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pi
2010-11
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I 1
1 31.1 pi . . . . . . . . . . . . . . . . . . . . . . . . . . .
31.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 5
1.2.1 pipi . . . . . . . . . . . . . . . . . . . . 71.2.2 . . .
. . . . . . . . . . . . . . . . . . . . . . . . 111.2.3 . . . . . .
. . . . . . . . . . . . . . . . . . . . 13
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 15
2 192.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 19
2.1.1 . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 . .
. . . . . . . . . . 212.1.3 . . . . . . . . . . . . 242.1.4 pi . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 . . . . . . . . . . . . . . . 282.3 . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 31
3 pi 353.1 . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
353.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.2 . . .
. . . . . . . . . . . . . . . . . . . . . . 42
3.3 . . . . . . . . . . . . . . . . . . . . . 453.3.1 . . . . .
. . . . . . . . . . . . . . . . . . 453.3.2 . . . . . . . . . . . .
. . . . . . . . . . . 46
3.4 . . . . . . . . . . . . . . . . . . . . . . . . 463.5 . . .
. . . . . . . . . . . . . . . . . . . 48
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iv
3.5.1 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . .
483.5.2 . . . . . . . . . . . . . . . . . . . . . . 49
3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 52
4 594.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 594.2 . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.1 Lipschitz . . . . . . . . . . . . . . . . . . . . . . . .
. 634.3 , , . . . . . . . . . . . . . . . 65
4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 654.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . .
664.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.4 pi . . . . . . . . . 694.4.1 Urysohn . . . . . . . . . . . .
. . . . . . . . . . . . . 694.4.2 . . . . . . . . . . . . . . . . .
. . . . . . . 704.4.3 . . . . . . . . . . . . . . . . . . . 71
4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 73
II pi 79
5 815.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 815.2 Cantor . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 855.3 Baire . . . . . . . . . . . . . . . . . . . . . . .
87
5.3.1 Baire . . . . . . . . . . . . . . . . . 895.4 * . . . . .
. . . . . . . . . . . . . . . . . . . . . . 925.5 Banach . . . . .
. . . . . . . . . . . . . 975.6 . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 99
6 pi 1036.1 pi . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1036.2 pi . . . . . . . . . . . . . . . . . . . . . . . .
1056.3 pi . . . . . . . . . . . . . . . . . . 1116.4 pi . . . . . .
. . . . . . . . . . . . . 1156.5 Cantor . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 1176.6 . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 120
III 127
7 1297.1 : . . . . . . . . . . . . . . . 129
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v
7.2 : . . . . . . . . . . . . . . . 1347.2.1 . . . . . . . . . .
. . . . . . . . . 1377.2.2 , pi . . . . . . . . . . . . . . . .
139
7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1427.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 146
8 pi 1518.1 C(K) . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 1518.2 pi Weierstrass . . . . . . . . . . . . . .
. . . . 1528.3 . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 157
IV 159
pi 161.1 pi . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 161.2 pi . . . . . . . . . . . . . . . . . . . . 163
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I
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1
1.1 pi
1.1.1 (). X . X : X X R pi :
(i) (x, y) 0 x, y X (x, y) = 0 x = y ( ).
(ii) (x, y) = (y, x) x, y X ( ).(iii) (x, z) (x, y) + (y, z) x,
y, z X ( ). X (X, ) . X .
1.1.2. () R
d(x, y) = |x y|, x, y R.() Rm, m- ~x = (x1, . . . , xm)pi , : ~x
= (x1, . . . , xm) ~y = (y1, . . . , ym) Rm,
2(~x, ~y) =
(mi=1
(xi yi)2)1/2
.
pi (pi 1.3).() X pi pi: : X X R
(x, y) =
{1, x 6= y0, x = y
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4
( pi (i), (ii) (iii) ). X.
() X pi pi : f : X R pi 1-1, pi df X :
df (x, y) = |f(x) f(y)|, x, y X. df X.
() n- Hamming.
Hn = {0, 1}n ={
(x1, x2, . . . , xn) xi = 0 1, i = 1, . . . , n}.
h : Hn Hn R, pi h(x, y) pi pi n- x = (x1, . . . , xn) y = (y1, .
. . , yn),
h(x, y) = card({
1 i n : xi 6= yi}).
h Hn. (Hn, h
) Hamming h Hamming.
1.1.3 ( ). (X, ) . A pipi pi X, pi A : AA R
A(x, y) = (x, y), x, y A( pi A A) A. A pi pi pi A.
pi, pi R pi .
1.1.4 (). () (X, ) . (X, ) pi C > 0 x, y X (x, y) C. ,
sup{(x, y) : x, y X}
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1.2 5
1.1.5. () R d(x, y) = |x y| .
() R pi pi arctan : R (pi2 , pi2 ),
(x, y) = | arctanx arctan y|, x, y R
diam (R, ) = pi. diam (R, ) pi pi
diam (R, ) | arctann arctan(n)| n N,
diam (R, ) limn | arctann arctan(n)| =
pi
2(pi
2
)= pi.
pipi pi | arctan t| < pi2 t R( ).
() R
(x, y) =|x y|
1 + |x y| , x, y R
pi , (x, y) < 1 x, y R. diam (R, ) = 1.() X, (X, ) (, pi pi ,
1).
1.2
pi pi . pipi, . pi , pi pi .
1.2.1 (). X pi . X : X R :() x 0 x X x = 0 x = ~0 ( ).
() x = || x R x X ( ).
() x+ y x+ y x, y X ( ).
X, (X, ) .
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6
1.2.2. () X, d :X X R
d(x, y) = x y, x, y X ( pi pi X pi ). ,
d(x, y) = x y 0 x, y X d(x, y) = x y = 0 x y = ~0 x = y.
d(y, x) = y x = (1)(x y) = | 1| x y = x y x, y X. x, y, z X
d(x, z) = x z = (x y) + (y z) x y+ y z = d(x, y) + d(y, z).
pipi, d :
d pi , d(x+ z, y + z) = d(x, y) x, y, z X.
d , d(x, y) = ||d(x, y) x, y X R.
, pi pi pi pi - pipi. , pi pi pi pi pi pipi pi pipi pi .
() pi . X 6= {0} pi pi : x X, x 6= 0, pi span({x}) = {x : R} X
pi pi ( , pi R). pi pi, pipi .
pi ( ) X pi pi pi pi . pi, X , pi : X R (x, y) = x y. pi pi: pi
, pi x X,x 6= 0,
nx = nx = (nx, 0) = 1 n N x = 1/n n = 1, 2, . . ., pi pi pi.
pipi .
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1.2 7
1.2.1 pipi
1. Rm supremum : Rm R : x =(x1, . . . , xm) Rm
x := max{|xi| : i = 1, . . . ,m}.
pi .
x+ y = |xi0 + yi0 |
pi i0 {1, . . . , n}. i0,
|xi0 + yi0 | |xi0 |+ |yi0 | x + y.
pi,x+ y x + y.
(Rm, ) `m.
2. Rm 1- 1 : Rm R
x1 := |x1|+ + |xm| =mi=1
|xi|.
pi pi R. (Rm, 1) `m1 .
3. Rm 2 : Rm R
x2 :=(
mi=1
|xi|2)1/2
.
pi pi pi pi CauchySchwarz.
1.2.3 ( CauchySchwarz). x1, . . . , xm y1, . . . , ym pi- .
,
mi=1
|xiyi| (
mi=1
|xi|2)1/2( m
i=1
|yi|2)1/2
.
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8
pi. pi pi pi Schwarz. B =mi=1 |xiyi|,
A =mi=1 |xi|2 C =
mi=1 |yi|2. pi B2 AC
(2B)2 4AC. p : R R
p() := (|x1|+ |y1|)2 + + (|xm|+ |ym|)2 0,
pi pi pi pi
p() = A2 + 2B+ C 0
R. A = 0 xi = 0 i = 1, . . . ,m pi ( ). pi pi A > 0 p() pi R.
pi pipi (2B)2 4AC 0, pi . 2
pi pi .
x+ y22 =mi=1
|xi + yi|2
=
mi=1
|xi|2 + 2mi=1
xiyi +
mi=1
|yi|2
x22 + 2mi=1
|xiyi|+ y22
x22 + 2x2y2 + y22pi pi CauchySchwarz. ,
x+ y22 (x2 + y2)2 = x+ y2 x2 + y2.
(Rm, 2) `m2 .
4. , Rm pi p-, 1 < p
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1.2 9
1.2.4 ( Holder). x1, . . . , xm y1, . . . , ym pi p, q > 1 1
1p +
1q = 1,
mi=1
|xiyi| (
mi=1
|xi|p)1/p( m
i=1
|yi|q)1/q
.
pi. pi log : (0,+) R , x, y > 0
log
(1
pxp +
1
qyq) 1p
log(xp) +1
qlog(yq)
log(xy) log(
1
pxp +
1
qyq).
pi log pi
() xy xp
p+yq
q x, y 0.
x1, . . . , xm y1, . . . , ym pi . pi pi (|x1|p+ +|xm|p)1/p 6= 0
(|y1|q+ +|ym|q)1/q 6= 0. x1 = = xm = 0 y1 = = ym = 0
mi=1 |xiyi| = 0 pi
pi.
ai =|xi|
(|x1|p + + |xm|p)1/p , i = 1, . . . ,m
bi =|yi|
(|y1|q + + |ym|q)1/q , i = 1, . . . ,m
pi ai, bi 0 mi=1
api =mi=1
bqi = 1.
pi () ai, bi
aibi api
p+bqiq
1 p q .
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10
pi i = 1, . . . ,m pi
mi=1
aibi 1p
mi=1
api +1
q
mi=1
bqi =1
p+
1
q= 1.
, mi=1 |xi||yi|
(|x1|p + + |xm|p)1/p(|y1|q + + |ym|q)1/q 1,
pi :
mi=1
|xiyi| (|x1|p + + |xm|p)1/p(|y1|q + + |ym|q)1/q.
2
1.2.5. Holder pi - CauchySchwarz: pipi pi p = q = 2.
1.2.6 ( Minkowski). x1, . . . , xm y1, . . . , ym pi p > 1,
(
mi=1
|xi + yi|p)1/p
(
mi=1
|xi|p)1/p
+
(mi=1
|yi|p)1/p
.
pi. pi pi mi=1 |xi+ yi|p > 0, pi
.
(+)
mi=1
|xi + yi|p =mi=1
|xi + yi|p1|xi + yi| mi=1
|xi + yi|p1|xi|+mi=1
|xi + yi|p1|yi|.
Holder mi=1 |xi + yi|p1|xi| pi
mi=1
|xi + yi|p1|xi| (
mi=1
|xi + yi|q(p1))1/q ( m
i=1
|xi|p)1/p
pi q p, 1p +1q = 1 q(p 1) = p. , pi
mi=1
|xi + yi|p1|xi| (
mi=1
|xi + yi|p)1/q ( m
i=1
|xi|p)1/p
.
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1.2 11
pi pi
mi=1
|xi + yi|p1|yi| (
mi=1
|xi + yi|p)1/q ( m
i=1
|yi|p)1/p
.
, pi (+)
mi=1
|xi + yi|p (
mi=1
|xi + yi|p)1/q ( m
i=1
|xi|p)1/p
+
(mi=1
|yi|p)1/p
(mi=1
|xi + yi|p)11/q
(
mi=1
|xi|p)1/p
+
(mi=1
|yi|p)1/p
.
pipi pi 1 1q = 1p . 2 x+ yp xp + yp p-
Minkowski (pi x = (x1, . . . , xm) y = (y1, . . . , ym)). (Rm,
p) `mp .
5. pi pi pi pi dp(x, y) =x yp Rm. x = (x1, . . . , xm) y = (y1,
. . . , ym) Rm,
dp(x, y) =
(mi=1
|xi yi|p)1/p
1 p 0 : n N |x(n)| M}
pi pi. ` supremum : ` R
x := sup{|x(n)| : n = 1, 2, . . .}.
pi :
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12
() x 0 x `. x = 0, |x(n)| 0 n N, x(n) = 0 n = 1, 2, . . .. pi, x
= 0.() x = supn |x(n)| = || supn |x(n)| = || x, R.() x, y ` n N.
,
|x(n) + y(n)| |x(n)|+ |y(n)| x + y. supremum pi n pi
x+ y = supn1|x(n) + y(n)| x + y.
2. c0 c0(N) ,
c0 ={x : N R
limnx(n) = 0
} pi ( pi ` - ) pi. supremum pi pi `.
3. `1 `1(N) 1- 2 ,
`1 =
{x : N R
n=1
|x(n)| < +}
pi c0. , n=1 |x(n)| < +
limnx(n) = 0. 1 : `1 R
x1 :=n=1
|x(n)|.
4. , 1 p < , `p `p(N) p- pi pi x : N R pi n=1 |x(n)|p < +.
`p p
xp :=( n=1
|x(n)|p)1/p
.
pi Minkowski pipi pi , pi p pi ( ).
2 pi pi pi .
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1.2 13
5. c00 c00(N) . , x c00 pi n0 n0(x) N x(n) = 0 n n0. pi pipi pi
p, 1 p .
1.2.3
1. C([0, 1]) pi [0, 1]
C([0, 1]) = {f : [0, 1] R | f }
pi pi. C([0, 1]) : C([0, 1]) R,
f = sup{|f(t)| : t [0, 1]}.
sup pi, |f | : [0, 1] R , max , pi , pi . .
2. C([0, 1]) pi pi 1-
f1 := 10
|f(t)| dt
, 1 p
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14
t [0, 1]. pi pi 10|f(t)|pdt = 1
0|g(t)|qdt = 1,
pi () 10
|f(t)g(t)| dt 1p
10
|f(t)|pdt+ 1q
10
|g(t)|qdt
=1
p+
1
q= 1
=
( 10
|f(t)|pdt)1/p( 1
0
|g(t)|qdt)1/q
.
pipi, pi f g, f1 := f/fp g1 := g/gq.
pi, pi Holder pi pi Minkowski pipi ,pi p: Minkowski . f, g : [0,
1] R 1 p
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1.3 15
1.3
1. (X, ) . pi x y x y x, y X.
2. (X, ) . :
() |(x, z) (y, z)| (x, y) x, y, z X.() |(x, y) (z, w)| (x, z) +
(y, w) x, y, z, w X.
3. R : RR R (a, b) = |a b|. pi (R, ) .
, : (X, ) d : X X R
d(x, y) =x y, x, y X,
(X, d) .
4. (X, d) . 1 = min{d, 1}, 2 = d1+d d = d
(0 < < 1) X.
5. d1, d2 X d1+d2, max{d1, d2}, min{d1, d2} X. d X, d2 X;
6. (X, d) . pi :
() diam(A) = 0 A = A (, A = {x} pi x X).() A B X diam(A)
diam(B).() A,B X
diam(A B) min{diam(A),diam(B)} max{diam(A),diam(B)} diam(A
B).
diam(A B) diam(A) + diam(B)
pi A,B X;
() (An) pi X diam(An) 0 n ,
n=1An pi ( pi ).
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16
7. pi A (X, ) pi x0 X r > 0 (a, x0) r a A.
8. A1, . . . , Ak pi (X, ). A1 A2 Ak pi .
9. () f : [0,) [0,) f(0) = 0 f(x) > 0 x > 0. pi pi f pipi,
. f(x+ y) f(x) + f(y) x, y 0. : d X f d X.
() pi f : [0,) R+, pi pipi f :
(i) f .
(ii) x 7 f(x)x , x > 0 .() pi () () 4 .
10. ( Holder ) f, g : [0, 1] R p, q (. p, q > 1 1p +
1q = 1). 1
0
|f(t)g(t)| dt ( 1
0
|f(t)|p dt)1/p( 1
0
|g(t)|q dt)1/q
.
11. (C([0, 1]), p)
fp =( 1
0
|f(x)|p dx)1/p
.
12. S pi . (mk) ,
kmk < +. pi d S :
x = (x(n)), y = (y(n)) S,
d(x, y) =
n=1
mn|x(n) y(n)|
1 + |x(n) y(n)| .
(S, d) , pi .
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1.3 17
13. P pi pi . p(x) =a0 + a1x+ + anxn pi pi P, p
h(p) = max{|ai| : i = 0, 1, . . . , n}.
() P pi h : P R P.() : P R,
(p) = |a0|+ |a1|+ + |an|
P.() h(p) (p) (n+ 1)h(p) pi p pi n.
14. (P, h) pi (c00, ). pi f : (P, h) (c00, )
p(x) = a0 + a1x+ + anxn f7 f(p) := a = (a0, a1, . . . , an, 0,
0, . . .)
pi pi. , f 1-1,pi pipi
(i) f(p+ q) = f(p) + f(q),
(ii) f(p) = f(p),
(iii) f(p) = h(p) p, q P R.
15. pi pi p Z . m,n Z m 6= n, p(n,m) p pi |nm|, m 6= n,
p(m,n) = max{k 0 : m nmod pk}.
p : Z Z R
p(m,n) =
{2p(m,n), m 6= n0, m = n
p Z (Z, p) .
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18
16. 6= A (0,+). pi pi (X, )
A = {(x, y) : x, y X, x 6= y}.
17. `p, 1 p c0.() : 1 p < q `p `q .() : 1 p
-
2
2.1
pi pi . pi x : N R (pi R). , xn := x(n) n- x {xn}n=1 {xn}
(xn).
(xn) R, (xn) pi x :
> 0 pi n0 n0() : n N n n0(), |xn x| < .
pipi, limxn = x limnxn = x , pi pi, xn x.
pi (xn) - (X, ). pi pi - pi : (xn) x X pi x {xn : n n0}. , (xn)
x pi xnpi x 0 n pi. pi pi pi . pi- pi pi pi .
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20
2.1.1
(X, ) . X x : N X. xn := x(n) n- x {xn}n=1 {xn} (xn) x = (x1,
x2, . . . , xn, . . .). 2.1.1 ( ). (xn) (X, ) x X pi ( -)
> 0 pi n0 n0() N n n0 (xn, x) < . xn
x pi xn x. x - (pi ) .
2.1.2. (xn) (X, ) x X. , xn
x ((xn, x))n pi .
pi. : ((xn, x))n R > 0 pi n0 n0() N n n0 (xn, x) = |(xn, x)
0| < . xn x. 2 2.1.3. (xn) (X, ). pi (xn), .
pi. pi xn x xn y, pi x, y X. x = y.
: n N 0 (x, y) (x, xn) + (xn, y).
> 0, pi n0 N , n n0,
(xn, x) 0.
Mmk = max{|xm(i) x(i)| : i = 1, . . . , k}., k N Mmk 0 m (;).
pi, pi k k() N 1
2k< 2 . k M
mk+1 0, pi m0(, k) m0 N
m m0 Mmk+1 < 2 . pi m m0,
d(xm, x) Mmk+1 +1
2k 0 pi n0 n0() N m,n n0 (xm, xn) < . 2.1.7. (X, ) . , X
Cauchy.
pi. (xn) . , pi x X xn x. > 0. xn
x, pi n0 N n n0 (xn, x) < 2 . m,n n0. ,
(xn, xm) (xn, x) + (x, xm) < 2
+
2= .
pi, (xn) Cauchy. 2
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2.1 25
2.1.8 ( ). (xn) (X, ). (xn) A = {xn : n N} pi X. , pi C > 0
(xm, xn) C m,n N. 2.1.9. (X, ) . , X .
, X .
pi. (xn) (X, ). , pi n0 > 1 m,n n0 (xn, xm) < 1. , (xn,
xn0) < 1 n n0.
C = max {(x1, xn0), . . . , (xn01, xn0), 1} > 0., n N
(xn, xn0) C.pi pi ( )
sup{(xm, xn) : m,n N} 2C.pi, (xn) .
pipi pi pi, . 2
2.1.10. () pi pi pi . pi (Q, | |) : (qn) pi qn = (1 + 1n )
n, , .
pi, (R, )
(x, y) = |f(x) f(y)|, x, y R,pi1 f(t) = t|t|+1 . xn = n - -., pi
(f(n)) pi ||-,
(n,m) = |f(n) f(m)| 0 m,n., (xn) -, pi x R (n, x)0. pi pi, f(n)
1,
(n, x) = |f(n) f(x)| |1 f(x)| . , |1 f(x)| = 0, x|x|+1 = 1.
pi.
1pi f 1-1 pi R pi (1, 1).
-
26
pi pi pi - R : , . pi pi .
() pi pi pi -pi pi . R , xn = (1)n , |xn xn+1| = 2 n = 1, 2, . .
..
2.1.4 pi
(X, ) (xn) X. k1 < k2 < < kn < (xkn) pi (xn).
2.1.11. () k : N N x : N X X, x k : N X pi (xn). , pi (xn) (xn)
.
() (kn) , kn n n = 1, 2, . . .. (pi) pi pi.
pi, pi pipi pi , xn
x pi (xkn) (xn) xkn x (4()). pi pi pi pi pi :
2.1.12. (X, ) (xn) X. (xn) pi, .
pi. (xn) xkn x, pi (xkn) pi
(xn).
. (xn) x.
, > 0. pi (xn) pi n1 N
(xn, xm) 0 pi (x0, ) > 0 : x X (x, x0) < (f(x), f(x0))
< .
f : X Y X X. f : (X, ) (Y, ), C(X,Y ). , Y = R C(X) C(X,R).
2.2.2. () X (Y, ) . f : X Y .pi. f : (X, ) (Y, ) x0 X. f x0. : >
0. pi = 12 > 0. pi , x X (x, x0) < = 12 x = x0, f(x) = f(x0)
(f(x), f(x0)) = 0 < . 2
() f : N X ( ).() I : (c00, ) (c00, 2) .pi. f : X Y x0 X pi
:
f : X Y x0 X pi > 0 : > 0 pi x X (x, x0) < (f(x),
f(x0)) .
-
2.2 29
pi I 0 = (0, 0, 0, . . .). : xn = (1n,
1n, . . . ,
1n
n
, 0, 0, . . .),
n N,
I(xn) I(0)2 = I(xn)2 = xn2 = 1
xn 0 = xn = 1n.
pi = 12 pi > 0 pi x c00 x < I(x) I(0)2 > 12 ( pi x = xn
pi n 1n< ). pi, I : (c00, ) (c00, 2) 0. 2
pi , pi pipi pi pi R.
2.2.3 ( ). (X, ) (Y, ) f : X Y x0 X. :() f x0.
() (xn) X xn x0 f(xn) f(x0).
() (yn) yn x0, (f(yn)) -.
pi. pi () ().
()() xn x0 > 0. pi f x0 pi > 0 x X (x, x0) < (f(x),
f(x0)) < . pipi, pi xn x0 pin0 N n n0 (xn, x0) < . pipi pi
(f(xn), f(x0)) < n n0, f(xn) f(x0).()() f x0. pi -pi. , pi 0
> 0 (xn) X xn
x0 (f(xn), f(x0)) 0 n = 1, 2, . . . ( ). pi pi f(xn)
f(x0), pi pi (, (f(xn), f(x0)) < 0). () ().
()() : yn x0, pi pi f(yn) f(x0), (f(yn)) -.
()() (xn) (X, ) xn x0.
yn = (x0, x1, x0, x2, x0, x3, . . .) yn =
{x0, n = 2k 1xk, n = 2k
,
-
30
pi x0. pi pi, pi y Y f(yn)
y. pipi, f(y2n1) = f(x0) f(x0), y = f(x0). , pi f(xn) =
f(y2n)
y = f(x0). 2pi pi
.
2.2.4 ( ). (X, ), (Y, ) (Z, ) . f : X Y g : Y Z . f x0 X g f(x0)
Y , g f : X Z x0.
pi. (xn) X xn x0. f x0, f(xn) f(x0). g f(x0) Y , (yn) Y yn f(x0)
g(yn) g(f(x0)).
, f(xn) Y f(xn) f(x0). pi,g(f(xn)) g(f(x0)).
(xn) X xn x0 (g f)(xn) = g(f(xn)) g(f(x0)) = (g f)(x0).
pi , g f x0. 2 pi
pi pi . pi , -pi pi .
2.2.5. f, g : (X, ) R, R x0 X. pi f, g x0. ,
() f + g, f fg x0.() pipi g(x) 6= 0 x X, fg X
x0.
pi. pi pi: pi, fg x0, , ,
(xn) X pi x0, ((
fg
)(xn)
)
(fg
)(x0). pi pi, f g x0. pi
f(xn) f(x0) g(xn) g(x0). g(xn) 6= 0 n N g(x0) 6= 0, (
f
g
)(xn) =
f(xn)
g(xn) f(x0)
g(x0)=
(f
g
)(x0).
-
2.3 31
pi f + g, f f g x0 . 2
2.2.6. (X, ) . C(X) - f : X R .
2.3
1. (X1, d1), . . . , (Xk, dk) pipi . pi pi X =
ki=1Xi:
(x, y) = max{di(x(i), y(i)) : i = 1, 2, . . . , k}
p(x, y) =
(ki=1
[di(x(i), y(i))]p
)1/p, 1 p
-
32
5. (X, ) . X X pipi d. : (X X, d) R (x, y) 7 (x, y) .
6. (xn) (X, ). pi pi x X : f : X R f(xn) f(x). xn x;
7. (Xn, dn), n = 1, 2, . . . dn(x, y) 1 x, y Xn, n = 1, 2, . .
..
X =n=1
Xn =
{x = (x(1), x(2), . . . , x(n), . . .) : x(n) Xn
}.
, X pi pi pi n- - Xn. d : X X R
d(x, y) =
n=1
1
2ndn(x(n), y(n)).
(X, d) d .
8. (Xn, dn)nN X =n=1Xn. d :
X X R d(x, y) =
n=1
1
2ndn(xn, yn)
1 + dn(xn, yn).
(X, d) d .
9. 1 p
-
2.3 33
12. (X, ) (xn) X xn 6= xm n 6= m.
A = {xn : n = 1, 2, . . .}. : xn x X 1-1 f : A A f(xn) x.
13. (xn) (X, ). (xn)
n=1
(xn, xn+1) < +.
pi :
() (xn) (, ). ;() (xn) pi .() (xn) pi pi -.
14. (xn) (X, ). (xn) pi- pi (xkn) (xkn , xkn+1) 0
B(x0, ) = {x X : (x, x0) < }.() -pi x0 > 0
B(x0, ) = {x X : (x, x0) }.() - x0 > 0
S(x0, ) = {x X : (x, x0) = }. pi pi , pipi pi B(x0, ), S(x0,
).pi.
3.1.2. () X . (X, ) r > 0
B(x, r) =
{ {x}, 0 < r 1X, r > 1
S(x, r) =
{X \ {x}, r = 1, r 6= 0, r 6= 1. .
-
36 pi
() R ,
B(x, ) = (x , x+ ), B(x, ) = [x , x+ ], S(x, ) = {x , x+ }.
() [0, 2] ,
B(1/2, 1) = [0, 3/2), B(1/2, 1) = [0, 3/2], S(1/2, 1) = {3/2}.()
R2 1, 2 . 1, 2, pi,
B1(0, 1) = {(x, y) : |x|+ |y| 1}( (1, 0) (0,1)),
B2(0, 1) = {(x, y) : x2 + y2 1}( (0, 0) 1)
B(0, 1) = {(x, y) : |x| 1, |y| 1}( (1,1)). 3.1.3 ( ). (X, ) A X.
x A (interior point) A pi x > 0 B(x, x) A. 3.1.4 ( ). (X, ) G X.
G - (open) x G pi x > 0 B(x, x) G., G .
3.1.5. () pi . : B(x, r) pi (X, ) y B(x, r). (x, y) < r, pi
> 0 < r (x, y). pi B(y, ) pi B(x, r), t B(y, ) (y, t)
<
(t, x) (t, y) + (y, x) < + (x, y) < r.pi pi t B(x, r),
B(y, ) B(x, r).() X. pi (X, ) . , A X. A : a A 0 < < 1 B(a, )
= {a} A.() (a, b] (R, | |), a < b . pi b > 0 pi , B(b, ) *
(a, b], b+ 2 B(b, ) b+ 2 / (a, b].() (R, | |), Q , pi .
-
3.1 37
pi pi pi pi- .
3.1.6. (X, ) . , :1
() X, .() (Gi)iI pi X
iI Gi
.
() G1, G2, . . . , Gn ni=1Gi = G1 Gn .
pi. () pi .
() x iI Gi. , pi i0 I x Gi0 . Gi0 ,pi 0 > 0 B(x, 0) Gi0
iI Gi. ,
iI Gi .
() x G1 Gn. , x Gi i = 1, . . . , n. Gi , i = 1, . . . , n pi i
> 0 B(x, i) Gi. = min{1, . . . , n} > 0. , i = 1, . . . , n
i,
B(x, ) B(x, i) Gi.
pi, B(x, ) ni=1Gi. 2 3.1.7. pi (X, ) . pi, R Gn =( 1n , 1n ), n
N, pi
n=1Gn = {0}, pi (
pi).
pi pi .
3.1.8. (X, ) G X. :() G pi X.
() x G (xn) X xn x pi n0 N n n0 xn G.pi. pi pi G pi X. x G (xn)
X xn
x. G , pi > 0 B(x, ) G. xn x, pi n0 N n n0 (xn, x) < .pi,
xn B(x, ) G n n0.
1 pi X pi pi X, , pi pipi , pi X. , 3.1.6 pi ( pi ) pi .
-
38 pi
pi (), G . ,pi x G > 0 pi B(x, ) pi G.
pi, n = 1, 2, ... pi xn B(x, 1n
) (X \G), xn / G (xn, x) < 1
n.
(xn) x G, pi (). 2
3.1.9. (X, ) . pi V X ( pi) pi pi X.
pi. V pi pi pi 3.1.6. , V . , x V pix > 0 B(x, x) V . V =
xV B(x, x). 2
* 3.1.1 ( pi ). U R .
pi. x U . U , pi x > 0 (x x, x+ x) U .
ax = inf{s : (s, x] U}. (ax, x] U : t (ax, x] t {s : (s, x] U},
pi s < t (s, x] U , t U . pi,
bx = sup{t : [x, t) U}pi [x, bx) U . pi x U (ax, bx) U , () U
=
xU
(ax, bx).
1. z, x U z (ax, bx) (ax, bx) = (az, bz)., [z, bx) U bx bz (ax,
z] U az ax. pi, (ax, bx) (az, bz). , x (az, bz), pi (az, bz) (ax,
bx). 2. x, y U (ax, bx) (ay, by) = (ax, bx) = (ay, by)., (ax, bx)
(ay, by) 6= . , z (ax, bx) (ay, by) pi pi pi
(ax, bx) = (az, bz) = (ay, by).
pi () U U =
jJ
Ij ,
-
3.1 39
pi Ij . , pipi -: : J Q : j J pi (j) qj Ij . pi , Ij . Q , J pi.
2
3.1.2
3.1.10 ( ). (X, ) F X. F - (closed) pi F c X \ F -. 3.1.11. ()
(X, ) {x}, x X ( ).
() pi B(x, r) . , X \ B(x, r) : y X \ B(x, r). , (x, y) > r.
pi 0 < < (x, y) r B(y, ) X \ B(x, r) , z B(y, ) (z, y)
<
(z, x) (y, x) (z, y) > (x, y) > r,
z X \ B(x, r)., R , [a, b]
( ).
() Q R , R\Q pi.
() X . pi (X, ) ( ).
() (X, d) . x X (xn) X, xn x.
E = {xn : n = 1, 2, . . .} {x} (X, d).
, y / E, = d(x, y) > 0. xn x, pi n0 N xn B(x, /2) n >
n0.
r = min{d(y, xi) : i = 1, 2, . . . , n0} > 0.
pi 0 < < min{r/2, /2}, B(y, ) X \ E. 3.1.12. pi pi pi, pi
(X, ) pi . pi, - pi (, ) pi (,).2
2 pi clopen.
-
40 pi
pi pi -. (X, ) G X, G : x G (xn) X xn xpi n0 N n n0 xn G. pi pi
: pi X pi pi :
3.1.13. (X, ) F X. -:
() F pi X.
() (xn) F xn x X, x F .
pi. pi pi F (xn) F pi pi x X. x / F . , x X \F X \F . pi , pi n0
N xn X \F n n0. pi: n n0 pi xn / F .
pi () F . , X \ F . pi, pi x X \ F : > 0, B(x, ) F 6= . pi =
1n , n = 1, 2, . . ., xn F (xn, x) < 1n , n = 1, 2, . . .. (xn)
F xn
x. (), pi x F . pi. 2 pi
pi pi pipi pi pi:
3.1.14. (X, ) . :() X, .() F1, F2, . . . , Fn pipi pi X,
ni=1 Fi .
() (Ei)iI pi X, iI Ei
.
pi. pi pipi pi pi De Morgan(iI
Ai
)c=iI
Aci
(iI
Ai
)c=iI
Aci
pi pi . 2
3.1.15. pi , . , R , Fn =
[1n , 1],
n = 2, 3, . . ., n=1 Fn = (0, 1] (0, 1] pi R.
-
3.2 41
3.2
3.2.1
3.2.1 ( ). A pi (X, ). (interior) A A intA ( A). ,
A intA = {x A | > 0 : B(x, ) A} . 3.2.2. A X A A ., x A. , pi
> 0 B(x, ) A. y B(x, ) , = (x, y) > 0 B(y, ) B(x, ) A. pi, y
B(x, ) A. , B(x, ) A. , x A.
3.2.3. () (a, b] R pi (a, b).
() Q R .() pi pi.
pi pi pi.
3.2.4. A,B pi (X, ). , :
() A A.() A =
{V A : V }. , A pi pi A.
() A = A A .() A B, A B.() (A B) = A B.() A B (A B).pi. () pi
.
() x A pi > 0 B(x, ) A.
x B(x, ) {V A : V }
B(x, ) pi pi A., x {V A : V }. , pi Vx A ,
x Vx. , pi > 0 B(x, ) Vx, B(x, ) A, pi x A.() pi pi A . pi, A
= A pi A .
-
42 pi
, A A = A. : pi () A A. , A , A, A A.() A B x A. , pi > 0
B(x, ) A B. pi x B.() A B A, (A B) A pi (). , (A B) B.pi,
(A B) A B., A A B B, A B A B. A B , pi ()
A B (A B). pipi pi (A B) = A B.() A A B, A (A B). , pi B (A B),
pi A B (A B). 2 3.2.5. pi . pi, R A = [0, 1] B = (1, 2) A B = (0,
1) (1, 2), (A B) = (0, 2).
pi A = Q B = R \ Q . A = B = A B = R. pi, A B = , (A B) = R.
3.2.2
3.2.6 ( pi). (X, ) A X. x X pi (contact point) A > 0 AB(x, )
6= ( pi x pi A).. x X pi A pi (an) A an
x. pi ( pi pi 3.1.13).
3.2.7. () (R, | |) A = (0, 1]. 0 pi A.
() (xn) (X, ) xn x, x
pi A = {xn : n N}.() . A (X, ), x X pi A x A. 3.2.8 ( ). (X, ) A
X. - (closure) A ( cl(A)) A pi . ,
A cl(A) = {x X : > 0, A B(x, ) 6= }.
-
3.2 43
3.2.9. A X A A ., (xn) A xn
x. n N pi an A (an, xn) < 1n , xn pi A. ,
(an, x) (an, xn) + (xn, x) < 1n
+ (xn, x) 0,
an x. (an) A an x. pi, x A.
3.2.10. () (R, | |) Q = R R \Q = R.() (R, | |), a, b R a < b
cl (a, b) = cl (a, b] = cl [a, b) = [a, b].() . A X A = A.
pi pi pi .
3.2.11. (X, ) A,B X. , :() A A.() A =
{F A : F }. , A pi X pi pi A.
() A = A A .
() A B, A B.() A B = A B.() A B A B.pi. () pi . A pi A.
() A A A. pi, {F A : F } A., F A F . x A pi
(xn) A xn x. , xn F F pi
x = limnxn F . , A F . pi A {F A : F }.
() A = A A A ., A , A pi ,
A ={F A : F } A, A A. A A, pi
pi A = A.
() A B A B B, A B. B pipi A, pi pi pi , A. , A B.() pi () A A B
B A B pi A A B B A B, A B A B. pipi, A A B B
-
44 pi
A B A B. A B , pi () pipi A B A B. , A B = A B.() AB A, pi A B
A. , A B B, A B AB. 2 3.2.12. pi . pi, (R, | |) Q Qc = , Q Qc = R.
pi pi :
3.2.13. (X, ) A X. :
() X \A = (X \A).() X \A = X \A.pi. () x X. , pi :
1. > 0 B(x, ) A 6= . , x A.2. pi > 0 B(x, ) A = , B(x, ) X
\ A. ,
x (X \A). pi A (X \ A) X.
pi X \A = (X \A).() pi X \A A, pi
X \X \A = (X \ (X \A)) = A. pi pi X \A = X \A. 2 pi G F .
3.2.14 ( G F). (X, ) A X.() A G pi X.() A F pi X.
3.2.15. () pi F. G.
() pi G. F.
() A G Ac F.
() (R, | |) (a, b] F G. , pi k N a+ 1k < b,
(a, b] =
n=k
[a+
1
n, b
]=
n=1
(a, b+
1
n
).
pi (), () () .
-
3.3 45
3.3
3.3.1
1 : A pi (X, ), pi A : A A R A(x, y) = (x, y), x, y A A. pi pi
pi (A, A).
3.3.1. (X, ) A X. :() G A (A, A) pi V pi X G = A V .() B A, A
intX(B) intA(B).
pi. () pi pi G pi (A, A)., pi pi A ,
G =xG
BA(x, x) =xG
(B(x, x
) A) = A (xG
B(x, x)
).
[ pi pi pi pi pipi pi pi pi, pi pi pi.]
V =xGB(x, x) G A V , pi V
pi X ( pi pi X)., V pi X G = AV . , x G
x V pi > 0 B(x, ) V . pi BA(x, ) = AB(x, ) AV , x G pi A. , G
pi (A, A).
() pi, A intX(B) A- pi A pi . , pi pi A- B. ,
A intX(B) intA(B).
2
3.3.2. pipi pi . pi, R , Z pi , intZ(N) = N intR(N) = . ,
= Z intR(N) $ intZ(N) = N.
-
46 pi
3.3.2
pi pi pi pi.
3.3.3. (X, ) (A, A) pi. :
() F A (A, A) F = AE pi E (X, ).
() B A clA(B) = A clX(B).pi. () F A A \ F A, A \ F = A G pi G X.
,A F c = A G.. F = A Gc. pi F AGc. F Gc. pi pi pi x F x G, x FG AG
= AF c., x / F , pi. pi, F Gc.
AGc F . pi pi,pi pi x A Gc x / F , x A \ F x / G. x A F c x / G,
pi pi A F c = A G.
Gc X, pi E = Gc .
() pi pi pi, A clX(B) A- B A clX(B). , pi 3.2.11() clA(B) A
clX(B).
, x AclX(B) > 0. B(x, )B 6= pipix A, BA(x, ) B = B(x, ) A B
6= , pi x clA(B). 2 3.3.4. () R A =(0, 1] {2}. (0, 1], {2} A.()
(R3, ) pi xy-pipi H ( (x, y, 0)). D2 xy-pipi (D2 = {(x, y, 0) R3
:x2 + y2 1}) H. pi F H H R3 ( ).
3.4
3.4.1 ( ). (X, ) A X. x X (accumulation point) A pi x pi A pi x.
, > 0
B(x, ) (A \ {x}) 6= . A A pi A.
-
3.4 47
3.4.2. (X, ) , A X x X. :() x A.() > 0 A B(x, ) pi .() pi
(an) A an
x an 6= x n = 1, 2, . . ..pi. () (): > 0. x A, pi y 6= x y
B(x, ). pi A (B(x, ) \ {x}) pipi pi. A (B(x, ) \{x}) = {y1, . . . ,
yk} = min{(x, y1), . . . , (x, yk)} > 0. x A, pi y 6= x y B(x,
). ,y B(x, ) \ {x} ( < y 6= x). pi, y = yi pi 1 i k. pi (x, y)
< (x, yi).() (): A B(x, 1) pi , pi a1 A a1 6= x (x, a1) < 1.
pi a1, . . . , an A ai 6= x (x, ai) < 1i i = 1, . . . , n. n+1 =
1n+1 . A B(x, n+1) pi , pi an+1 A an+1 6= x (x, an+1) < n+1 =
1n+1 . pi, (an) A, pi x, (x, an) < 1n , pi pi
pi an x.
pi () () pi (). 2 3.4.3. (X, ) A X. ,
A = A A.
pi, A pi .
3.4.4 (). (X, ) A X. x X (boundary point) A pi xpi A Ac. , >
0 B(x, )A 6= B(x, )Ac 6= . A (boundary) A bd(A) (A).
3.4.5. (X, ) A X. ,() bd(A) = bd(Ac).() A = bd(A) int(A).() X =
int(A) bd(A) int(Ac).() bd(A) = A \ A. , bd(A) = A X \A. , bd(A)
.() A bd(A) A.pi. . 2
-
48 pi
3.5
3.5.1 pi
3.5.1 (pi pi). (X, ) D X. D pi (dense) X, D = X.
3.5.2. () Q, R \Q pi (R, | |).() c00 pi (`1, 1).pi (). 1 pi pi .
a = (an) `1,
n=1 |an| < +, > 0.
pi Cauchy pi n0 N
n=n0+1
|an| < .
x = (a1, ..., an0 , 0, ...) c00. ,
d1(a, x) = a x1 =
n=n0+1
|an| < ,
x Bd1(a, ). , c00 Bd1(a, ) 6= . > 0 , a c00. 2() (
Kronecker). R \Q.
D() := {(cos(2pin), sin(2pin)) : n N}
pi S1 = {(x, y) R2 : x2 + y2 = 1} (). 3.5.3. (X, ) D X. :
() D pi X.
() F D F , F = X.() G X G D 6= .() x X > 0 D B(x, ) 6= .() x
X pi (xn) D xn x.() (X \D) = .pi. ()() F pi X D F . , D F X F .()()
pi pi G X G D = . , D Gc. Gc , pi pi Gc = X, G = , pi.
-
3.5 49
()() , pi .()() x X. , n N D B(x, 1n ) 6= . , pi (xn) D xn B(x,
1n ) n N. (xn, x) 0, xn x.()() pi int(X \ D) 6= . , pi x X \ D >
0 B(x, ) X \ D. , B(x, ) D = . pi pi pi (xn) D xn
x. , pi n N (xn, x) < . , xn B(x, ) xn D pi pi, B(x, ) D =
.()() pi pi 3.2.13 X \ D = (X \ D). pi (X \D) = , X \D = . , D = X.
2 3.5.4. Qn pi `np , 1 p .pi. pipi 1 p < ( pipi p = ).
x = (x(1), . . . , x(n)) `np > 0. Q pi R, i = 1, . . . , n pi
q(i) Q |q(i) x(i)|p < pn . q = (q(1), . . . , q(n)). , q Qn , pi
p,
dp(x, q) =
(ni=1
|x(i) q(i)|p)1/p
0 x = (xn) `p. n=1 |xn|p < + pi Cauchy pi n0 N
n=n0+1
|xn|p < p
2.
i = 1, . . . , n0, pi pi Q R pi qi Q |xi qi|p < p2n0 . q =
(q1, . . . , qn0 , 0, . . .) q D
dpp(x, q) =
n0n=1
|xi qi|p +
n=n0+1
|xn|p < n0 p
2n0+p
2= p,
, dp(x, q) < . pi, D pi `p. 2
pi pi, (`, ) (pi 3.5.11()).
() Hilbert, H .pi. E - [1, 1] pi H. pi x H > 0. pi n0 N
nn0
12n 0 B(x, ) G. pi pi Q R pi q Q 0 < 2q < . D pi X, pi
B(x, q) pi D, xn. x B(xn, q) ( xn B(x, q)) B(xn, q) G. , y B(xn,
q)
(x, y) (x, xn) + (xn, y) < q + q < , y B(x, ) G.()(). pi
pi O pi X pi pi (). 6= U O pi xU U . , D = {xU : U O} pi pi X ( ).
2 3.5.8. (X, ) . pi A X pi .
pi. X , pi O -pi X : G X x G pi U O x U G. , OA = {U A : U O}
pipi pi pi A . , (A, A) . 2
pi pi .
3.5.9. (X, ) A pi pi X : pi > 0 (x, y) x, y A x 6= y., (X, )
.
pi. pi (X, ) . , pipi D. pi B(x, 2 ), x A. pi pi. D pi, D B(x, 2
) 6= x A, pi dx D dx B(x, 2 ). pi A 3 x 7dx D, pi 1-1. , A pi pi D.
pi, D , A pi. 2
3.5.10. (X, ) pi pi pi .
pi. . 2
pi pi pi -.
3.5.11. () (R, ) , R , .
-
52 pi
pi. pi 3.5.9 A = R. (x, y) = 1 x 6= y R pi. pi, (R, ) , 2
, pi S (S, ) .
() ` (`(N), ) .pi. ` S = {A : A N}, pi A A N. , A(n) = 1 n A
A(n) = 0 n / A., S pi {0, 1}N pi pi (pi ). pipi, A 6= B pi x pi ,pi
|A(x) B(x)| 1 A B 1 pi pi B
(A,
12
)
. pi pi 3.5.9 pi ` . 2
3.6
1. (X, ) F,G pi X. F G , F \G G \ F .
2. (X, ) . pi A X pi (X, ).
3. f : R R . G = {x R : f(x) > 0} pi R F = {x R : f(x) = 0}
pi R.
4. R R .
5. pi pipi pi .
6. pi . pi ;
7. (X, d) , x X > 0. , pi
B(x, ) = {y X : d(x, y) }.
[pi: A X A A.]
-
3.6 53
8. (X, d) . X X = {(x, x) :x X}. pi X X pi d2, pi
d2((x1, y1), (x2, y2)) =d2(x1, y1) + d2(x2, y2).
, pi pi X X.
9. pi pi pi R pi pi pi ;pi pi R pi pi pi ;
10. A,B pi (X, d). pi :() A B = X, A B = X.() A B = , A B =
.
11. (X, d) . pi :() (A \B) A \B A,B X.() A \B A \B A,B X.pi
;
12. (X, ) 6= A X. diam(A) = diam(A). A;
13. () A pi (X, ) G A. G A X.
() A pi (X, ) G A. G A X;
14. pi pi R \Q pi .
15. (X, ) . B(x, r) = B(x, r) x X r > 0.
16. c0 pi `. pi pi c00; pi `; pi `;
17. (X, ) . :() G .() A X, G A G A.() A X, G A = G A.
-
54 pi
18. pi R pi .
19. pi R pi pi ( pi R) pi .
20. () n Z, Fn pi (n, n + 1). F =nZ Fn. pi F R.
pi: pi n pi n > 0 |xy| n pipix Fn y Fm, n 6= m.() R pi .
21. (X, d) . pi :() X pi pi , pi G X, G 6= X \G 6= .() X pi , pi
G X G X \ G pi.
22. (X, ) x, y X x 6= y. pi U, V x U , y V U V = .
23. (X, ) , x X F pi X x / F . pi U, V x U , F V U V = . pi pi ,
pipi, U V = ;
24. (X, ) A X. A pi A, A. pi :
() A = A A. pi A pi .
() A .
() A B X A B.() A = (A). , A A .
() (A) A. pi A R .
25. :() pi A R A = N.() pi A R A = Z.() pi A R A = Q.
-
3.6 55
26. (X, ) . A,B X, pi A pi B :
dist(A,B) = inf{(a, b) : a A, b B}.pi pi:
() A B 6= , dist(A,B) = 0.() dist(A,B) = dist(A,B).
() dist(A,B C) = min{dist(A,B),dist(A,C)}.() pi pi A,B (X, ) pi
pi.
27. (X, ) A X. x X pi xpi A pi {x} A:
dist(x,A) = inf{(x, a) : a A}.pi :() dist(x,A) = 0 x A.()
|dist(x,A) dist(y,A)| (x, y) x, y X.() {x X : dist(x,A) < } , {x
X :dist(x,A) } .() A B A, dist(x,A) = dist(x,B) x X.
28. (X, ) A X. pi A = {x X : dist(x,A \ {x}) = 0} .
29. (X, ) . pi pi X pi X .
30. (X, ) A X. pi A:
() bd(A) = bd(Ac).
() cl(A) = bd(A) A.() X = A bd(A) (X \A).() bd(A) = A \A bd(A) =
AX \A. pi, .
() A bd(A) A.
31. (X, ) A,B X. pi :
-
56 pi
() A pi X bd(A) .
() A B = bd(A B) = bd(A) bd(B).
32. pi A R (bd(A)) = R.
33. A pi (X, ). G H A, pi U V X G = A U H = A V .
34. (X, ) . - pi X pipi .
35. (X, ) . :
() D pi pi X, D G = G piG X.
() G pi pi X D pi pi X, G D pi pi X. G pi;
() pi pi X pi pi X;
36. (X1, d1), . . . , (Xn, dn) . (X, d) X =
ni=1Xi d = max1in di. :
() Gi di- Xi, i = 1, . . . , n, ni=1Gi d-
X.
() Fi di- Xi, i = 1, . . . , n, ni=1 Fi d-
X.
() Di pi Xi, i = 1, . . . , n, D =ni=1Di pi
X.
, (Xi, di), i = 1, ..., n (X, d) -.
37. (X, ) P X. P . pi :
() P (X, ) P = P .() ( ) R ( ) . pi, R pi R2.() pi P R pi. [pi.
P pi. , P = {xn : n N}.
-
3.6 57
[an, bn] , n N, [an, bn]P 6= xn / [an, bn].]
38. A R x R. x pi A > 0 A (x , x+ ) pi. pi :() A pi.
() A pi P pi A P = P A \ P .() A pi R pi P Z A = P Z P Z = .
39. (X, ) (xn) X. x X (xn) pi pi (xkn) (xn) xkn
x. L(xn) (xn). pi
() xn x L(xn) = {x}. ;
() A = {xn : n N} X A L(xn) A. pi pi .
() L(xn) pi X.
() A , L(xn) 6= . pipi, (xn) -Cauchy, -.
() x (xn) > 0 n Npi m n xm B(x, ).
40. ; pi (X, d) pi pi pi A X G A pi A.
41. (X, ) . pi :
() X pi .
() S pi pi X pi S, pi S.
42. R \Q. D() := {(cos(2pin), sin(2pin)) : n N}
pi S1 = {(x, y) R2 : x2 + y2 = 1}.
43. (X, ) . A X pi pi int(A) = .pi :
() A X pi pi A (X \A).
-
58 pi
() A X pi pi X \A pi .
() A pi X, A pi pi A = bd(A).
() A pi pi pi X X \ B pi X \ (A B) pi X.() pipi pi pi pi pi X
pipi pi X.
44. (qn) Q.
In =
(qn 1
2n, qn +
1
2n
), n N.
U =n=1 In pi pi R U c
pi pi.
45. (X, ) A X. pi :() A pi pi.
() A pi .
() pi X pi pi A.
() pi X pi pi pi A.
46. (Xn, n), n = 1, 2, . . . n(x, y) 1 x, y Xn, n = 1, 2, . . ..
(X, ), pi X =
n=1Xn
(x, y) =n=1 2
nn(x(n), y(n)). pi = ((n)) X.
Dm = {x = (x(n)) X : x(n) = (n), n > m}, m = 1, 2, . . .
D :=
m=1
Dm.
pi D pi X.
47. A,B , pi pi R. pi f : A B pi , 1-1 pi.
-
4
4.1
(X, ) (Y, ) . 2.2 f : X Y pi x0 X: f x0 > 0 pi (x0, ) > 0
x X (x, x0) < (f(x), f(x0)) < .
4.1.1. pi : > 0 pi > 0
f(B(x0, )) B(f(x0), )., > 0 pi > 0 pi ( X) x0 pi, f , pi (
Y ) f(x0) .
pi pi - f : X Y pi X Y (pi f x X). 4.1.2. f : (X, ) (Y, ). :() f
.
() G pi Y , f1(G) pi X.() F pi Y , f1(F ) pi X.
pi. () () G pi Y . f1(G) pi X. f1(G) pi .
-
60
, x f1(G). , f(x) G G , pi pi > 0 B(f(x), ) G. f x, pi > 0
f(B(x, )) B(f(x), ) G, B(x, ) f1(G). pi, f1(G) .
() () pi f1(Y \A) = X\f1(A): F pi Y . , Y \F pi Y . pi pi , f1(Y
\F ) pi X. , f1(Y \ F ) = X \ f1(F ). X \ f1(F ) , f1(F ) .() () x
X. f x. > 0. pi B = B(f(x), ). Y \ B pi pi f1(Y \ B) = X \ f1(B)
pi , f1(B) .pipi, x f1(B) f(x) B. , pi > 0 B(x, ) f1(B)., f(B(x,
)) B = B(f(x), ). 2
pi pi - f : X Y : 4.1.3. f : (X, ) (Y, ). :() f .
() A X f(A) f(A).() B Y f1(B) f1(B).() C Y f1(C) (f1(C)).pi. ()
() A X y f(A). , pi x A y = f(x). x A, pi (xn) A xn x. f x, f(xn)
f(x) = y. , f(xn) f(A) n N. pi, y f(A).() () B Y , A = f1(B) ()
f(f1(B)) f(f1(B)) B. pipi pi f(f1(B)) B pi f : X Y B Y .
, pi f(f1(B)) B pi f1(B) f1(B).() () C Y . pi 3.2.13
X \ (f1(C)) = X \ f1(C) = f1(Y \ C) pi pi f1(Y \ C) f1(Y \ C)
pi
X \ (f1(C)) f1(Y \ C).pipi,
f1(Y \ C) = f1(Y \ C) = X \ f1(C).
-
4.2 61
pi,X \ (f1(C)) X \ f1(C),
f1(C) (f1(C)).() () G pi Y . C = G () pi
f1(G) = f1(G) [f1(G)]
G = G. pi f1(G) . pi pi 4.1.2 f . 2
4.2
4.2.1 ( ). (X, ) (Y, ) . f : X Y > 0 pi () > 0 x, y X (x,
y) < (f(x), f(y)) < . pi . ., p : R R p(x) = x2 : > 0, pi
x = 1 y =
1 +
2 , |x y| <
|p(x) p(y)| = 1 + 2
4> 1.
pi pi p .
4.2.2. () f : (X, ) (Y, ) pi X pipi .
pi. > 0. x, y X (x, y) < 12 pi x = y, (f(x), f(y)) =0 <
. 2
pi pi a : N Y .
() A pi (X, ). pi pi A dA : X R
t 7 dist(t, A) inf{(t, a) : a A}.
dA : pi
() |dA(t) dA(s)| (t, s)
t, s X. , > 0, pi = t, s X (t, s) < pi |dA(t) dA(s)| <
= .
-
62
pi (): t, s X. a A
dA(t) (t, a) (t, s) + (s, a).
pi, dA(t) (t, s) {(s, a) : a A}. pi dA(t) (t, s) dA(s). ,
dA(t) dA(s) (t, s).
pi
dA(s) dA(t) (t, s),
pi pi pi (). 2() f : R R pi pi, limx f(x) = 0, ( pi pi ).
(pi pi pi ) pi :
4.2.3. (X, ), (Y, ) f : X Y . :
() f .
() (xn), (zn) X (xn, zn) 0, (f(xn), f(zn)) 0.
pi. pi pi f (xn), (zn) X (xn, zn) 0. > 0. pi , pi > 0
x, z X (x, z) < (f(x), f(z)) < .
(xn, zn) 0, pi n0() N : n n0 (xn, zn) < . n n0. , (xn, zn)
< xn, zn X, pi (f(xn), f(zn)) < . > 0 , pi (f(xn), f(zn))
0.
: f , pi > 0 :
> 0 pi x, z X (x, y) < (f(x), f(y)) .pi = 1, 12 , . . .
,
1n , . . ., xn, zn X (xn, zn) 0. f , pi > 0 x, y X (x, y)
< (f(x), f(y)) < . (xn) , pi pi n0 N m,n n0 (xn, xm) < .
pipi pi m,n n0 (f(xn), f(xm)) < .() (). x X. f x. (xn) X xn
x f(xn) f(x). (yn) = (x, x1, x, x2, x, x3, . . .). (yn) x () .
pi pi (f(yn)) pi . , pi(f(y2n1)) (f(yn)) f(x), pi f(x). (f(yn)) pi,
, f(x)( pipi pi). 2
4.2.5. () pi pi pi pi ( pi () () ). f : (0,+) (0,+) f(x) = 1x
,
(1n
)nN
(0,+) pi , f( 1n) = n.() pi, pi () (), pi pi - . pi, p : R R
p(x) = x2 . , (xn) R ,pi pi , , pix R xn x , pi p , p(xn) p(x),
(p(xn)) .
4.2.1 Lipschitz
4.2.6. f : (X, ) (Y, ) C > 0. f CLipschitz (, pi Lipschitz C
> 0) x, y X
(f(x), f(y)) C (x, y).
f Lipschitz pi Lipschitz pi C > 0.
-
64
4.2.7. () Lipschitz . . pi f : [0,) R f(t) =
t, pi Lipschitz
().
() Lip(X,Y ) Lipschitz f : X Y . pipi Y = R, pi Lip(X) Lip(X,R).
Lip(X,Y ) pi pi . Lip(X) pi , , pi pi : A pipi pi X, pi dA : X R
dA(x) = dist(x,A) 1Lipschitz.
() f, g Lip(X), f + g Lip(X) R f Lip(X). ,
(Lip(X),+, ) pi C(X).
() f Lip(X,Y )
fLip = inf{C > 0 : (f(x), f(y)) C (x, y), x, y X}.
fLip = sup{(f(x), f(y))
(x, y)
x, y X,x 6= y} (f(x), f(y)) fLip(x, y) x, y X.
4.2.8. () pi f : R R pi - pi Lipschitz.
pi. pi pi C > 0 |f (x)| C x R. pi x, y x < y. f [x, y],
(x, y)
f(x) f(y)x y = f
().
, f(x)f(y)xy C, , |f(x) f(y)| C |x y|. 2
, sin, cos, arctan Lipschitz.
() X . : X R 1Lipschitz, .
() f(x) = sinx, x R, fLip = 1. pi pi: fLip 1 pi |f (x)| = |
cosx| 1 x R, fLip 1 pi lim
t0+sin tt = 1.
-
4.3 , , 65
4.2.9. (X, ), (Y, ) f Lip(X,Y )., f pi pi X pi Y . , f
CLipschitz, pi A X,
diam(f(A)) Cdiam(A).
pi. A X . , diam(A)
-
66
4.3.3. () () pi u : R R u(x) = x+ u u : R R u(x) = x + u (pi u
R) . , Ty : `n2 `n2 Ty(x) = x+ y, pi y Rn, pi.() n < m `n2
isom `m2 .
, pi i : `n2 `m2
i(x1, x2, . . . , xn) = (x1, x2, . . . , xn, 0, . . . , 0 m
)
.
4.3.2
4.3.4 ( ). X , - X. ( ) .
xn x xn x.
4.3.5. X , X. :
() , .
() I : (X, ) (X,) . , I I1 pi.() ( Hausdorff) > 0 x X pi 1, 2
> 0 B(x, 1) B(x, ) B(x, 2) B(x, ).() G X .() F X .pi. () (). pi
pi pi I I1.() (). > 0 x X. I , pi 1 > 0 I(B(x, 1)) B(I(x), )
B(x, 1) B(x, ). , I1 pi 2 > 0 B(x, 2) B(x, ).() (). pi G . . x
G, G pi > 0 B(x, ) G. pi pipi > 0 B(x, ) B(x, ). , B(x, ) G.
x G , G . .
() (). pi: pi F pi ().
-
4.3 , , 67
() (). (xn) X xn x. xn x. , pi 0 > 0 pi (xkn) (xn) (xkn , x)
0 n = 1, 2, . . .. F = {y X : (y, x) 0}. F pi pi pi . pipi, xkn F
() n = 1, 2, . . . xkn
x. pi x F , (x, x) 0, pi.pi, xn
x. . 2 pi pi,
pi pipi . pi pi.
4.3.6. X, pi X pi .
pi. : X X R (x, y) = (x,y)1+(x,y) , x, y X. , (xn, x) 0 (xn, x)
0. 2
4.3.3
4.3.7 (). f : (X, ) (Y, ). f - (homeomorphism) 1-1, pi (. f f1
). pi f : (X, ) (Y, ), (X, ) (Y, ) X
hom Y X ' Y . 4.3.8. () - .
() X. , , (X, ) (X,) ( pi). : pi X , X (X, ) (X,) , , . X = c00
T : c00 c00
T (x1, x2, . . . , xn, 0, . . .) = (x1, 2x2, . . . , nxn, 0, . .
.).
T c00 : yT = Ty y c00. , (c00, ) (c00, T ) ( T ) pipi c00
().
4.3.9. f : (X, ) (Y, ) 1-1 pi. :
() f .
() (xn) X x X, xn x f(xn) f(x).
-
68
() G X f(G) Y .() F X f(F ) Y .() d(x, y) = (f(x), f(y)) X .
pi. . 2
4.3.10. (X, ) .
pi. pi pi pi 4.3.6 pi 4.3.8(). 2
4.3.11. 1 Hilbert H.pi. (X, ) . pi F : X H pi 1-1 F1 : F (X) X
pi ., X ' F (X) H.
H y : N [1, 1] d(y, y) =
n=1 2
n|y(n) y(n)|. pi pi pi (X, )
(x, y) 1 x, y X ( pi pi). D = {xn : n = 1, 2, . . .} pi pi X.
piF : X H F (x) = y pi y(n) = (x, xn), n N.. F 1-1.
x, y X F (x) = F (y) , (x, xn) = (y, xn) n = 1, 2, . . .. >
0. D pi, pi xn D (y, xn) = (x, xn) < 2 . ,
(x, y) (x, xn) + (y, xn) <
pi (x, y) = 0, x = y.
. F ( Lipschitz).
d(F (x), F (y)) (x, y) x, y X, pi d Hilbert.
d(F (x), F (y)) =
n=1
2n|(x, xn) (y, xn)| n=1
2n(x, y) = (x, y).
, F1 : F (X) X am, a X F (am)
d F (a) am a ( pipi pi ).1 X Y pi f : X Y , 1-1 f1 : f(X) X
pi . , X pi Y .
-
4.4 pi 69
: F (am)d F (a). d ,
limmF (am)(n) = F (a)(n) n N, limm (am, xn) = (a, xn) n N. >
0. D pi X, pi n0 N (a, xn0) 0. f x, pi > 0 f(B(x, )) B (f(x),
2). pi
diam(f(B(x, ))) diam(B(f(x),
2
)) .
,f (x) = lim
r0diam(f(B(x, r)))
, > 0 , pi f (x) = 0.
() (). pi f x. , pi 0 > 0 : > 0 pi x X (x, x) < (f(x),
f(x)) 0. ,f (B(x, )) 0 > 0. ,
f (x) = lim0
f (B(x, )) 0.2
-
4.5 73
4.4.11. f : (X, ) (Y, ). C(f) X pi f . , C(f) Gpi X.
pi. pi pi f x X f (x) = 0. , C(f) = {x X : f (x) = 0}
C(f) =
n=1
{x X : f (x) < 1
n
}.
, n N, Bn ={x X : f (x) < 1n
} .
x Bn.
f (x) = lim0
f (B(x, )) 0 f (B(x, )) < 1n .
. B(x, 2
) Bn (, Bn )., y B (x, 2). B (y, 2) B(x, ). , z B (y, 2) (z, x)
(z, y) + (y, x) < 2 + 2 = . , pipi
f (y) f(B
(y,
2
)) f (B(x, )) < 1
n,
y Bn. 2 4.4.12. f : (X, ) (Y, ). D(f) f . , D(f) Fpi X.
pi. D(f) pi C(f), pi G. pi, F.
4.5
1. f, g : (X, ) (Y, ) D pi pi (X, ). :
() E = {x X : f(x) = g(x)} .() f(x) = g(x) x D, f g.
2. f : (X, ) (Y, ) x0 X. f x0 > 0 pi > 0 x, y X (x, x0)
< , (y, x0) < (f(x), f(y)) < .
-
74
3. (X, ) G X. G pi f : (X, ) R V R , G = f1(V ).
4. () f : (X, d) R Z(f) f ,
Z(f) = {x X : f(x) = 0}.
: f Z(f) X.
() F X. F pi f : (X, ) R Z(f) = F .
5. (X, ) A X. A A, pi A : X R
A(t) =
{1, t A0, t / A .
pi A A (X \A), bd(A) A A (clopen).
6. f : (X, ) (Y, ). f
Gr(f) ={
(x, f(x)) : x X} X Y. , f , Gr(f) f X Y pi . pi pi .
7. f : (X, ) (Y, ) A pi X (, (A, A) ). f(A) pi Y .
8. pi , f : R R pi . pi ;
9. pi pi pi -, pi.
10. f : (X, ) (Y, ) . (X, ) (Y, ) .
-
4.5 75
11. f : (X, ) (Y, ). , A X f(A) (f(A)), f . ;
12. f : R (Y, ), pi Y . f .
13. f : (X, ) (Y, ) pi (locally bounded) x X pi pi Ux x f |Ux
.() f : (X, ) (Y, ) . f pi . ;
() f : R R. :(i) f .(ii) f pi .
14. f : (X, ) (Y, ) D pi pi X. pi .
() f |D , f .() f |D , f .() f |D 1-1, f 1-1.
15. (X, ), (Y, ) f : X Y . :
() f .
() > 0 pi = () > 0 : A,B X dist(A,B) < , dist(f(A),
f(B)) < .
16. (X, ) A,B X . f : X [0, 1] Urysohn, f(x) =
dist(x,A)dist(x,A)+dist(x,B) , pi :
() dist(A,B) = 0, f .
() dist(A,B) = > 0, f 1Lipschitz.
17. (X, ) A,B X dist(A,B) > 0 f1 : A R,f2 : B R () . pi f :
AB R
f(x) =
{f1(x), x Af2(x), x B
() .
-
76
18. (X, ) A X. f : A R , pi f pi F : AR.
19. pi.
() R Z.() R Q.() Q Z.() Z N.
20. (X1, d1), . . . , (Xk, dk) ki=1Xi
d = max{di : 1 i k}. (X, d) f : X ki=1Xi f = (f1, . . . , fk),
pi fi : X Xi i = 1, . . . , k. :() f fi, i = 1, . . . , k .
() f Lipschitz fi Lipschitz.
() f fi, i = 1, . . . , k .
() f fi ;
() f fi ;
21. F pi R f : F R . pi g : R R g(x) = f(x) x F .
22. (X, ), (Y, ) f : X Y . 0 (modulus of continuity) f :
f () = sup{(f(x), f(y)) : d(x, y) , x, y X}.
() f : [0,) [0,] , 0 1 < 2 f (1) f (2).() f : X Y f () 0 0+.
[pi. (f(x), f(y)) f (d(x, y)) x, y X].
23. f : (X, ) (Y, ). f G X f(G) pi Y . , f F X f(F ) pi Y .
-
4.5 77
() pi: pi , - pi , pi , pi .
() f : (X, ) (Y, ) 1-1 pi, : (i) f , (ii) f , (iii) f1 .
pi, f ( ) .
24. (Xi, di) , i = 1, . . . ,m X =mi=1Xi
d =mi=1 di. ipi pii : X Xi pi
:pii(x1, . . . , xi, . . . , xm) = xi.
pi pii , pi .
25. f : (X, ) (Y, ). f f(A) (f(A)) A X. pi , f :X Y pi A X f(A)
pi (f(A)).
26. (X, ) . pi :
() X.
() X .
() X .
() Y , f : X Y .() G X .
27. () f : (X, ) R t R {x X : f(x) t} pi X. f , (xn) X xn x
X,
f(x) lim infn f(xn).
pi pi .
() f : (X, ) R f .pi pi , pi pi ().
28. pi pi (X, ), (Y, ) pi - pi : pi f : X Y, g : Y X pi , 1-1
pi.
-
II
pi
-
5
5.1
5.1.1 (pi ). (X, ) pi(complete) (xn) X .
5.1.2 (pi ). () pi-pi X, (X, ) pi. : (xn) X, . pi, .
() (R, | |) pi . pi pi pi .
() (Rm, 2), pi 2 , pi .pi. (xn) Rm. xn = (xn(1), . . . ,
xn(m)),pi xn(i) R.
> 0. (xn) , pi pi n0 N n, s n0
() 2(xn, xs) = mj=1
(xn(j) xs(j))21/2 < .
, i = 1, . . . ,m,
|xn(i) xs(i)| mj=1
(xn(j) xs(j))21/2 .
pi, n, s n0, i = 1, . . . ,m |xn(i) xs(i)| < .
-
82
: i = 1, . . . ,m (xn(i)) R. pi pi R pi pi x(1), . . . , x(m)
R
xn(i) x(i), i = 1, . . . ,m
n . x = (x(1), . . . , x(m)) Rm 2(xn, x) 0 n.
pi (): n, s n0 mj=1
(xn(j) xs(j))21/2 < .
pi n n0. pi limsxs(j) = x(j), j = 1, . . . ,m, m
j=1
(xn(j) x(j))21/2 = lim
s
mj=1
(xn(j) xs(j))21/2
supsn0
mj=1
(xn(j) xs(j))21/2
(). , n n0
2(xn, x) =
mj=1
(xn(j) x(j))21/2 .
> 0 , pi 2(xn, x) 0. 2() {(Xi, di)}ki=1 pipi . (
ki=1Xi,
ki=1 di)
pi (Xi, di) pi i = 1, 2, . . . , k. pi pi pi pi.
() {(Xn, dn)}n=1 dn(x, y) 1 x, y Xn, n = 1, 2, . . . X =
n=1Xn
d(x, y) =
n=1
1
2ndn(x(n), y(n))
pi x = (x(1), . . . , x(n), . . .) y = (y(1), . . . , y(n), . .
.) X. Xn pi (X, d) pi . pi pipi pi pi () .
-
5.1 83
() Baire (NN, ) pi
(x, y) =
{1
2(x,y), x 6= y
0, x = y
(x, y) = min{n N : xn 6= yn} x = (xn), y = (yn). Baire pi
().
5.1.3 ( pi ). 2.1.3 pi pi pi pi .
() Q , d(x, y) = |x y|. (Q, d) pi: pipi Q = R. R \ Q pi (qn) Q
qn . (qn) ( R) R Q. , pi q Q |qn q| 0, q = , pi pi.
() (R, ) (x, y) = | arctanx arctan y| pi. - xn = n - -.
5.1.4 ( Banach). (X, ) . X Banach pi pi pi pi pi , (X, d) pi
d(x, y) = x y pi .
pi pi Banach: pi ` c0 pi. pi `p,1 p < . pi (C([0, 1]), ) , ,
pi. 5.1.5. . `() - x : R,
d(x, y) = sup{|x() y()| : }
pi.
pi. (xn) `(). > 0. (xn) , pi n0 N n, s n0 sup{|xn() xs()| : }
< .pi, n, s n0
() |xn() xs()| < .
(xn())n R., pi x() R
limnxn() = x(), .
-
84
x : R pi x().pi pi x `().
, |xn() xs()| |xn() x()| s . , pi () pi
() |xn() x()| n n0 .
pi, ,
|x()| |xn0()|+ xn0 + .
pi sup |x()| xn0 + 0 , xn x pi d. 2 pi pipi = N, :
5.1.6. ` ,
d(x, y) = sup{|x(i) y(i)| : i N}
pi.
5.1.7. (X, ) pi F X. F pi X (F, |F ) pi .pi. pi pi F pi X. pi pi
(X, ). (xn) F . , (xn) X , X pi, pi pi x X xn x. , pi pi F , pi x F
. , F pi pi.
F pi . pi X. (yn) F yn y X. (yn) , pi pi F . , F . pi , yn y, y
F . pi F pi X. 2
5.1.8. c0 pi pipi ` pi .
1: (xn) , . M = supn xn, n N |xn()| M pi |x()| M , x M
-
5.2 Cantor 85
pi. c0 pi `. pi, pi `.
x = (x(i)) c0 . pi x c0, limi
x(i) = 0.
> 0. pi y = (y(i)) c0 y x < . , i N,|x(i) y(i)| < .
y = (y(i)) c0, , pi i0 N , i i0,|y(i)| < .
pi i i0,|x(i)| |x(i) y(i)|+ |y(i)| < + = 2.
, x(i) 0 i, x c0. 2
5.2 Cantor
Cantor pi . pi, R Cantor, pi Frechet. pi pi.
5.2.1. (X, ) {An} pi X diam(An) 0. ,
n=1An pi pi .
pi. pi pi x, y X x 6= y x, y n=1An. diam(An) 0, pi n N diam(An)
< (x, y). pi, x, y An. 2 5.2.2. (X, ) (xn) X.
Rn = {xk : k n} n = 1, 2, . . .. :
() (xn) .
() diam(Rn) 0 n.pi. pi (xn) . > 0. pi n0 N m,n n0 (xn, xm)
< 2 . pi Rn pi diam(Rn0) 2 . {Rn} (R1 R2 ), n n0
diam(Rn) diam(Rn0) < .pi diam(Rn) 0.
-
86
, pi (diam(Rn)) . > 0. pi n0 N diam(Rn0) < . m,n n0 xn, xm
Rn0 , pi
(xn, xm) diam(Rn0) < pi, (xn) (X, ). 2
5.2.3 (CantorFrechet). (X, ) . :
() (X, ) pi.
() {Fn}nN , pi X diam(Fn) 0,
n=1 Fn = {x} pi x X.
pi. pi pi X pi {Fn}pi pi. Fn , pi pi xn Fn n. , (xn).
Rn = {xk : k n}. {Fn} , Rn Fn. , diam(Rn) diam(Fn). diam(Fn) 0,
pi diam(Rn) 0 pi pi (xn) . (X, ) pi, pi pix X (xn, x) 0.
x n=1 Fn. m N. , (xn+m)nN Fm pi (xn). pi, x pi Fm pi x Fm.
pi 5.2.1 n=1 Fn pi , .
, pi (). (xn) X. Rn = {xk : k n} pi 5.2.2. , Rn , pi X diam(Rn)
0. pi pi Rn pi pipi . - diam(Rn) = diam(Rn) 0. , pi pi, pi x X
n=1Rn = {x}. , n N xn, x Rn,
(xn, x) diam(Rn) 0
n, xn x. 2 5.2.4. () pi Fn pi pi. pi (R, | |) Gn = (0, 1n )
diam(Gn) = 1n 0,
n=1Gn = . Gn
R.() pi diam(Fn) 0 pi pi. pi (R, | |) En = [n,+) En En+1 n = 1,
2, . . ., n=1En = . diam(En) = n.
-
5.3 Baire 87
() 5.2.3, pi pi, pi {Fn}nN , pi X diam(Fn) 0 . pi, (R \ Q, | |)
Fn = [ 1n , 1n ] Qc Fn Fn+1 n =1, 2, . . . diam(Fn) 0 n,
n=1 Fn = .
5.3 Baire
pi pi Cantor pi Baire. Baire pi (pi , pi, ). pi , pi Osgood pi
Banach pi pi pi . pi pi pi , pi.
(X, ), pipi G1, G2, . . . , Gmpi pi pi X,
mi=1Gi pi ( , -
). pi pi pi . pi, (Q, | |) (qn) Q Gn = Q \ {qn}. , Gn pi Q, . Q
pi . Baire pi , pipi pi ( pi pi ).
5.3.1 (Baire). (X, ) pi (Gn) pi pi X. ,
n=1Gn pi X.
,n=1Gn 6= .
pi. , V X. V (n=1Gn) 6= . G1 pi, V G1 6= . V G1 , pi
0 < r1 < 1 x1 V G1 B(x1, r1) V G1. B(x1, r1) , G2 B(x1,
r1)
. pi x2 G2 B(x1, r1) 0 < r2 < 12
B(x2, r2) G2 B(x1, r1) V G1 G2.
pi, pi pi pi B(xn, rn)pi pi :
diam(B(xn, rn)) 2n
-
88
B(xn, rn) Gn B(xn1, rn1), B(xn, rn) B(xn1, rn1) B(xn, rn) V G1
Gn.
pi Cantor, pi x X x n=1 B(xn, rn). x V G1 . . . Gn n N. pi, x V
(
n=1Gn). pi
. 2
5.3.2. G pi Gpi R. , G pi-.
pi. pi pi. pi G = {x1, x2, . . . , xn, . . .} pi Gn pi R G =
n=1Gn. Gn
pi ( G pi). pi, n N Vn = R \ {xn} ( {xn} ) pi ( int{xn} = ).
,(
n=1
Vn
)( n=1
Gn
)= (R \G) G = .
Baire. 2
5.3.3. Q Gpi R.
5.3.4. pi f : R R C(f) = Q ( ).
pi. pi pi pi 5.3.3 4.4.11: f :(X, ) (Y, ), C(f) f Gpi X. 2
pi Baire :
5.3.5 (Baire). (X, ) pi Fn pi X X =
n=1 Fn. , pi k N int(Fk) 6= .
pi. pi n N int(Fn) = . , Gn = X \ Fn pi, X \ Fn = X \ int(Fn) =
X. pi,
n=1
Gn = X \n=1
Fn = .
pi 5.3.1. 2
pi pi Baire pi :
-
5.3 Baire 89
5.3.6. (X, ) .
() pi A X pi pi int(A) = .() pi B X pi ( X) - pi pi pi X, pi En,
n = 1, 2, . . .pi pi pi X, B =
n=1En.
() pi C X ( X) pi.
, Baire pi : pi ( ).
5.3.1 Baire
5.3.7 (Osgood). fn : [0, 1] R, n N, .pi t [0, 1] (fn(t)) . ,
pi[a, b] [0, 1] M > 0 , t [a, b] n N,
|fn(t)| M.
, (fn) pi [a, b] [0, 1].pi. m N
Am = {t [0, 1] : n N, |fn(t)| m}.
:
(i) Am : pi
Am =
n=1
{t [0, 1] : |fn(t)| m}
pi {t [0, 1] : |fn(t)| m} fn .
(ii) [0, 1] =m=1Am: t [0, 1]. pi pi, (fn(t)) ,
pi Mt > 0 , n N, |fn(t)| Mt. pim = m(t) N m Mt, m t Am.
[0, 1] pi , pi Baire pi Am0 , pi pi [a, b] Am0 . , (fn) [a, b]:
t [a, b] n N, |fn(t)| m0. 2
-
90
* 5.3.1. C([0, 1]) [0, 1] d(f, g) = maxt[0,1] |f(t) g(t)| ( pi
pi ). M f C([0, 1]) pi pi [0, 1] pi C([0, 1]).
pi pi Baire :
* 5.3.1. f : [0, 1] R > 0, pi pi g : [0, 1] R d(f, g) <
.
[ g pi pi -, pi P = {0 = t0 < t1 < < tN = 1} [0, 1]
g(t) = ait+ bi (ti1, ti), i = 1, . . . , N .]
pi . pi f : > 0 pi > 0 : t, s [0, 1] |t s| < |f(s)
f(t)| < /2.
N pi pi 1/N < [0, 1] N . P =
{0 = t0 < t1 < < tN = 1
} ti = iN . g g(t) = ait + bi [ti1, ti],i = 1, . . . , N pi ai,
bi pi pipi f : 2
g(ti) = f(ti), i = 0, 1, . . . , N.
t [0, 1]. pi 1 i N ti1 t ti. ,
() |f(t) g(t)| |f(t) f(ti)|+ |f(ti) g(t)|.
|t ti| < |f(t) f(ti)| < 2 , pi g [ti1, ti] |ti ti1| = 1N
< , pi
|f(ti) g(t)| = |g(ti) g(t)| |g(ti) g(ti1)| = |f(ti) f(ti1)| <
2.
pi () pi |f(t) g(t)| < 2 + 2 = . t ,d(f, g) < . 2
pi . n N
Dn =
{f C([0, 1]) : t [0, 1] y (t 1
n, t+
1
n
) (0, 1) : |f(y) f(t)| > n|y t|}.. f n=1Dn , pi pi .
2g(t) = f(ti) +f(ti)f(ti1)
titi1 (t ti), t [ti1, ti] .
-
5.3 Baire 91
pi. f n=1Dn. t [0, 1] n N pi yn = yn(t) (0, 1) |t yn| < 1n
|f(yn) f(t)| > n|yn t|, yn 6= t. yn t
limn
f(yn) f(t)yn t =,
f (t) pi. 2 pi
n=1Dn pi C([0, 1]).
Baire, Dn pi.
. Dn pi C([0, 1]).pi. pi pi Dcn Dn . fk Dcn fk f pi d.
fk Dcn, pi tk [0, 1] y (0, 1) |y tk| < 1/n () |f(y) f(tk)|
n|y tk|.
tk [0, 1], (tk) pi: pi t [0, 1] pi- (tkm) (tk) tkm t.
y (0, 1) |y t| < 1/n |f(y) f(t)| n|y t| (pi,f Dcn.) > 0 y
(0, 1) |y t| < 1/n. :(i) ykm = y + (tkm t), ykm y. , m ykm (0,
1) |ykm tkm | = |y t| < 1/n. pi, pi (),
|fkm(ykm) fkm(tkm)| n|ykm tkm | = n|y t|.(ii) f tkm t. pi, ykm y
= tkm t 0. , m
|f(t) f(tkm)| < |f(y) f(ykm)| < .(iii) m, d(fkm , f) <
( fkm f).
m pi pi (i), (ii) (iii),
|f(y) f(t)| |f(y) f(ykm)|+ |f(ykm) fkm(ykm)|+ |fkm(ykm)
fkm(tkm)|+|fkm(tkm) f(tkm)|+ |f(tkm) f(t)|
< + d(f, fkm) + n|y t|+ d(fkm , f) + < n|y t|+ 4.
> 0 , |f(y) f(t)| n|y t|. y (0, 1) |y t| < 1/n, f Dcn. 2.
Dn pi pi C([0, 1]).
-
92
pi. f C([0, 1]) > 0. pi , pi g : [0, 1] R, pi, d(f, g) <
/2. pi, h Dn d(g, h) < /2.
g pi, pi 0 = t0 < t1 < . . . < tN = 1 g pi (ti1, ti).
li g (ti1, ti).
w : [0, 1] R : (i) 0 w(t) /2 [0, 1] (ii) w ( pi ) piQ = n +
max{|lj | : j = 1, . . . , N}. pi : [0, 1] pi /2Q, w pi 0 /2 , pi w
pi. h = g + w, pi
d(g, h) = maxt[0,1]
|w(t)| 2.
h Dn. t [0, 1]. pi i N pi t [ti1, ti]. pi |s| < 1/n t, t+ s g
w pi ( s pi ). y t, t+ s, y (0, 1), |y t| < 1/n |g(y) g(t)| =
|li||y t|,
|h(y) h(t)| |w(y) w(t)| |g(y) g(t)|> Q|y t| |li||y t| =
(n+ max
j|lj |)|y t| |li||y t|
= n|y t|.pi, h Dn. 2
5.4 *
pi pi pi X pi pi pi X pi pi X. pi : pi pi .
5.4.1 (pi ). (X, ) . pi (Y, ) pi X pi T : X Y pi T (X) pi pi Y
.
pi pi pi.
5.4.2 (pi pi). (X, ) . , pipi (X, ) T : X X T (X) pipi X.
-
5.4 * 93
pi pi. pi pi (pi `(X) f : X R. d pi d(f, g) = sup{|f(x) g(x)| :
x X}, pi ). 5.4.3. (X, ) . pi T :(X, ) (`(X), d).pi. pi pi a X. x
X, fx : X R pi pi
fx(t) = (t, x) (t, a), t X. fx X. , t X |fx(t)| =|(t, x) (t, a)|
(x, a). fx `(X) fx|| (x, a). pi pi
T : X `(X) x 7 fx. pi . x, y X, t X |fx(t) fy(t)| = |(t, x) (t,
y)| (x, y), sup{|fx(t) fy(t)| : t X} (x, y) sup{|fx(t) fy(t)| : t
X} |fx(y) fy(y)| = (x, y), pi .
d(T (x), T (y)) = (x, y).
2
pi 5.4.2. X T (X) X (`(X), d) pi T pi . T , T (X), pi pi X. X pi
pi (`(X), d) pi pi pi . 2
pi 5.4.2. pi .
(i) (X, ). X: (xn) (xn) (xn) (xn)
limn (xn, x
n) = 0.
. pi (yn) pi y X zn = y:
yn y X (yn) (y, y, . . .). X . ( X) x, y, z, . . . (xn) x (xn)
pipi x.
-
94
, X. x, y X. pipi (xn), (yn) x, y,
(x, y) = limn (xn, yn).
pi pi pi. (xn), (yn) :
pi ,
|(xn, yn) (xm, ym)| (xn, xm) + (yn, ym)pi pi pipi ((xn, yn)) R,
limn (xn, yn) pi.
pi pi (x, y) pi pi pipi (xn) x (yn) y. pi (xn) (xn) (yn) (yn)
pi
|(xn, yn) (xn, yn)| (xn, xn) + (yn, yn) 0pi
limn (x
n, yn) = lim
n (xn, yn).
, pipi pi . . (xn) x, (yn) y (zn) z. , pi, (x, y) = 0 lim
n (xn, yn) = 0, (xn) (yn) x = y. pi,
(x, y) (x, z) + (z, y) pi
(xn, yn) (xn, zn) + (zn, yn)., (X, ).
(ii) X X. y X (y, y, . . .), , pi X.
T : (X, ) (W, ) : b b
pi b (b, b, . . .).
(b1, b2) = limn (b1, b2) = (b1, b2)
b1, b2 X, pi pi T . W = {b : b X}.
-
5.4 * 95
W
= X ( X pi X). , x X (xn) pipi x, (xn) x. > 0 pin0 N (xm, xn)
< /2 m,n n0 ( (xn) ). T (y) W y = (ym) pi ym = xn0 m.,
(x, T (y)) = limm (xm, yn) = limm (xm, xn0)
2< .
, x X > 0 W B(x, ) 6= . , W pi X.
(iii) (X, ). (xn) X. W pi X, n N pi yn = T ((yn, yn, . . .)) W
(xn, yn) < 1/n. pi ,
(ym, yn) = (ym, yn) (ym, xm) + (xm, xn) + (xn, yn) 0, pi n
pi
pi ( ). pi, xn y. , (X, ) pi.
2
pi pi pipi pipi .
5.4.4 ( pi). (X1, 1) (X2, 2) pi (X, ). , pi : X1 X2, pi. , Ti :
X Xi (i = 1, 2) , (T1(x)) = T2(x) x X.
pi 5.4.4 pi pi pi . , :
-
96
5.4.5. (M,) D pi pi M . (N, ) pi f : D N , pi F : M N .pipi, f ,
F .
pi. pi f , M \D. x M .pi D pi M pi (tn) D tn
x., (tn) . f , pi , (f(tn)) . pi, (N, ) pi, pi pi y N f(tn) y.
pi pi F :M N x 7 F (x) := limn f(tn), pi (tn) D tn x. 1. F .
(an), (bn) D an x bn x, y1 = lim f(an), y2 = lim f(bn). , (an,
bn) (x, x) = 0. pi
(y1, y2) = limn(f(an), f(bn)) = 0,
f . pi, y x pi pi pi pi x.
2. F pi f , F |D = f . t D tn = t, n = 1, 2, ... tn t, F (t) =
limn f(tn) =
f(t).
3. F .
> 0. f , pi > 0 t1, t2 D (t1, t2) < , (f(t1), f(t2))
< 3 . x, y M (x, y) < 3 . (F (y), F (x)) < . pi a D (a, x)
< 3 (F (x), f(a)) < 3 (). , pi b D (b, y) < 3 (F (y),
f(b)) < 3 . pi pipi (a, b) < . pi, (f(a), f(b)) < 3 . pi
(F (y), F (x)) < .
f
(F (x), F (y)) = limn(f(an), f(bn)) = limn (an, bn) = (x, y)
pi (an), (bn) D an x bn y. F pi ( 4.1). 2
pi 5.4.4. T1 : X X1 T2 : X X2 pi X pi X1, X2 . ,
X1 X2
-
5.5 Banach 97
T1(X) T2(X)
T1 T2X
i X pi T2 T11 : T1(X) T2(X) X2 pi
T1 pi T1(X) T2 pi T2(X).pi, T2 T11 pi T1(X) X2. pipi, T1(X) pi
X1 , pi pi 5.4.5 pi T2 T11 pi X1.
. : X1 X2 pi. y X2. T2(X) pi X2, pi (xn) X
T2(xn) y. , (T2(xn)) . pi (xn) (T1(xn)) . pi, pi x X1 T1(xn) x.
, pi
(x) = limn (T1(xn)) = limn(T2 T
11 (T1(xn))) = limn(T2(xn)) = y.
, pi X2. pi. 2
5.5 Banach
pi pi Banach pi pi pi . pi - , pi pi .
5.5.1 ( ). f : X X x0 X. x0 f f(x0) = x0.
Fix(f) f .
5.5.2. f : (X, ) (X, ) . , Fix(f) pi X.
pi. (xn) f xn x.
x f . pi f f(xn) f(x).
f(xn) = xn x. pi f(x) = x,
x Fix(f). 2 5.5.3 (Banach). (X, ) pi T : X X : pi 0 < c <
1
(T (x), T (y)) c (x, y)
-
98
x, y X. , pi z X T (z) = z.pi. pi x X. pi - pi x0 = x xn+1 = T
(xn) n = 0, 1, . . . . ,
(xn) = (x, T (x), T2(x), . . .).
. (Tn(x))n .
n 1 (Tn(x), Tn+1(x)) c (Tn1(x), Tn(x))
pi pi
(Tn(x), Tn+1(x)) cn (x, T (x)) n = 0, 1, ... , m > n pi
(5.1)
(Tn(x), Tm(x)) (cn + cn+1 + + cm1)(x, T (x)) cn
1 c(x, T (x)) 0
m,n, 0 < c < 1. , (Tn(x))n . pi pi X pi pi z X Tn(x) z. ,
T Lipschitz. pi, Tn+1(x) = T (Tn(x)) T (z). (Tn+1(x)) pi x, pi
(Tn(x)), pi T (z) = z.
pi , pi pi z T ,
(z, z) = (T (z), T (z)) c (z, z) 0 < c < 1, pi (z, z) = 0,
z = z. , z T . 2
5.5.4. () Tn(x) pi pi pi pi z. Tn(x) z pi pi x n- pi pi c
n
1c(x, T (x)). , m > n
(Tn(x), Tm(x)) cn
1 c (x, T (x)).
m pi
(Tn(x), z) = limm (T
n(x), Tm(x)) cn
1 c (x, T (x)).
-
5.6 99
() (T (x), T (y)) < (x, y) T pi , pi pi . T : R R T (x) =
log(1 + ex) pi , |T (x)| < 1 x R, .() pi (X, ) pi pi: f : (0, 1)
(0, 1) f(x) = x2 |f(x) f(y)| 23 |x y| x, y (0, 1), f . ((0, 1), |
|) pi .
5.6
1. N d(m,n) = |m n| (m,n) =| 1m 1n |.() (N, d) pi (N, ) pi.()
{n} d- -.() d (, (N, d) (N, ) -).
2. R d(x, y) = | arctanx arctan y|. d R (R, d) pi.
3. d1 d2 X. pi pi a, b > 0: x, y X,
ad1(x, y) d2(x, y) bd1(x, y).
(xn) X (X, d1) (X, d2).
4. (X, ) D pi pi X. : (xn) D pi x X, X pi.
5. (X, ) pi pi
B(x, ) = {z X : (z, x) },
pi x X > 0, pi pi X.
6. (X, ) pi D pi pi X X \D pi pi. pi D, X \D F.
-
100
7. : (Ln) R2 int (n=1 Ln) = .
8. () (`p, p), 1 p 0 diam([n,)) n N.
14. X pi B(xn, rn) pi pi. pi
n=1 B(xn, rn) 6= .
15. (X, ) pi f : X Y . pi (En) pi X, diam(En) 0,
f
( n=1
En
)=
n=1
f(En).
-
5.6 101
16. (X, ) pi G pi X.
(x, y) = (x, y) +
1dist(x,X \G) 1dist(y,X \G)
G G. (G, ) pi |G.
17. (Gn) pi pi R. G =
n=1Gn pi.
18. f : R R. f pi pi pi R.
19. pi d Q d (Q, d) pi.
20. (X, ) pi f, g : X X. pi:
() pi k N fk = f f , pi x X f(x) = x.() f f g = g f , pi x X
f(x) =g(x) = x.
21. (X, d) pi , E pi Gpi X. pi h : X X E h(E) 6= .
22. pi fn : R R
x R, limn fn(x) = Q(x).
23. () A = {a1, a2, . . . , an, . . .} pi R. FA : R R
FA(x) =
{n:anx}2n
, pi pi A.
() A pi pi R. pi g : R R D(g) g R \A.
-
102
() E pi R. G = E Q h : R R h(x) = E(x) G(x). pi D(h) = E.() E
=
n=1En Fpi R. fE : R R
fE(x) =
1
min{n:xEn} , x Q E 1min{n:xEn} , x (R \Q) E0, x R \ E
pi D(fE) = E.
24. f : [0,) R . pi pi (a, b) [0,) : y (a, b) limn f(ny) = 0.pi
limx f(x) = 0.
25. f : R R pi pi . pi : x R pi nx N , n nx, f (n)(x) = 0. f
pi.
-
6
pi
6.1 pi
pi pi pi, pi pi pi- pi pi . pi -pi pi, pi BolzanoWeierstrass [a,
b] pi pi . pi :
(X, ) () pi - (xn) X pi (xkn) pi pi x X.
pi pi, pi pi pi pi pi - pi pi pi pipi . pi pi pi pi ( ).
6.1.1 (). X A X. (Ui)iI pi X A
A iI
Ui.
pi J I A iJ Ui, (Ui)iJ pi (Ui)iI A.
6.1.2 ( ). (X, ) (Ui)iI pi X. (Ui)iI X, X =
iI Ui.
, A X, (Ui)iI A, A iI Ui.
-
104 pi
6.1.3 (pi). (X, ) . X pi(compact) X pipi pi. , :
(Ui)iI pi X pi pi X =
iI Ui pi m N i1, . . . , im I X =
Ui1 Uim . pi K X pi, pi . : (Vi)iI K pi pi X, pi pipi pi (Vij
)
mj=1, K m
j=1 Vij ().
6.1.4. () (X, ) pi X pipi .
() pi R pi . , {(n, n)}nN, pipi pi.
() S` = {x = (xn) ` : x = 1} pi pi `. {B(x, 1/2) : x S`}, pi
pipi pi, en em = 1 n,m N, n 6= m.() (X, ) , x X (xn) X xn x.
K = {xn : n = 1, 2, . . .} {x}
pi X. (Gi)iI K. pii0 I x Gi0 . xn x Gi0 , pi N N xn Gi0 n > N
. 1 j N pi ij I xj Gij . ,K Nj=0Gij . 6.1.5. (X, ) K pi pi X., K
X.
pi. y X \K. x K x := (x,y)2 . pi {B(x, x) :x K} pi K. , pi x1,
x2, . . . , xm K K mj=1B(xj , xj ). = min{xj : j = 1, 2, . . . ,m}
> 0. , B(y, ) X \K., x K pi 1 j m (x, xj) < xj . ,
(x, y) (y, xj) (xj , x) > 2xj xj ,
x / B(y, ). X \K , pi K . 2 6.1.6. (X, ) K , pi pi- X. , K pi
X.
-
6.2 pi 105
pi. pi x X {B(x, n) : n N}. X, K:
K n=1
B(x, n).
K pi, pi n1, . . . , nm N
K B(x, n1) B(x, nm).
r = max{n1, . . . , nm} B(x, nj) B(x, r) j = 1, . . . ,m,
K B(x, r).pi, K . 2
6.1.7. (X, ) pi F pi X. , F pi.
pi. (Ui) F . {X \ F} {Ui : i I}pi X ( ). X pi, pii1, . . . , ik
I X = Ui1 . . . Uik (X \ F ). , F Ui1 . . . Uik . 2
6.2 pi
pi pi pi . pi , (X, ) pi BolzanoWeierstrass: , pi piA X X (A 6=
). pi pi: (X, ) pi pi . pi pi .
6.2.1 ( pi ). (X, ) . X pi (sequentially compact) (xn) X pi
(xkn) pi pi x X. 6.2.2 ( ). (X, ) (totally bounded) > 0 pi m N
x1, . . . , xm X
X =
mi=1
B(xi, ).
, > 0 pi pipi pi pi > 0 pi .
-
106 pi
, (X, ) A X, A > 0 pi x1, . . . , xm X
A mi=1
B(xi, ).
pi , A X B A pi ( ).
pi pi pi xi A:pi, pi A > 0.pi x1, . . . , xm X A
mi=1B(xi, /2). pi pi
B(xi, /2) A ( pi pi ).pi pi ai B(xi, /2) A, i = 1, . . . ,m. ,
a1, . . . , am A B(xi, /2) B(ai, ) ( ). pi,
A mi=1
B(ai, ).
6.2.3. () (R, ||) . , pi x1 < x2 < < xk R R =
ki=1(xi1, xi+1) (x11, xk+1),
pi.
() (X, ) X pipi ( ).
() n Hamming Hn Hilbert H, ().
6.2.4 ( pi). (X, ) . :
(i) (X, ) pi.
(ii) pi pi A X X(, A 6= ).
(iii) X pi.
(iv) X pi .
pi. (i) (ii): A pi X pi - X. , x X pi x > 0 B(x, x) (A \ {x})
= . {B(x, x) : x X} X. X pi,pi pipi pi. , pi x1, . . . , xm X
X = B(x1, x1) B(xm, xm).
-
6.2 pi 107
,A = (A B(x1, x1)) (A B(xm, xm)).
, i = 1, . . . ,m, pi B(xi, xi) (A \ {xi}) = pi A B(xi, xi)
{xi}.
pi A {x1, . . . , xm}
A pipi .
(ii) (iii): (xn) X. (xn) pi.
A = {xn : n N} (xn). A pipi, pi x A k1 < < kn < kn+1
< xkn = x n N. , (xn) pi pi.
pi pi A (xn) pi. , pi x X pi A. pi, pi x pipi (xn) ( pi pi A).
pi = 1, 12 , . . . ,
1n , . . . pi x, pi
(kn) (x, xkn) 0 : m N u1, . . . , um X
() X \mj=1
B(uj , ) 6= .
pi () pi (xn) X : pi x1 X pi () pi
x2 X \B(x1, ). (x2, x1) .
pi pi x1, . . . , xn , i, j {1, . . . , n} i 6= j (xi, xj) . ,
pi pi (), pi
xn+1 X \B(x1, ) B(xn, ). (xn+1, xi) i = 1, . . . , n.
-
108 pi
pi, (xn) X : n 6= m (xn, xm) . (xn) pi (). pi, pi.
(iv) (i): pi pi. pi X pi. , pi (Ui)iI X pi pipi pi. ,X =
iI Ui , m N i1, . . . , im I
X \ (Ui1 Uim) 6= .
pi pi X , x11, . . . , x1N1 X
X =
N1j=1
B(x1j , 1/2).
. pi j0 {1, . . . , N1} , m N i1, . . . , im I
B(x1j0 , 1/2) \ (Ui1 Uim) 6= .[, B(x1j , 1/2) pi pi pipi pi -
(Ui)iI X =
N1j=1B(x1j , 1/2) pi pi pipi
pi (Ui)iI , pi]. x1 := x1j0 . , B(x1, 1/2) X, pi x21, . . . ,
x2N2
X
B(x1, 1/2) N2j=1
B(x2j , 1/22).
pi pi B(x1, 1/2) B(x2j , 1/22) 6= j = 1, . . . , N2, pipi B(x2j
, 1/22) pi pi B(x1, 1/2).
pi pi , j1 {1, . . . , N2} , m N i1, . . . , im I
B(x2j1 , 1/22) \ (Ui1 Uim) 6= .
x2 := x2j0 .
(x1, x2) 0. pi n0 N 1n0 < . i, j n0 Ci, Cj Cn0 xmi , xmj Cn0
. pi (xmi , xmj ) diam(Cn0) 1n0 < . 2 6.2.6. xn = (1)n . , pi pi
pi pi.
6.2.7. (X, ) (xn) X pi.
pi. (X, ) (xn) - X. A = {xn : n N} pi X pi 6.2.5() pi .
, pi X pi. - (X, ) . , pi 0 > 0 n N x1, . . . , xn X X \
ni=1B(xi, 0) 6= . pi
x1 X. , X \ B(x1, 0) 6= , pi x2 X (x1, x2) 0. ,
-
6.3 pi 111
X \ 2i=1B(xi, 0) 6= , pi x3 X (x3, xi) 0 i = 1, 2. - pi, pi (xn)
(xn, xm) 0 n 6= m. , pi- pi. 2
6.2.8. (X, ) pi pi, pi X .
pi. pi pi (X, ) pi. A pi, pi X. pi (an) A an 6= am n 6= m. B =
{an : n = 1, 2, . . .} pi A, . pi 6.2.5() (an) pi (akn). X pi, pi x
X akn x. (akn) pi pi pi A. pi x A., pi pi, pi X . (xn) X. pi pipi
pi pi (ii) (iii) 6.2.4 (xn) pi. , . , (X, ) pi. 2
6.3 pi
pi pi pi pi . K pi pi (X, ) :
(i) K pi X.
(ii) (xn) K pi (xkn) pi pix K.
pi, pi pi . pi pi .
6.3.1. . , pi .
pi. (X, ) . , > 0 pi pipi pi F X pi F > 0 pi X. 1, 12
,
13 , . . . pi D1, D2, D3, . . . pipi pi X
X =xDn
B(x,
1
n
)
-
112 pi
n = 1, 2, . . .. D =nNDn. D
pipi .
. D pi X.
pi D. B(x, ) pi X. , pi n N 1n < . pi, X =
yDn B(y,
1n ). pi,
x yDn B(y, 1n ) pi y Dn x B(y, 1n ). , x B(y, ), , y B(x, ). ,
Dn B(x, ) 6= , D B(x, ) 6= . 2 6.3.2 ( pipi ). X (Fi)iI pi X.
(Fi)iI pipi pipi J I
iJFi 6= .
pi, {(, x] : x R} pi R pipi . pi A N pi N \A pipi ( ). 6.3.3.
(X, ) . :
(i) (X, ) pi.
(ii) (Fi)iI pi X pi pipi- ,
iIFi 6= .
pi. (i) (ii): pi pi (Fi)iI pi X pi pipi ,
iI Fi = . ,
X =iI
(X \ Fi).
, (X \Fi)iI X. X pi,pi i1, . . . , im I X =
mj=1(X \ Fij ). ,
mj=1 Fij = , pi
pi.
(ii) (i): pi X pi. , pi (Gi)iI X pi pi pipi pi. Fi = X \Gi, i I.
m N i1, . . . , im I X 6= Gi1 Gim ,
Fi1 Fim = (X \Gi1) (X \Gim) = X \mj=1
Gij 6= .
-
6.3 pi 113
, (Fi)iI pipi . pi pi,iI Fi 6= ,
iIGi = X \
iI
Fi 6= X,
pi pi. 2
6.3.4 ( Lebesgue). (X, ) . (Vi)iI X Lebesgue, pi > 0 :
E X diam(E) < pi i I E Vi. pi pi pipi Lebesgue . pi X
Lebesgue (Vi)iI X Lebesgue.
pi, (Z, d) d(n,m) = |n m| Lebesgue., (Vi)iI Z. , = 1 : E Z Z 6=
diam(E) < 1 pi pi n Z E = {n}. , pi in I n Vin , E Vin . , (X, )
, Lebesgue.
(R, | |) Lebesgue. r : R (0,) x 7 rx := 11+|x|
Ux = B(x, rx) = (x rx, x+ rx), x R > 0 pi E R diam(E) < x
R E * Ux. , > 0, pi x0 > 0 rx0 < /4. y0 > x0 + 1. E =
[y0, y0 + /2]. , x R E * Ux. , pi x R E Ux,
x rx < y0 < y0 + /2 < x+ rxpi pi rx > /4, x < x0
( rx0 < /4). , y0 0 ix I B(x, x) Uix . {B(x, x/2)}xX X. X pi, pi
x1, x2, . . . , xk X X =
kj=1B(xj , xj/2).
:=1
2min{xj : j = 1, 2, . . . , k} > 0
-
114 pi
A X diam(A) < , pi i I A Ui. A X , diam(A) < . a A pi xj X
a B(xj , xj/2)., A Uixj . , z A
(z, xj) (z, a) + (a, xj) < + xj/2 xj, z B(xj , xj ) Uixj .
2
pi , pi pipi, pi pi .
6.3.6. () (Xi, di)mi=1 pi . X =mi=1Xi
Xi d pipi X, (X, d) pi.
() (Xn, dn), n = 1, 2, . . . pi dn(x(n), y(n)) 1 x(n), y(n) Xn,
n = 1, 2, . . .. ,
n=1Xn
d(x, y) =n=1 2
ndn(x(n), y(n)) pi.
pi. () (X, d) pi. pi 2.1.13. xn = (xn(1), . . . , xn(m)) (X, d).
X1 pi, (xn(1)) pi(xkn(1)):
xkn(1) x(1) X1. X2 pi, (xkn(2)) pi (xkn (2)):
xkn (2) x(2) X2.
xkn (1) x(1), xkn(1) x(1) (xkn (1)) pi xkn(1). , pi(xkn ) pi .
pipi m- pi m pi (xn) pi pi . d X, (xn) pi. 2
() . 2
6.3.7. Hilbert H pi . 2 ( 2.1.13) Rm (
) pi. pi pi pipi pi pi Rm.
6.3.8. Rm, m 1, . pi- K Rm pi .
-
6.4 pi 115
pi. : , pi . , K pi Rm. K pi. (xn) K. K , (xn) . pi 2.1.13, pi
pi (xkn) (xn) pi pi x Rm., K (xkn) pi K. , x = lim
nxkn K. K pi pi K, K
pi. 2
. , [a, b] pi pi R., Rm = [a1, b1] [am, bm] pi B2(x, ) pi pi
(Rm, 2).
pi pi pi: pi 6.1.4() , (`, ), S` , pi B(0, 1), pi .
6.4 pi
pi pi pi pi - f : [a, b] R pi pi .
6.4.1. (X, ) pi , (Y, ) f : X Y . , f .
pi. pi. pi -pi Lebesgue, pi.
pi. > 0. f , x X pi x =x() > 0 f(B(x, x)) B(f(x), /2).
{B(x, x) : x X} pi X, pi Lebesgue > 0 : A X diam(A) < pi x X
A B(x, x). z1, z2 X (z1, z2) < ., A = {z1, z2} (z1, z2) < ,
pi pi x X A B(x, x). , pi f x pi
(f(z1), f(z2)) (f(z1), f(x)) + (f(z2), f(x)) < 2
+
2= .
pi, f . 2
pi. pi f . , pi- > 0 (xn), (yn) X (xn, yn) 0
-
116 pi
(f(xn), f(yn)) n N. pi () pi X pi pi (xkn) (xn)
