) A X. B ( ) T, BA = {BA : B B} () TA.
-
38
. B T, BA TA. U TA. U = AG G T. B T, (Bi)iI B G =
iI Bi.
U = A
(iI
Bi
)=iI
(A Bi)
(ABi)iI BA. , BA TA. .
4.1.3. () (X, ) A X. (T)A = Td d = |AA . , :
B(x, ) A = Bd(x, ) x A, > 0,
{Bd(x, ) : x A, > 0} Td {B(x, ) : x A, >0} (T)A. , G A (T)A x G A, G T > 0 B(x, ) G, x B(x, ) A G A.
() 4.1.2, [0, 1] R
B = {(a, b) [0, 1] : a, b R a < b}= {(a, b) : 0 a < b 1} {[0, b) : b [0, 1]} {(a, 1] : a [0, 1]} {[0, 1]}.
() (X,T) A B X. (TB)A = TA. ,
(TB)A = {U A : U TB}= {(G B) A : G T}= {G A : G T}= TA
4.1.4. (X,T) A X.
(i) F A, F A K, (X,T), F = A K.
(ii) B A clAB = (clXB) A.
(iii) x A NAx = {U A : U Nx}.
(iv) (x) A x A, x x TA x x T.
. (i) F A. A \ F A, A \ F = A G G T. A:
F = A \ (A G) = A \G = A Gc,
K := Gc X., F = A K, K X. -
A :
A \ F = A \ (A K) = A \K = A Kc,
-
4.1. 39
G := Kc X. A \ F TA F A.
(ii) B A. (i), (clXB) A A B. (clXB)A clAB. F A F B, F (clXB) A. , (i), F = A K K X.
B F K = clXB K = (clXB) A K A = F.
(iii) x A U Nx. x U A U A, U A TA. U A NAx . NAx {U A : U Nx}.
, V NAx . x intAV = A G G T. U = G V . U Nx ( x G U G T)
U A = (G A) (V A) = (intAV ) V = V.
NAx {U A : U Nx}. NAx = {U A : U Nx}.
(iv) (iii).
4.1.5. (X,T) B A X.
() intAB (intXB) A, (intXB) A TA B. . ,
Q = intQQ % (intRQ) Q = .
() A X, B A B T. ,
() B = A G G T. B X, .
() B = A B, B A.
() A X, B A B X.
4.1.6. (X,T), (Y,S) f :X Y .
(i) f A X, f |A : (A,TA) (Y,S) .
(ii) f(X) B Y , f : (X,T) (B,SB) f : X Y .
(iii) I X =iI Ai, Ai X
i I, f f |Ai i I.
(iv) X =ni=1 Fi, Fi i = 1, 2, . . . , n, f
f |Fi i = 1, 2, . . . , n.
-
40
. (i) G Y , (f |A)1(G) = A f1(G), f1(G) X. , (f |A)1(G) A f |A .
(ii) f : (X,T) (B,SB) . G Y Y f1(G) = f1(B G), B f(X). B G SB , f1(G) T. f : (X,T)(Y,S) .
f : (X,T)(Y,S) , V SB G S V = B G. f1(G) T. f1(V ) = f1(B G) = f1(G). f : X B B- X-, .
(iii) :
() , (i).() G Y . :
f1(G) =
(iI
Ai
) f1(G) =
iI
(Ai f1(G)) =iI
(f |Ai)1(G),
(f |Ai)1(G) Ai, X ( 4.1.5()). f1(G) , . f .(iv) (iii).
4.1.7 ( ). (X,T), (Y,S) . X = F1 F2 fi : Fi Y (i = 1, 2) f1(x) = f2(x) x F1 F2, g : X Y
g(x) =
{f1(x) x F1f2(x) x F2
.
. f1(x) = f2(x) x F1 F2 g. g g|Fi = fi, i =1, 2. , (iv) , .
4.1.8. F1, F2 . , , .
g(x) =
{x 1 x < 0x+ 1 x 0
4.1.9. (X,T) , A X : A X, (x) = x x A. :(i)
(ii) TA A .
. (i) G T, 1(G) = GA TA .(ii) T A, . G T,1(G) = G A T. TA T. TA .
-
4.2. 41
4.2
X, - , - (fi)iI ( fi X Xi). X ( ), . , .
4.2.1. X , (Xi,Ti) fi : X Xi, i I.
C = {f1i (G) : G Ti, i I}
T C (fi)iI .
4.2.2. () T X fi ( T C).
() T
B =
{nk=1
f1ik (Gk) : ik I, Gk Tik k = 1, . . . , n N
}.
, Bi Ti i I,
B =
{nk=1
f1ik (Gk) : ik I, Gk Bik {Xik} k = 1, . . . , n N
}
T. , B B T B B x B B B x B B: B =
nk=1 f
1ik
(Gk), Gk Tik k = 1, 2, . . . , n, fik(x) Gk k, Bk Bik fik(x) Bk Gk k. x
nk=1 f
1ik
(Bk) nk=1 f
1ik
(Gk), nk=1 f
1ik
(Bk) := B B.
4.2.3. (X,T) , T fi : X Xi, i I.
(i) Z g : Z X,
g fi g : Z Xi i I.
(ii) (x) X x X,
x x T fi(x) fi(x) Xi, i I.
(iii) A X, TA (fi|A)iI .
-
42
. (i) g . fig , . fi g . i I G Xi g1(f
1i (G)) = (fi g)1(G) Z.
C = {f1i (G) : G Xi , i I} T, 3.2.4() g .
(ii) x x, fi(x) fi(x) i I, fi . , fi(x) fi(x) i I G x. B ( 4.4.2()) T, n N, i1, i2, . . . , in I G1 Xi1 , G2 Xi2 , . . . , Gn Xin , x
nk=1 f
1ik
(Gk) G. fik(x) Gk k = 1, 2, . . . , n. fik(x) fik(x) k = 1, 2, . . . , n, k fik(x) Gk k k = 1, 2, . . . , n. 0 0 1, 0 2, . . . , 0 n. 0 fik(x) Gk k = 1, 2, . . . , n, x
nk=1 f
1ik
(Gk) G. x x.
(iii) i I G Ti f1i (G)A = (fi|A)1(G). {f1i (G) A : i I, Gi Xi } TA {(fi|A)1(G) : i I, Gi Xi } (fi|A)iI .;;
4.3
(X,T) (Y,T) , - X Y , , , X Y . - . XY X Y ( {U V : U T, V T} ). , - , . , , - . , 4.3.1. , .
n N X1, X2, . . . , Xn , Xi :
ni=1
Xi = {(x1, x2, . . . , xn) : x1 X1, x2 X2, . . . , xn Xn}
=
{x : {1, 2, . . . , n}
ni=1
Xi : x(i) Xi i = 1, 2, . . . , n
}
4.3.1. I (Xi)iI .
-
4.3. 43
X =iI
Xi =
{x : I
iI
Xi x(i) Xi i I
}
(Xi)iI . iI Xi
x = (xi)iI , xi = x(i) i I. xi x. i : X Xi i(x) = xi x X i. Xi 6= i I,
iI Xi 6= . Xi = Y i I,
iI Xi
Y I Y .
4.3.2. (Xi,Ti)iI , I - . X =
iI Xi
(i)iI , X (Xi)iI .
. T X i : X Xi . T
C = {1i (G) : G Ti, i I}. T
B ={
1ik (Gk) : ik I, Gk Tik , k = 1, 2, . . . , n n N}
=
{iI
Gi : Gi Ti i I |{i I : Gi 6= Xi}|
-
44
(iii) Ai Xi i I A =iI Ai, TA
A T A Ai
Ti. A =(
iI Ai)
=iI Ai.
(iv) i I i : X Xi , .
. (i), (ii) , 4.2.3(i), (ii) .
(iii) 4.2.3(iii), TA i|A : A Xi, i I. , i|A : A Ai, i I (;), A.
(
iI Ai)
=iI Ai. x
iI Ai
i(x) = xi Ai, 1i (Ai) ( i ).
iI Ai
iI Ai =
iI
1i (Ai), ,
, iI
Ai iI
Ai.
, x = (xi)iI iI Ai B B, B
T, x B. B iI Ai 6= . B =
iI Gi,
Gi Ti i I |{i I : Gi 6= Xi}| < . , i I,xi Ai ( x
iI Ai) xi Gi ( x
iI Gi). AiGi 6=
i I ,iI(Ai Gi) 6= , (
iI Ai) (
iI Gi) 6= .
(iI Ai) B 6= .
(iv) B B B 6= . B =iI Gi, Gi Ti Gi 6=
i I. i(B) = Gi Xi. i . 1 : RR R ( ).
F = {(x, 1/x) : x > 0} R R 1(F ) = (0,+) R.
4.3.4. (X1, 1), (X2, 2) , (X1,T1) (X2,T2) T -, ((x, y), (x, y)) = max{1(x, x), 2(y, y)},
B((x, y), ) = B1(x, )B2(y, ) (x, y) X1 X2, > 0
{B((x, y), ) : (x, y) X1X2, > 0} {B1(x, )B2(y, ) :x X1, y X2, > 0} T T .
4.3.5. Xn, n N . X =
nNXn
.
. n N, n Xn Tn Xn. n 1 n N ( n ). : X X R
(x, y) =
n=1
1
2nn(xn, yn), x = (xn), y = (yn) X.
-
4.4. 45
X (;). T = T, T X.
n N x, y X n(n(x), n(y)) = n(xn, yn) 2n(x, y). n : (X,T) Xn . T X n : (X,T) Xn , T T.
, G T. G T, x G B B ( B T) x B G ( G B, G T). x = (xn) G. G T, > 0, B(x, ) G. B =
kn=1
1n (Bn(xn, /2)),
k 1/2k < /2. B B, x B y =(yn) B
(x, y) =
kn=1
1
2nn(xn, yn) +
n=k+1
1
2nn(xn, yn)
2
kn=1
1
2n+
n=k+1
1
2n
0 B(x, )B(y, ) = ,x B(x, ), y B(y, ) B(x, ), B(y, ) .
() (X,T) T2 T X T T, (X,T) T2.
() RS T2, () - R.
() X T X. (X,T) T1 ( 5.1.4()), T2 ( , , X).
5.2.6. X Hausdorff X ( - ).
. () , , (p) X x, y X x 6= y, p x p y. X Hausdorff, G1, G2 x G1 y G2. p x, 1 p G1 1. p y, 2 p G2 2. 0 0 1 0 2, p0 G1 p0 G2, , G1 G2 = .
() , , X Hausdorff. x, y X x 6= y, U, V x U y V U V 6= . = Nx Ny = {(U, V ) : U Nx, V Ny} :
(U1, V1) (U2, V2) U2 U1 V2 V1.
(,) . (U, V ) p(U,V ) U V . (p(U,V ))(U,V ) X. p(U,V ) x., U0 Nx. (U0, X) (U, V ) (U, V ) (U0, X) p(U,V ) U V U U0. p(U,V ) x. p(U,V ) y, , x 6= y.
5.2.7. X T2 Y . f : X Y 1-1, , 1 Y T2. , Y X, T2 .
. x, y Y x 6= y u = f1(x), v = f1(y) X. X T2 , U V , u U v V . f , f(U) f(V ) Y x f(U), y f(V ). f 1-1, f(U) f(V ) = f(U V ) = . Y T2.
5.2.8. X T2 A , A T2.
1 , g : Y X 1-1,
-
5.3. 55
. x, y A. , X T2, - G1, G2 X x G1 y G2. U = G1 A V = G2 A A, x U y V . A T2.
5.2.9. (Xi)iI T2 . X =
iI Xi T2.
. x, y X x 6= y. i0 I xi0 6= yi0 . Xi0 T2, G1, G2 , Xi0 , xi0 G1 yi0 G2. 1i0 (G1)
1i0
(G2) X ( i0 ), x 1i0 (G1) y
1i0
(G2). X T2 .
5.3
. , , .
5.3.1. X T3 F X x X \ F , G1, G2 x G1, F G2 G1 G2 = .
5.3.2. T3 T1 T2 .
5.3.3. X :
(i) X .
(ii) x X F X x / F , U Nx U F = .
(iii) x X U Nx, V Nx V U .
(iv) x X F X x / F , G1, G2 x G1, F G2 G1 G2 = .
. (i) (ii) x X F X x / F . X T3, U V x U , F V U V = . U Nx U F = ( y F , y V , V U V = ).
(ii) (iii) x X U Nx, x U. F = X \ U, x / F F . (ii), V Nx V F = , V U U .
(iii) (iv) x X F X x / F . X \ F Nx. (iii), V Nx ( V ) V X \F . (iii), W Nx W V . G1 = W G2 = X \V . G1, G2 , x G1, F G2 G1 G2 = ( G1 W G1 W G2 = X \ V = (X \ V ) X \ V = X \ V X \W ).
(iv) (i) .
-
56
5.3.4. (ii) (iii) 5.3.3, Nx Bx x Bx = {U X : U x U}.
5.3.5. X x X .
. () x X Bx ={U : U Nx
}. Bx Nx
5.3.3(i)(iii), Bx x.
() 5.3.3(i)(iii), X (iii).
5.3.6. () (X, ) , x X, {B(x, ) : > 0} x .
() X = {a, b, c} T = {, {a}, {b, c}, X}. (X,T) T1, {b} ( T2), T3, X .
() (X,T) A X A = . S X T{Ac}, (X,S) T3.: A X T, x clTA \ A. A S x / A. U, V S x U A V . U V 6= ( x A S).
S B = T {G \A : G T},
S = {G1 (G2 \A) : G1, G2 T}.
G1, . . . , G4 T U = G1 (G2 \ A) V = G3 (G4 \ A). A V , A G3. x U , x G1 x G2.
x G1, G1 A 6= , x clTA. U V 6= .
x G2, G2 A 6= , x clTA. A G3, G2G3 6= T. G2G3 * A, intTA = . (G2 \ A) G3 6= . G2 \ A U G3 V , U V 6= .
() T R A = { 1n : n N}. S R T {Ac}. (R,S) T2 , T3. , (R,T) T2 ( ) S T, (R,S) T2. A R intTA = . , (), (R,S) T3 ( 0 / A, A S, 0, A S).
5.3.7. Y , X D X. f : X Y f |D{x} x X, f .
. x X (x) X x x. f(x) f(x). , , f(x) 9 f(x). ( Y T3) V , f(x), , f(x) / V .
-
5.3. 57
f |D{x} , U0 x,
f(U0 (D {x})) V. (1)
x x, 0 0 x U0. 1 1 0 f(x1) / V .
V c () f(x1) f |D{x1} , U1 x1 ,
f(U1 (D {x1})) V c. (2)
U1 U0 ( x1 U1 U0), U1 U0 D 6= ( D X). , ,:
U1 U0 D U1 D = f(U1 U0 D) V c
U1 U0 D U0 D = f(U1 U0 D) V
(1) (2). f(x) f(x) , f .
5.3.8. X T3 Y - , X, Y T3 .
. : X Y X Y , y Y . X T3, x = 1(y) X Bx . , {(V ) : V Bx} y . Y T3.
5.3.9. X T3 A , A T3.
. x A F A, A, x / F . K X ( X) F = K A. x / K , X T3, G1, G2 , X, x G1 K G2. U = G1 A V = G2 A A, x U F = K A V . A T3.
5.3.10. (Xi)iI T3 . X =
iI Xi T3.
. x X U Nx. V Nx V U . x U, , B B (B ) x B U U . B =
nk=1
1ik
(Uk), Uk Xik , k = 1, . . . , n. x B, xik Uk k = 1, . . . , n. Xik T3, k Vk Xik , xik Vk V k Uk. V =
nk=1
1ik
(Vk). V x V , V Nx.
V =
(nk=1
1ik (Vk)
)=iI
Ai =iI
Ai =
nk=1
1ik (V k) nk=1
1ik (Uk) = U
Ai =
{Vk, i = ik, k = 1, . . . , n
Xi, .
-
58
5.4
, - , . Urysohn Tietze, (, ). -, .
5.4.1. X T4, F1, F2 X F1 F2 = G1, G2 X F1 G1, F2 G2 G1 G2 = .
5.4.2. T4 T1 T3 .
5.4.3. X :
(i) X .
(ii) F1, F2 X F1 F2 = , U X F1 U U F2 = .
(iii) F X G X F G, U X F U U G.
(iv) F1, F2 X F1 F2 = , G1, G2 F1 G1, F2 G2 G1 G2 = .
. 5.3.3 .
5.4.4. X F F = {x X : f(x) = 0} f : X R, X . , .
. F1, F2 X F1 F2 = . f1, f2 : X R F1 = f11 ({0}) F2 = f
12 ({0}).
: X R (x) = |f1(x)||f1(x)|+ |f2(x)|
.
( |f1(x)| + |f2(x)| x F1 F2) (x) = 0 x F1 (x) = 1 x F2. U =1((, 1/2)) V = 1((1/2,+)). U, V , ( ) F1 U, F2 V . X T4.
(X,T) , X T = T. F X . f : X R f(x) = d(x, F ) ( x F ). f (|f(x) f(y)| (x, y) x, y X) x X
f(x) = 0 d(x, F ) = 0 x F = F,
F = f1({0}). X T4.
-
5.4. 59
5.4.5. X {U1, U2 . . . , Un} - X. {V1, V2 . . . , Vn} X, V k Uk k = 1, 2, . . . , n.
{V1, V2 . . . , Vn} {U1, U2 . . . , Un}.
. V1 X V 1 U1 V1 U2 U3 Un = X, . X \ (U2 Un) U1. X \ (U2 Un) , U1 X T4, 5.4.3(iii), V1
X \ (U2 Un) V1 V 1 U1.
X = V1 U2 U3 Un V 1 U1.
5.4.6. () X = {a, b, c} ( a, b, c )
T = {, {a}, {b, c}, X}.
5.3.6(), (X,T) T1 ( T2) T4 ( T3), X .
() X = {a, b, c}( a, b, c ) - c,
T = {, {a}, {b}, {a, b}, X}.
(X,T) T4. , X X, , {b, c},{a, c}, {c}. F1, F2 X , F1, F2 F1. F1, F2 G1 = G2 = X. (X,T) T3. , F = {c}, a / F F X. a F ., (X,T)
T x0, (X,T) T4 T3.
() RS T4. F1, F2 RS . F1 RS \ F2, RS \ F2 RS . x X, {(a, x] : a R, a < x} x, x F1 ax < x (ax, x] RS \F2. F2 RS \F1, y F2 by < y (by, y] RS \ F1.
G1 =xF1
(ax, x] G2 =yF2
(by, y].
G1, G2 RS , F1 G1 F2 G2. , , G1G2 6= . x F1 y F2 (ax, x](by, y] 6= . x < y y < x ( F1 F2 = ). x < y, x (by, y] ( (ax, x] (by, y] 6= ) (by, y] G2 RS \ F1, x RS \ F1, . y < x. G1 G2 = .
5.4.7. X T4 Y - , X, Y T4 .
-
60
. : X Y X,Y , F1, F2 Y . 1(F1), 1(F2) ( ). X T4, G1, G2 X , 1(F1) G1 1(F2) G2. F1 (G1) F2 (G2) ( ), (G1), (G2) Y ( ) ( 1-1). Y T4.
5.4.8. X T4 A , A T4.
. F1, F2 A A. A X, F1, F2 X. , ( X) G1, G2 F1 G1 F2 G2. G1 A,G2 A A , F1 G1 A F2 G2 A.
5.4.9. 5.4.8 A X. , .
5.4.10 ( Sorgenfrey). RS . Sorgenfrey RS RS -. Sorgenfrey ( ).
.
A = {(x,x) : x R}
RS RS , A R R RS R. , A ( RS RS , x R {(x,x)} = ((x 1, x] (x 1,x]) A. A RS RS .
F1 = {(x,x) : x Q} F2 = {(x,x) : x R \Q}.
, F1, F2 ( ) RS RS . F1, F2 .
G1, G2 RSRS , F1 G1 F2 G2 G1G2 6= . x R\Q, Bx, RSRS , (x,x) Bx G2. Bx (x 1nx , x] (x
1nx,x],
nx N (;). x R \Q nx N,
(x,x) (x 1
nx, x
](x 1
nx,x
] G2.
n N, An = {x R \Q : nx = n}
R \Q =n=1
An R =
( n=1
An
)
qQ{q}
.
-
5.4. 61
Baire R, - n0, (An0)
6= ( - R). a, b R a < b, (a, b) An0 . q (a, b).
(q,q) F1 G1, > 0
(q,q) V := (q , q] (q ,q] G1
< 1n0 .
q An0 , t An0 |qt| < 3 . (q, q] (t , t] ( q < t q (t , t] t < q t (q , q]). x (q , q] (t , t]. y (q ,q] (t ,t].
, (x, y) V , W :=(t 1n0 , t] (t
1n0,t] (t,t),
t 1n0
< t < x t t 1n0
< t < y t.
t An0 W G2 ( An0), (x, y) V W G1 G2 G1 G2 6= .
5.4.11. RS . -, RS , RS RS , . ( ). , , RSRS . QQ , RSRS . .
A = {(x,x) : x R}
. , A ( RSRS) , , x R {(x,x)} = (x1, x] (x1,x]A. A , , .
5.4.1
- , . - Urysohn, , ., -.
5.4.12 ( Urysohn). X A, B X. f : X [0, 1] f(a) = 0 a A f(b) = 1 b B.
-
62
. ( - X) X, () [0, 1]. - [0, 1]:
=
n=0
n, n =
{k
2n: k = 0, 1, . . . , 2n
} n = 0, 1, 2, . . .
0 = {0, 1}, 1 = {0, 12 , 1} = 0 {12}, . . .
(n)n0
n+1 \n ={
k
2n+1: k , k {0, 1, . . . , 2n+1}
}.
U(d) d , a, b a < b, U(a) U(b). , U(d) , - . U(d) . , U(d) d n, n, (i) A U(d) Bc, d .(ii) d, d d < d, U(d) U(d).
n = 0 U(0), U(1). U(1) = Bc. X , U(0)
A U(0) U(0) U(1).
, U(d) d n. U(d) d n+1 \ n. k {0, 1, . . . , 2n+1}. k12n+1 d. , D (d,+) D(x), f ,f(x) = inf D(x) d.
() x / U(d) f(x) d. , x / U(d), x / U(r) r D r < d. , D(x) (d,+), f ,f(x) = inf D(x) d.
x0 X > 0. U x0,
f(U) (f(x0) , f(x0) + ).
D [0,+), d1, d2 D
f(x0) < d1 < f(x0) < d2 < f(x0) + .
U = U(d2) \ U(d1). U x0 U . : f(x0) < d2, () x0 U(d2) f(x0) > d1, () x0 / U(d1)), U Nx0 . x U . x U(d2) U(d2) , (), f(x) d2. , x / U(d1) , (), f(x) d1. ,
f(x) [d1, d2] (f(x0) , f(x0) + ).
f(U) (f(x0) , f(x0) + ). , , f . .
5.4.13. Urysohn [0, 1] - [a, b], a, b R a < b, f , , g = a+ (b a)f .
Urysohn , Tietze.
5.4.14 ( Tietze). X - F X.
(i) f : F [a, b] ( a, b R, a < b) , g : X [a, b] g|F = f ( g f).
(ii) f : F R , g : X R g|F = f .
. , X F f .
g : X R, , f F . , f [r, r] ( r > 0). g : X R,
|g(x)| 13r x X |g(y) f(y)| 23r y F.
-
64
[r, r] 23r :
I1 =
[r,1
3r
], I2 =
[1
3r,1
3r
], I3 =
[1
3r, r
].
A = f1(I1), B = f1(I3) F, f . A, B X. Urysohn ( 5.4.13), g : X [ 13r,
13r], g(a) =
13r a A g(b) =
13r
b B. |g(x)| 13r x X. y F |g(y) f(y)| 23r. :
y A, f(y), g(y) I1, 23r.
y B, f(y), g(y) I3, 23r.
y / A B, f(y), g(y) I2, 23r.
|g(y)f(y)| 23r. g.(i) , [a, b] [1, 1]. f : F [1, 1] . , ( r = 1), g1 : X R
|g1(x)| 13 x X |g1(y) f(y)| 23 y F.
f g1, F [ 23 ,23 ].
, f g1 ( r = 23 ) g2 : X R
|g2(x)| 13 23 x X |f(y) g1(y) g2(y)|
(23
)2 y F.
(gn) . , g1, g2, . . . , gn : X R |gk(y)| 13 (
23 )k1 k = 1, 2, . . . , n |f(y)
nk=1 gk(y)| (
23 )n
y F . , f g1 g2 gn( r = ( 23 )
n) gn+1 : X R
|gn+1(x)| 1
3(
2
3
)nx X
f(y)n+1k=1
gk(y)
(
2
3
)n+1y F.
n N supxX |gn(x)| 13 (23 )n1
n=1
1
3(
2
3
)n1= 1
-
5.4. 65
(|nk=1 gk(x)| 1 n N) (
). , y F f(y)nk=1
gk(y)
(
2
3
)n n N.
, n, f(y) = g(y). g|F = f .
(ii) (1, 1) R, f (1, 1). (i) h : X [1, 1] f . g : X (1, 1) f . K = h1({1, 1} X. K ( h ) F K = ( h(F ) = f(F ) (1, 1)). Urysohn : X [0, 1] (x) = 0 x F (y) = 1 y K. g = h : X R. g x F
g(x) = h(x)(x) = h(x) 1 = h(x) = f(x).
g f . g(X) (1, 1). x K g(x) = h(x) 0 = 0. x / K, |g(x)| |h(x)| < 1. , g(x) (1, 1).
5.4.15. X {U1, U2, . . . , Un} - X. i : X [0, 1], i = 1, 2, . . . , n, () , {U1, U2, . . . , Un},
(i) {x X : i(x) 6= 0} Ui i = 1, 2, . . . , n.
(ii) 1(x) + 2(x) + + n(x) = 1 x X.
5.4.16 ( ). X {U1, U2, . . . , Un} X. {U1, U2, . . . , Un}.
. 5.4.5, {V1, V2, . . . , Vn} - {U1, U2, . . . , Un}. , {W1,W2, . . . ,Wn} {V1, V2, . . . , Vn}. {W1,W2, . . . ,Wn} X Wk Vk, Vk Uk k. Urysohn f1, . . . , fn : X [0, 1], f(W k) = {1} f(X \ Vk) = {0} k = 1, . . . , n. {W1,W2, . . . ,Wn} () X, f1(x) + f2(x) + + fn(x) > 0 x X. k
k : X [0, 1] k =fk
f1 + f2 + + fn.
f1k (R \ {0}) Vk, {x X : fk(x) 6= 0} V k Uk, {x X : i(x) 6= 0} Uk k = 1, . . . , n. 1, 2, . . . , n .
-
66
5.4.17. X A X. f : X [0, 1] A = f1{0} A G X (. X).
. () f : X [0, 1] A = f1{0}. A , f . , n N {
x X : f(x) < 1n
}= f1
((, 1
n
)) A =
n=1{x X : f(x) 0. A = f1({0}).
5.5
Urysohn , . , , Urysohn, .
Ury-sohn - . , . X, B X x Bc. , Urysohn, U(1) = Bc U(0) -, x U(0) U(0) U(1) ( X). , , . U( 12 ),
U(0) U( 12 ) U(12 ) U(1).
, X .
.
5.5.1. X T3 12 x X F X x / F , f : X [0, 1] f(x) = 0 f(y) = 1 y F .
-
5.5. 67
5.5.2. () , x X F X x / F , f : X [0, 1] f(x) = 0 f(y) = 1 y F , x f1((, 1/2)) F f1((1/2,+)) f1((, 1/2)), f1((1/2,+)) .
() T1 X . , F X x X \ F , {x} X( {x} F = ). , Urysohn, f : X [0, 1], f(x) = 0 f(y) = 1 y F . X .() [0, 1] , [a, b]. , a, b R, f g = (ba)f+a, g : X [a, b] g(x) = a g(y) = b y F .
5.5.3. X T3 12 Y - X, Y T3 12 .
. y Y F Y y / F . : X Y X, Y f : X [0, 1] 1(y) 1(F ), f 1 : Y : [0, 1] .
5.5.4. X T3 12 A , A T3 12 .
. x A F A x / F . K, X F = KA. X T3 12 , f : X [0, 1] f(x) = 0 f(y) = 1 y K. f |A : A [0, 1] .
5.5.5. (Xi)iI T3 12 . X =
iI Xi T3 12 .
. x X F X x / F . x F c.
nk=1
1ik
(Vk), Vk Xik k, x
nk=1
1ik
(Vk) F c. ik(x) = xik Vk xik / V ck , Xik k. Xik T3 12 , fk : Xik [0, 1] fk(xik) = 0 f(V ck ) {1}. fk ik : X [0, 1], k = 1, . . . , n, , f : X [0, 1] f(y) = max
1knfk(ik(y)) :
f(x) = max1kn
fk(ik(x)) = max1kn
fk(xik) = 0, fk(xik) = 0
k = 1, . . . , n.
y F , y /nk=1
1ik
(Vk), k0 {1, . . . , n} yik0 = ik0 / Vk0 , yik0 V
ck0, fk0(yik0 ) = 1. f(y) = 1.
-
68
5.5.6. () RS T4 T1, T3 12 .
() T4 T1, T3 12 .
() Sorgenfrey RS RS T3 12 , T3 12 , T4.
T3 12 T3. T3 . T3 12 , .
. 2
2 L. A. Steen, J. A. Seebach, Counterexamplesin Topology (Springer, 1978) 90. Tychonoff Corkscrew.
-
6
6.1
, , . , . , . .
6.1.1. X , , A,B X, X = A B. , .
6.1.2. X , , A,B X, X = A B.
, . .
6.1.3. X. :
(i) X .
(ii) X ,X.
(iii) f : X {0, 1} ( {0, 1} ). , - f : X {0, 1} .
. (i) (ii) A X . X = Ac A, A,Ac . X , A = Ac = . A = A = X.
(ii) (iii) f : X {0, 1} , A = f1({0}) . f , A 6= A 6= X, . , f .
-
70
(iii) (i) X , , A,B X, X = AB. , A, A : X {0, 1}, (;), . X .
6.1.4. () R, Q = (Q (, a)) (Q (a,+)) a R \Q.
() Sierpinski ( 1.1.5()) , - , , .
() RS , (, a] , a R.
6.1.5. a, b R, a b, [a, b] ( ) .
. a = b, [a, b] , - . a < b. , , [a, b] , , - A,B [a, b], [a, b] = A B. A , , s = supA (a s b) s A. s 6= b, A , > 0, [s, s + ) A. , s < s + 2 A. s = b A. , s2 := supB s2 B s2 = b. b A B, . [a, b] .
6.1.6. - X, , . , Ai X , i I,
iI Ai 6= ,
iI Ai
.
. iI Ai 6= , x0
iI Ai.
f :iI Ai {0, 1}. Ai , f |Ai
f(x0) ( x0 Ai, i I). f(x) = f(x0) x
iI Ai f . , ,
f :iI Ai {0, 1}.
iI Ai .
R ( ).
6.1.7. R . R ( ).1
1 I, R, x, y I, x y, z R, x < z < y, z I. R : [a, a] = {a} a R, [a, b], (a, b],[a, b), (a, b) a, b R a < b, (a,+), [a,+) a R, (, b), (, b] b R R = (,+).
-
6.1. 71
. R , R =n=1[n, n]
n=1[n, n] =
[1, 1] 6= ( R ). A R, , x, y A x < y z R x < z < y, z / A. A1 = A (, z),A2 = A (z,+) A, , (x A1 y A2) A = A1 A2. A . , R .
, .
6.1.8. X A,B X.
(i) A A B A, B .
(ii) A , A .
. (i) f : B {0, 1}. A , f |A . , f(A) {0, 1} . f(A) . f , f(B) f(A) (, x B x A (xi) xi A i xi x, f(x) = limi f(xi) f(A)). , f(B) , f . B .
(ii) (i), B = A.
6.1.9. 6.1.8(ii) , - , . - Q () .
. , X -, X . .
6.1.10. - .
. X,Y , X . f : X Y . g : f(X) {0, 1} . g f : X {0, 1} , X . g , f(X) .
6.1.11 ( ). X f : X R . f(X) R .
. 6.1.10, R, ( 6.1.7).
(Xi)iI , Xi ( - i ). .
-
72
6.1.12. (Xi)iI - . X =
iI Xi .
. I . , |I| = 2 ( - I ). X Y (x0, y0) X Y . x X
Ax = ({x} Y ) (X {y0}).
Ax , . {x} Y , X {y0} ( X Y ) : (x, y0). , (x0, y0)
xX Ax XY =
xX Ax. , XY
. , z = (zi)iI X
J I ,
XJ = {x = (xi)iI X : xi = zi i I \ J}.
XJ iJ Xi,
, . , XJ . , z XJ J I, XJ .
Y ={XJ : J I }
. Y X (;). 6.1.8 X .
, - R.
6.1.13. X x, y X. X, x y, f : [a, b] X, a, b R a < b, f(a) = x f(b) = y. f f([a, b]). X X.
6.1.14. () X , . , , X = A B ( A,B X A B = ). X, f : [a, b] X. f([a, b]) X, , A, B (;). , A B ( ), .
() - . , ( ).
-
6.2. 73
() ( R ).
6.1.15. () Bn :={x Rn :x21} Rn . , x, y Bn 2
f : [0, 1] Rn f(t) = (1 t)x+ ty.
f t [0, 1] f(t)2 (1t)x2+ty2 1, f([0, 1]) Bn. f Bn x y. Rn ., Rn - .
() (the topologists sine curve) R2,
S = {(x, sin( 1x ) : x (0, 1]}.
S (0, 1], . , , S R2 . , S = S({0} [1, 1]). S , 6.1.14() .
, , f : [a, c] S S. {t [a, c] : f(t) {0} [1, 1]} , b, b < c (;)., f [b, c] b f(b) {0} [1, 1], f((b, c]) S. g : [0, 1] [b, c] S g(t) = f(tc+ (1 t)b), g(0) = f(b) g((0, 1]) = f((b, c]) S. g(t) = (x(t), y(t)). x(0) = 0, t > 0 x(t) > 0 y(t) = sin( 1x(t) ). , (tn)nN tn 0 y(tn) = (1)n ( ), y(tn) , g.
6.2
X, ( ) .
6.2.1. X . X:
x y X x, y.
2 f x y.
-
74
X (;) X.
.
6.2.2. X X. X. X .
. , X. A X ( A 6= ). , x A, A x. , , A , C1, C2, x1 A C1 x2 A C2. x1 x2. C1 = C2. , A X ( ).
X . C X x0 C. x C, x0 x, Ax X x0, x. Ax C, Ax C. C =
xC Ax. Ax x0
, C .
6.2.3. () 6.2.2 X, . C X A, C A X, A = C.() , .
.
6.2.4. X x p y X, x y. p X.
p X. x p x x X, f : [0, 1] X f(t) = x t [0, 1] x .
x, y X x p y. f : [0, 1] X f(a) = x f(b) = y. g : [0, 1] X g(t) = f(1 t). g X, y x. y p x.
x, y, z X x p y y p z. f1 : [0, 1] X, f2 : [1, 2] X f1(0) = x, f1(1) = y = f2(1) f2(2) = z. g : [0, 2] X
g(t) =
{f1(t), t [0, 1]f2(t), t (1, 2]
.
-
6.2. 75
g ( 4.1.6(iv)), g(0) = x g(2) = z. g X, x z, , x p z.
6.2.5. X , X. X. X .
. 6.2.2 .
6.2.6. X ( )., X , (;). , . , Q ( R) , Q.
- . , -, .
6.2.7. S 6.1.15() . {0} [1, 1] S. S S , {0} [1, 1] .
, .
6.2.8. X. X
(i) x X, x .
(ii) , .
(iii) x X, x .
(iv) , .
6.2.9. () R .
() [1, 0) (0, 1] R , .
() S , .
() Q , .
-
76
6.2.10. X U X, U X.
. X U X. C U x C, V x, V U . V , V C. C .
, . x X U Nx. C U x, C ( ) x C U . x , X x. x X , X .
:
6.2.11. X - U X, U X.
- :
6.2.12. X, X. X - , .
. C X x C. P X x, P , P C. X , P = C. , , P $ C. X C . Q X ( C) P , C = P Q. X , X . , P ( ) Q ( ) ( ). C, C. , P = C.
-
7
7.1
, , , 1 2. . , Bolzano-Weierstrass3:
.
- . , .
7.1.1. {Ui : i I} X ( X iI Ui = X) , J I
X =iJ Ui. F X , F
.
7.1.2. () -. , X = {x1, x2, . . . , xn} =
iI Ui, Ui ,
k = 1, 2 . . . , n ik I xk Uik . X =nk=1 Uik .
() X X ( X ) (;).
1 f : [a, b] R , x1, x2 [a, b] f(x1) f(x) f(x2), x [a, b].
2 - .
3 ( ) .
-
78
- , . , - , .
7.1.3. X F X. F - {Ui : i I} F X ( {Ui : i I} X F
iI Ui)
.
. .
7.1.4. X F X , F .
. (Ui)iI F X. (Ui)iI {X \ F} () X. i1, . . . ik I X = Ui1 Uik (X \F ). F Ui1 Uik , F .
- , .
7.1.5. (Fi)iI X 6= J I iJ Fi 6= .
7.1.6. X (Fi)iI X -
iI Fi 6= .
. X (Ui)iI X X =
iI Ui, J I X =
iJ Ui.
( Fi = U ci i I) (Fi)iI X
iI Fi = , J I
iJ Fi = .
, (Fi)iI X ,
iI Fi 6= .
, . , . .
7.1.7. X X ( , X ).
. () (x) X. F = {x : }. F F1 F2 2 1., (F) , 1, 2, . . . , n , 1, . . . , n, F
ni=1 Fi .
-
7.1. 79
X , 7.1.6 x F.
x (x). , U Nx , , x F = {x : }, U {x : } 6= , x U .
() (Fi)iI X, . 7.1.6,
iI Fi 6= .
H I xH iH Fi.
= {H I : H }
H1 H2 H1 H2.
, (,) (xH)H X. , x X, (xH).
i0 I U Nx. x (xH), H {i0} xH U .
xH iH
Fi Fi0 .
, xH Fi0 U Fi0 U 6= . x F i0 = Fi0 ( Fi0 ). , i0 I, x
iI Fi. ,
iI Fi 6= X .
7.1.7 , .
7.1.8. X (x) X . (x) .
Hausdorff , -.
7.1.9. X K X .
(i) F X x K x F , K F .
(ii) X Hausdorff L X K, K L .
(iii) X F X K, K F .
. (i) x K , Ux, Vx x Ux F Vx. (Ux)xK K. , x1, . . . , xn K K
ni=1 Uxi . :
U =
ni=1
Uxi V =ni=1
Vxi .
-
80
U ,V , K U F V . U V = . , x U , i {1, 2, . . . n} x Uxi x / Vxi . U V = .
(ii) x K y L x 6= y , X Hausdorff, x, y . L , (i), x K, x L . K , (i), K L .
(iii) x K, x F , x / F X T3. , (i) K F .
7.1.10. X Hausdorff.
(i) X , .
(ii) X .
. (i) F1, F2 X X. , F1 F2 ( 7.1.4) , 7.1.9(ii), F1, F2 . X .
(ii) K X x X \K. {x} K X. 7.1.9(ii), ., U V x U K V ,
x U X \ V X \K.
X \K Nx, x X \K. X \K , K .
7.1.11. 7.1.10 ;; Hausdorff , .
- .
7.1.12. X Y - . f : X Y , Y . , .
. (Vi)iI Y . f , {f1(Vi) : i I} X. X , i1, i2, . . . , ik I,
X = f1(Vi1) f1(Vi2) f1(Vik).
f , Y = Vi1 Vi2 Vik . , Y .
-
7.2. 81
( 7.1.13) , ( 7.1.14).
7.1.13. X - f : X R. f .
. 7.1.12, f(X) R . f(X) , f . sup f(X) R sup f(X) f(X). f(X) , R. sup f(X) f(X) = f(X) (sup f(X) = max f(X)). f . .
7.1.14. X Y Hausdorff. f : X Y , 1-1 , f1 ( f ).
. f . F X. X , F . 7.1.12, f(F ) Y , Y Haus-dorff, f(F ) Y . f .
7.2
, . ., .
7.2.1 ( ). X Y . x0 X U X Y , {x0} Y U , V x0
{x0} Y V Y U.4
. y Y , (x0, y) U , VyWy XY (x0, y) VyWy U . {x0}Y , Y , ,(Vy Wy)yY , . y1, y2, . . . yk Y
{x0} Y (Vy1 Wy1) (Vy2 Wy2) (Vyk Wyk) U.
V = Vy1 Vyk . V x0 {x0} Y V Y U . , (x, y) V Y , i {1, 2, . . . , k} (x0, y) Vyi Wyi .
(x, y) V Wyi Vyi Wyi U.4 V Y {x0} Y .
-
82
7.2.2. Y . , X = Y = R
N =
{(x, y) R2 : |x| < 1
y2 + 1
},
N R2 {0}R, V R {0} R.
4.3.3 (iv), (Xi)iI - X =
iI Xi ,
i- () , . - , .
7.2.3. X Y . ( ) 1 : X Y X .
. F X Y . 1(F ) , X \ 1(F ) . x X \ 1(F ). ({x} Y ) F = , {x} Y (X Y ) \ F .
(X Y ) \ F , , V x,
{x} Y V Y (X Y ) \ F.
x V X \ 1(F ). X \ 1(F ) .
7.2.4. X Y Haus-dorff. f : X Y
Gr(f) = {(x, f(x)) : x X}
f X Y .
. () (x, f(x)) Gr(f), (x, f(x)) (x, y) XY . x x f(x) y. f , f(x) f(x), Y Hausdorff, y = f(x). (x, y) = (x, f(x)) Gr(f). Gr(f) .5
() Gr(f) f . F Y , 2 : X Y Y , 12 (F ) X Y , Gr(f) 12 (F ) . , 7.2.3, 1 , 1(Gr(f) 12 (F )) X. 1(Gr(f) 12 (F )) = f1(F ) ( ). f , .
5 Y . Hausdorff, .
-
7.2. 83
7.2.5. X,Y . X Y . , - , , .
. U = (Ui)iI X Y . x X, {x} Y U , Ux U {x}Y , {x} Y
Ux. , Vx
x,
Vx Y Ux.
, x X Vx x, Vx Y Ux U .
V = (Vx)xX X, . V , x1, x2, . . . , xk X, X = Vx1 Vxk . Ux1 ,Ux2 , . . . ,Uxk ,
U0 = Ux1 Uxk U
X Y ,
X Y =
(ki=1
Vxi
) Y =
ki=1
(Vxi Y ) ki=1
(Uxi)
=U0.
, U , X Y .
. (z) XY . (z) . (z) = (x, y), (x) X, (x())M , x X. : M (y())M (y). Y , (y()) (y((j)))jJ , y Y . (z((j)))jJ = (x((j)), y((j)))jJ (z)( : J M ((J)) ) (x, y) XY , (x((j))) x ( (x())) (y((j))) y.
- 7.2.5 . , . ( 6.1.12), - . , . .
-
84
7.3 Tychonoff
7.3.1. (Xi)iI , X =
iI Xi . Xi
( i : X Xi ).
, X =iI Xi ,
Xi . : X . Zorn, , U0. ( 7.3.2) Xi U0 .
7.3.2. X . U0 X, ,
n N V0, V1, . . . Vn X , nk=1 Vk V0
V0 U0, k {1, . . . , n} Vk U0.
.
= {U : U X }
6= , X . (,) . Zorn, (. ) . C = {Uj : j J} .
U =jJUj = {U X : j J U Uj}.
, U , U .
U X {U1, U2, . . . Un} U Uj0 C {U1, U2, . . . Un} Uj0 ( C ) Uj0 , {U1, U2, . . . Un} () X. U U .
, Zorn , U0.. U X U / U0, m N U1, . . . Um U0 X = (U1 Um) U .: {U}U0 X, U0. U0, U0 {U} / , {U} U0 . , m N U1, . . . Um U0 X = (U1 Um) U .
U0 . n N V0, V1, . . . , Vn X ,
nk=1 Vk V0. Vk / U0
k = 1, . . . n V0 / U0. Vk / U0,
-
7.3. Tychonoff 85
nk N Uk1 , . . . Uknk U0, X = (Uk1 Uknk) Vk,
k = 1, . . . n. ( ):
X =
nk=1
(Uk1 Uknk)
(nk=1
Vk
)=
nk=1
(Uk1 Uknk) V0.
Uk1 , . . . Uknk U0 k = 1, . . . n, U0 -
, V0 / U0.
7.3.3. (Xi)iI X =
iI Xi . , i0 I,
Xi0 ={W Xi0 : W Xi0 1i0 (W ) U0},
U0 X 7.3.2.
. , , . , i I, yi Xi
yi Xi \{W Xi : W Xi 1i (W ) U0}.
y = (yi)iI X. U0 X, V U0 y V . V , B ( ), y B V . B
B =
nk=1
1ik (Wk), Wk Xik , k = 1 . . . n,
nk=1
1ik
(Wk) V . , V , 1ik (Wk) ( k = 1, . . . n), X V U0. U0, k {1, . . . , n} 1ik (Wk) U0. y B, y 1ik (Wk) yik Wk, yik .
7.3.4 (Tychonoff). (Xi)iI - , X =
iI Xi .
. X , i0 I Xi0 . X , 7.3.3, i0 I,
W = {W Xi0 : W Xi0 1i0 (W ) U0}
Xi0 . {W1, . . . ,Wn} - , 1i0 (W1), . . . ,
1i0
(Wn) U0 U0 ,
X 6=nk=1
1i0 (Wk) = 1i0
(nk=1
Wk
).
nk=1Wk 6= Xi0 , i0(X) = Xi0 . W
, Xi0 .
-
86
-
8
, . , . . , , . .
8.1
- . . , ( Jones).
8.1.1. X D X X.
8.1.2. () RS , Q RS , Q (a, b] 6= a, b R, a < b.
() X , X ( X X).
8.1.3. X . :
(i) X .
(ii) X .
. D X .(i) U X . D U U (
-
88
U ( X) D) . U .
(ii) Y f : X Y . f(D) Y , f D X,
f(D) f(D) = f(X) = Y.
f(D) Y , Y .
8.1.4. () X (Vi)iI , , X, I . , D X . i I D Vi . xi D Vi : I D (i) = xi. 1-1 ( Vi ), |I| |D| . I .
() X =iI Xi ( Xi 6= i I)
, Xi ( 8.1.3, i : X Xi i I).
() X1, X2 , X1X2 -. , D1 X1, D2 X2 . D1 D2 X1 X2, D1 D2 = D1 D2 = X1 X2. - X1X2 . , X1, X2, . . . , Xn ,
ni=1Xi .
() Xn, n N , X =n=1Xn
. , n N Dn Xn . t = (tn)nN
n=1Dn. :
D =
{x = (xn)
n=1
Dn : {n N : xn 6= tn}
}
D =
m=1
{x = (xn)
n=1
Dn : xn = tn, n > m
}
=
m=1
(mn=1
Dn {tm+1} {tm+2}
)
, . D . U , X. U U =
mk=1
1nk
(Uk), Uk Xnk , , k = 1, . . . ,m. ank Uk Dnk 6= k = 1, . . . ,m x = (xn) X :
xn =
{an n {n1, . . . , nm}tn n / {n1, . . . , nm}
x U D, U D 6= . D X.
-
8.1. 89
-. . , , - . , 8.1.6 , .
8.1.5. (Xi)iI , |I| c, X =
iI Xi .
. |I| c, 1-1 : I R, I R ( I (I)). t = (ti)iI X i I Di = {di,n : n N} Xi. n N, n (I1, . . . , In) R n (m1,m2, . . . ,mn) Nn x(n, I1, . . . , In,m1, . . . ,mn) = (xi)iI X
xi =
{di,mk , i Ik k {1, . . . , n}ti, i /
nk=1 Ik
( , i Ik, Ik ). D X. D X. U =
nk=1
1ik
(Uk) , X, i1, . . . , in I . I1, . . . , In i1 I1, i2 I2 . . . ,in In. k {1, . . . , n} Dik Uk 6= , mk N di,mk Uk. x = x(n, I1, . . . , In,m1, . . . ,mn) D U , k = 1, . . . , n ik Ik, ( x)xik = dik,mk Uk. D U 6= D .
8.1.6. (Xi)iI Hausdorff , X =
iI Xi . |I| c.
. D X . i I Ui, Vi , Xi. : I P(D) (i) = D 1i (Ui), i I. 1-1. , i, j I, i 6= j.
(i) \ (j) = D 1i (Ui) \ 1j (Uj)
D 1i (Ui) 1j (Vj) ( Uj Vj = )
6=
1i (Ui) 1j (Vj) D
X. (i) 6= (j). 1-1, |I| |P(D)| |P(N)| = c.
8.1.7. I , {0, 1}I ( {0, 1} R) |I| c (, 6.1.5 6.1.6).
8.1.8 (Jones). X . F X .
-
90
. F F ( F ) X ( F ). X -, A F U(A), V (A) X, A U(A) F \A V (A) ( A, F \A , X). D X .
: P(F ) P(D) (A) = D U(A).
1-1. , A, B P(F ) A 6= B. A \B 6= (A) \ (B) 6= ( B \A 6= (B) \ (A) 6= ).
(A) \ (B) = D U(A) \ U(B) D U(A) V (B).
U(A) V (B) ,
U(A) V (B) A (F \B) = A \B 6= .
D , D U(A) V (B) 6= . (A) 6= (B). 1-1 D ,
|P(F )| |P(D)| P(N)| = c.
|F | < c, |F | < |P(F )|.
8.1.9. Sorgenfrey . , RS , Sorgenfrey. 5.4.10, A = {(x,x) : x R} RS RS . |A| = c, R 3 x 7 (x,x) A 1-1 . Jones, RS RS T4.
8.1.10. . , Sorgenfrey , A , .
8.2
. , . , ( ).
8.2.1. X ( ) X - .
8.2.2. () A X . , x X Bx x X, Bx = {B A : B Bx} x A, .
-
8.2. 91
() X, Y , X Y . , (x, y) X Y (Un)nN x X (Vn)nN y Y , {Un Vm : n, m N} (x, y) X Y . , Xn, n N , X =
n=1Xn
.
8.2.3. () X . X , x X, {{x}} x.
() , x X {B(x, 1n ) : n N
} x.
8.2.4. X x X.
(i) A X, : x A (xn) A xn x.
(ii) f : X Y ( Y ) , : f x (xn) X xn x, f(xn) f(x).
. Bx = {Bn : n N} x. - Bn Bn+1 n N. (xn) X xn Bn n N, xn x. , U Nx n0 N Bn0 U ( Bx x). n n0, xn Bn Bn0 U xn x.(i) x A. A Bn 6= n N. (xn) A xn ABn n N. xn Bn n xn x. , .
(ii) , . , , (xn) X xn x, f(xn) f(x). , , f x. V Nf(x) f1(V ) / Nx. Bn * f1(V ) n N, Bn \ f1(V ) 6= n. , xn Bn \ f1(V ) n, (xn) x ( xn Bn n N) (f(xn)) f(x)( f(xn) / V n N), .
8.2.5. 1 = N {U} ( U N) . , () U , U N. (xn) N, xn U .
, , . , xn 6= xm, n 6= m (;). A B , .
A = {x2n : n N} B = {x2n1 : n N}.
xn U , x2n U x2n1 U . , U U A U 6= 6= B U ( U).
1
-
92
U {A}, U {B} , , N ( 2.2.7). U , A, B U . = A B U , . , 8.2.4 , .
8.2.6. X , Hausdorff, |X| c. , .
. D X . x X - (xn)nN D xn x, 8.2.4 (i). : X DN (x) = (xn)nN. 1-1. , x, y X (x) = (y) = (xn), xn x xn y. X Hausdorff, x = y. ,
|X| |DN| |NN| = c.
8.2.7. X , Y f : X Y , . Y .
. y Y x X f(x) = y. Bx = {Bn : n N} x X. f , By = {f(Bn) : n N} y . y. U Y y. f , Bn Bx x, f(Bn) U . , f(Bn) By. By () y , , Y .
8.2.8. X - X X . X X X ( - ), . , f 8.2.7 .
8.3
, . -, . , , .
8.3.1. X ( ) X .
-
8.3. 93
8.3.2. () X . , B X x X, Bx = {B B : x B} x .
() A X -. , B X, B = {B A : B B} A, .
() X, Y , X Y . , (Un)nN X (Vn)nN Y , {Un Vm : n, m N} X Y ., Xn, n N , X =
n=1Xn .
8.3.3. () X , X X , X . , .
() RS , ( x RS , {(a, x] : a Q, a < x} x) . , RS , RS RS , A = {(x,x) : x R} , , A .
8.3.4. X , Y f : X Y , . Y .
. B = {Bn : n N} X. f , BY = {f(Bn) : n N} Y . Y . U Y . f , f1(U) X, I N,
iI Bi = f
1(U).,
Uf
===== f(f1(U)
)= f
(iI
Bi
)=iI
f(Bi)
{f(Bi) : i I} BY . BY () Y , , Y .
8.3.5. 2 = N {U} - . ( ) ( ) - . , f 8.3.4 .
8.3.6. X .2
-
94
. B = {Bn : n N} X. Bn 6= n N. xn Bn n D = {xn : n N}. D . , U X , , n0 N Bn0 U , xn0 U . D U 6= , X .
8.3.7. X .
. () X D X .
B = {B(x, ) : x D Q, > 0}.
B . B X. U X x U . B B x B U . U , Q, > 0 B(x, ) U . D , y B(x, /2) D. B(y, /2) B, x B(y, /2) B(y, /2) U , z B(y, /2) (x, z) (x, y) + (y, z) < /2 + /2 = ( ) z B(x, ) U . B X , X .
() , 8.3.6.
8.3.8. A X .
. 8.3.7, X , A () ( 8.3.2()). A .
8.3.9. X . (Gi)iI X, J I -,
iI Gi =
iJ Gi.
. G =iI Gi B X.
B = {B B : i I B Gi}.
G =B. , B B i I B Gi
B G. B G. , x G i I x Gi.
B B x B Gi. B B x B.
G B. B B iB I B GiB .
B
BB
GiB G
G =BB GiB . J = {iB : B B}. J
( B ) G =iJ Gi.
8.3.10. X B . B X.
-
8.4. Lindelof 95
. (Bn)NN X. Bn - B 8.3.9 Bn =
m=1 Cn,m,
Cn,m B m N. {Cn,m : n, m N} B X (;).
8.4 Lindelof
- ( ).
8.4.1. X Lindelof ( X Lindelof) X , {Ui : i I} X X =
iI Ui, J I X =
iJ Ui.
8.4.2. () X Lindelof X .
() X Lindelof X - .
() A X Lindelof A X ( {Ui : i I} X A
iI Ui J I
A iJ Ui).
8.4.3. Lindelof X.
(i) X Lindelof.
(ii) Y f : X Y -, Y Lindelof.
. (i) F X {Vi : i I} F X. , {Vi : i I} {X \ F} X. X Lindelof, , {Vin : n N}{X \F}. F
{Vin : n N}
F Lindelof.
(ii) {Ui : i I} Y . {f1(Ui) : i I} X, , {f1(Uin) : n N}. , f , {Uin : n N} {Ui : i I} Y .
8.4.4. Lindelof.
. 8.3.9.
8.4.5. RS Lindelof.. {(ai, bi] : i I} RS . C =
iI(ai, bi). R ,
-
96
8.3.9 J I C =iJ(ai, bi).
x RS \ C, ix I x = bix .
RS = C (RS \ C)
(iJ
(ai, bi]
) xRS\C
(aix , bix ]
., RS \ C .. C = {(aix , bix ] : x RS \ C} . , x, y RS \ C x 6= y
(aix , bix ] (aiy , biy ] = (aix , x] (aiy , y] 6= .
x < y, x > aiy x (aiy , y) = (aiy , biy ) C, x RS \ C. , y < x.
C RS RS , 8.1.4() RS \ C .
8.4.6. Lindelof Lindelof. , RS Lindelof, SorgenfreyRS RS Lindelof, A = {(x,x) : x RS} - RSRS Lindelof ( ).
8.4.7. (X, ) Lindelof.
. () , 8.4.4.
() n N X =xX B(x,
1n ) X Lindelof,
An X X =xAn B(x,
1n ). D =
n=1An.
D X. , y X > 0. n0 N 1n0 < . X =
xAn0
B(x, 1n0 ),
x An0 y B(x, 1n0 ). (x, y) 13n .
, En(V ) En(W ), v Tn(V ), w Tn(W ), (x, v) < 13n (y, w) 1
n 1
3n 1
3n
=1
3n
(3) U A, En(U) U .
, y En(U), y B(x, 13n ) x Tn(U). Tn(U) x Sn(U) B(x, 1n ) U . (x, y) < 13n 1
3n 1
6n=
1
6n.
z / B(x, 16n ) B(x,1
6n ) En. En . (4),(5),(6) E .
-
104
, X - . n N An = {B(x, 1n ) : x X} X, Stone -
En =mNEnm.
9.2.6. A En a, b A, (a, b) < 2n . , En An, A B(x, 1n ) x X. a, b B(x, 1n ),
(a, b) (a, x) + (x, b) < 1n
+1
n=
2
n.
:E =
nNEn =
n,mN
Enm.
E Enm. E X. U X. x U E E x E U . > 0 B(x, ) U . n N 2n < . E
n X, E En x E. 7.2.6, y E, (x, y) < 2n < ., y B(x, ) , x E B(x, ) U . E X. T1, , , .
9.2.7. (X, ). X - ( (i)(ii) 7.2.3).
T1 X - . X . :
9.2.8. A = {Ai : i I} - X A =
iI Ai. A =
iI Ai.
. i I Ai A A, Ai A. iI
Ai A 2
, x A. A - , U Nx A, A1, A2, . . . , Ak. , , x A1, A2, . . . , Ak, x /
ki=1Ai, .
U \ki=1Ai x, A.
2 .
-
9.2. Nagata-Smirnov-Bing 105
x / A, . x Ai i {1, 2, . . . , k}. ,
A iI
Ai.
, .
9.2.9. X - B, X .
. , , . , - - X. :
. G X . (Un)nN,
G =nN
Un =nN
Un.
, B - , B =nN Bn,
Bn X. n N,
Cn = {B Bn : B G}
Cn Bn, Cn .
Un =Cn =
BCn
B.
Un ( ) 7.2.8, Un =BCn B,
Cn . , B G B Cn, Un G.
nNUn
nN
Un G.
G nN Un. x G. Gc
x / Gc. Q, U V , x U Gc V ,
x U V c G.
n N, B Bn x B U V c. V c , B V c G, B Cn.
x B BCn
B = Un nN
Un.
G nN Un.
X . A,B X . Bc , , (Un)nN Bc =
nN Un =
nN Un.
, Un B A nN Un. ,
-
106
(Vn)nN B V n A.
, nN Un,
nN Vn
A B, . , , . n N, :
Un = Un \ni=1
V i Vn = Vn \ni=1
U i.
Un, Vn . :
U =nN
Un V =nN
Vn
U V , . A B .
x A. x Un n N, A nN Un.
x / V n n N.
x Un \ni=1
V i = Un U .
A U . , B V .
, , U V 6= . , , x U V . m,n N
x Um = Um \mi=1
V i x Vn = Vn \ni=1
U i.
m n. x Um, x / U1, . . . , Um, . . . , Un, . n m. , U V .
, U V A B . X .
9.2.10. 7.2.9 - F3 , , G.
9.2.11. T1 . X - B, X ( (iii)(i) 7.2.3).
3 X F . F G.
-
9.2. Nagata-Smirnov-Bing 107
. X [0, 1]B, (p, q) = supBB |p(B)q(B)|, p, q [0, 1]B. , .
(i) B B |p(B) q(B)| 0. (p, q) 0.
(p, q) = 0 |p(B) q(B)| = 0 B B p(B) = q(B) B B p = q.
(ii) (p, q) = supBB |p(B) q(B)| = supBB |q(B) p(B)| = (q, p).
(iii) 4
(p, r) = supBB|p(B) r(B)|
supBB{|p(B) q(B)|+ |q(B) r(B)|}
supBB|p(B) q(B)|+ sup
BB|q(B) r(B)|
= (p, q) + (q, r)
. F X . f : X [0, 1] F = f1({0}).
, 7.2.10, F G . , 7.2.9, X , 5.4.17, f : X [0, 1] F = f1({0}).
, F , - X [0, 1]., (;), f [a, b], a, b R ( a < b). B - , B =
nN Bn, Bn -
. , Bn Bm = n 6= m (;). B B n N B Bn , fB : X [0, 1n ] f
1B ({0}) = Bc. :
: X [0, 1]B (x) = (fB(x))BB.
1 1: x, y X x 6= y. X T1, U, V X x U y V . , B B x B U V c. fB(x) > 0 fB(y) = 0, (x) 6= (y).
: x X. , (x) U x (U) V . , > 0 U x (U) B((x), ). > 0. n N, Bn , Un x, Bn. B Bn Un, x fB(x) = 0. fB(y) = 0 y Un. |fB(x) fB(y)| = 0.
4 U, V R, supU +supV = sup(U +V ) ( U + V := {u + v : u U, v V }) K L R, supK supL.
-
108
B Un 6= . fB : X [0, 1n ] ,
(fB(x) 2 , fB(x) +2 ) [0,
1n ] [0,
1n ]
Wn(B) x,
fB(Wn(B)) (fB(x) 2 , fB(x) +2 ) [0,
1n ].
B, B1, . . . , Bk. Vn = Un
ki=1Wn(Bi), x. ,
y Vn Wn(Bi) |fBi(x) fBi(y)| < 2 . y Vn |fB(x) fB(y)| < 2 , B Bn.
, N N 1N N B Bn |fB(x)fB(y)| < 1n 0} [0, 1]B, B0. V . , h V . B(h, h(B)), g B(h, h(B))
(h, g) < h(B0) = |h(B) g(B)| < h(B0) B B= |h(B0) g(B0)| < h(B0)= g(B0) > 0.
g V B(h, h(B)) V . V . (X), W = V (X).
B0, fB0(x) > 0. (x) V g(x) W . W (U). p W . p = (y) y X. p W = V (X) V , fB0(y) = p(B0) > 0. y B0 U , p = (y) (U). W (U), (U) . ,, U X , (U) (X). , 1 .
X , .
7.2.7
