1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete sp

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1 1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete space T1 T2 =, X = X, X X = X T3 =, X =, X X = X 2 X T X X T = 2 X 2 X X (X, T ) (discrete space) T1 X, X X 2 X X 2 X = T, X 2 X = T T2 {O λ λ Λ} λ Λ X λ Λ O λ T T3 O 1, O 2,, O n (n < ) O 1 O 2 O n X O 1 O 2 O n T X Λ Λ λ X A λ A = {A λ λ Λ} Λ (index set) X

2 3 X X X O X\O X T1 X\ = X X\X = X T2 {O λ λ Λ} X λ Λ X\O λ X De Morgan X\ O λ = (X\O λ ) λ Λ λ Λ λ Λ X\O λ X\ O λ = (X\O λ ) = X = X λ Λ λ Λ λ Λ λ 0 Λ X\O λ0 X\ O λ = (X\O λ ) X\O λ0 λ Λ λ Λ X\O λ X\ λ Λ O λ T3 O 1, O 2,, O n (n < ) k = 1,, n X\O k X De Morgan n n X\ O k = (X\O k ) k=1 k=1 k 0 (= 1,, n) X\O k0 = X n n X = X\O k0 X\ O k = (X\O k ) X k=1 k=1 X\ n k=1 O k = X k X\O k n n X\ O k = (X\O k ) k=1 k=1 = (X\O 1 ) (X\O 2 ) (X\O n ) n k=1 O k

3 (X, d) X x r x r S r (x) = {y X d(x, y) < r} X O O x ε S r (x) O 1.1 (X, d) ( ) T1 x ε S ε / X x ε S ε (x) X, X T2 {O λ λ Λ} X λ Λ O λ x x O λ0 λ 0 Λ O λ0 ε S ε (x) O λ0 λ Λ O λ S ε (x) O λ0 O λ λ Λ T3 O 1, O 2,, O n (n < ) nk=1 (O λ ) x k x O k O k k ε k S εk O k ε =min{ε 1, ε 2,, ε n } S ε (x) S εk (x) O k n S ε (x) n k=1 O k O k k=1

4 1.2 S r (x) ( ) S r (x) z x S r (x) d(x, z) < r ε = r d(x, z) ε > 0 S ε (x) S r (x) w S ε (x) d(x, w) < ε d(x, w) < ε = r d(x, z) d(z, w) + d(x, z) < r d(x, w) d(x, z) + d(z, w) < r w S r (x) S ε (x) S r (x) S r (x) S ε (x) S r (x) Q.E.D 1 R n x = (x 1, x 2,, x n ), y = (y 1, y 2,, y n ) d(x, y) = n (x k y k ) 2 (R n, d) k=1 2 R n x = (x 1, x 2,, x n ), y = (y 1, y 2,, y n ) d(x, y) =max{ x 1 y 1, x 2 y 2,, x n y n } (R n, d)

5 3 (X, d) S r (x) = {y X d(x, y) < r} (1)0 < r 1 S r (x) = x (2)r > 1 S r (x) = x d(x, y) = { 0 (x = y) 1 (x y) y = S r (x) d(x, y) < r 1 d(x, y) = 0 y = S r (x) d(x, y) < r y = x d(x, y) = 0 d(x, y) = 1 y = x y x y X (Hausdorff ) X X x, y U, V x U, y V, U V = X Hausdorff X X Hausdorff X Hausdorff X x, y U, V x U y V U V = U X\U X X\U = X U = x U X\U X\V (X\U) (X\V ) = X\(U V ) = X\ = X X (X\U), (X\V ) =

6 1.3 Hausdorff ( ) x, y X ε = d(x, y) x y ε > 0 U = S ε (x), V = S ε (y) 2 2 U, V x U, y V z U V z U z V d(x, z) < ε 2, d(y, z) < ε 2 ε = d(x, y) d(x, z) + d(z, y) < ε 2 + ε 2 = ε ε < ε U V = Q.E.D 1.4 (X, T X ) X Y Y T Y T = {O Y O T X } T Y Y (Y, T Y ) (X, T X ) ( ) T1 T X = Y T Y X T X Y = X Y Y T Y T2 {U λ λ Λ} Y λ Λ O λ T X λ Λ U λ = O λ Y U λ = (O λ Y ) λ Λ = ( O λ T X λ Λ O λ T X λ Λ O λ ) Y T3 U 1, U 2,..., U n Y k = 1, 2,..., n U k = O k Y X O k n U k = n (O k Y ) k=1 k=1 = n ( O k ) Y k=1 nk=1 T X n k=1 U k T Y X Y U Y U = O Y X O Q.E.D

7 2 X, Y f : X Y Y O f 1 (O) X X, Y f : X Y X A f(a) = {f(x) x A} f Y B f 1 (B) = {x X f(x) B} f B 1 X 1 X : X X 1 X (x) = x (x X) 1 X X 1 X X A A = f 1 X (A) x 1 1 X (A) 1 X (x) A x A X O 1 1 X (O) = O 1 1 X (O) X 1 X 2 X Y i : Y X i(y) = y (y Y ) i i O X i 1 (O) = {y i(y) O} = {y Y y O} = O Y i 1 (O) i

8 3 X, Y p Y c p : X Y c p (x) = p (x X) c p c p O (1) p / O c 1 p (O) c 1 (O) x 0 x 0 c 1 p (O) c p (x) O p O c 1 p (O) X c p (2) p O c 1 p (O) = {x X c p (x) O} c 1 p (O) X x X c p (x) = p O x c 1 p (O) X c 1 p (O) X = c 1 p (O) X c p c 1 p p 4 X Y f : X Y Y O f 1 (O) X X f 1 (O) f

9 X, Y f : X Y f (1)(2)(3) f X Y X Y X Y (1)f (2)f (3)f f 1 : Y X 2.1 X, Y, Z f : X Y g : Y Z gf : X Z gf(x) = g(f(x)) (x X) ( ) z O (gf) 1 (O) = f 1 (g 1 (O)) g g 1 (O) Y f f 1 (g 1 (O)) X (gf) 1 (O) X gf (gf) 1 (O) = f 1 (g 1 (O)) X A (gf) 1 (A) = {x X (gf)(x) A} x (gf) 1 (A) (gf)(x) A g(f(x)) A f(x) g 1 (A) x f 1 (g 1 (A)) Q.E.D

10 2.2 (X, d X ), (Y, d Y ) f : X Y (1),(2) (1)f (2)f ( ) (1) (2) O Y f 1 (O) a a f 1 (O) f(a) O O Y ε S Y ε (f(a)) = {y Y d Y (y, f(a)) < ε} O f a X ε δ d X (x, a) < δ = d Y (f(x), f(a)) < ε S X δ (a) = {x X d X (x, a) < δ} x S X δ (a) d X (x, a) < δ d Y (f(x), f(a)) < ε f(x) S Y ε (f(a)) S Y ε (f(a)) O x f 1 (O) f 1 (O) X f (2) (1) ε S Y ε (f(a)) Y f 1 (S Y ε (f(a))) X f(a) S Y ε (f(a)) a f 1 (S Y ε (f(a))) δ S X δ (a) f 1 (S Y ε (f(a))) d X (x, a) < δ = x Sδ X (a) = x f 1 (Sε Y (f(a))) = f(x) Sε Y (f(a)) = d Y (f(x), f(a)) < ε f Q.E.D

11 3 X X A (closed set) X\A 3.1 X C1 X C2 C3 ( ) C1 = X\X X X = X\ X C2 A 1, A 2,..., A n A 1 A 2... A n De Morgan X\(A 1 A 2... A n ) = (X\A 1 ) (X\A 2 )... (X\A n ) k = 1, 2,..., n X\A k X\(A 1 A 2... A n ) A 1 A 2... A n C3 {A λ λ Λ} X De Morgan X\ λ Λ A λ = (X\A λ ) λ Λ λ Λ A λ X\A λ X\ λ Λ A λ λ Λ A λ Q.E.D (X, d) r 0, a X A = {x X d(x, a) r} X X\A x X\A x / A d(x, a) < r r > 0 x S r (a) X\A = S r (a) X\A r = 0 d(x, a) < 0 X\A = X\A

12 3.2 X, Y f : X Y f Y A f 1 (A) ( ) = A Y Y \A f f 1 (Y \A) x f 1 (Y \A) f(x) Y \A f 1 (Y \A) = X\f 1 (A) f(x) Y f(x) / A x f 1 (Y ) x / f 1 (A) x X\f 1 (A) X\f 1 (A) f 1 (A) = O Y Y \(Y \O) = O Y \O Y f 1 (Y \O) X f 1 (Y \A) = X\f 1 (A) X\f 1 (A) X\(X\f 1 (A)) = f 1 (A) f Q.E.D

13 3.3 (X, d) X A A {a n } lim n a n = a a A ( ) = a / A A X\A a / A a X\A a X\A ε S ε (a) X\A lim a n = a n ε N n N d(a n, a) < ε N n N d(a n, a) < ε d(a n, a) < ε a n S ε (a) n N a n S ε (a) a n X\A a n / A {a n } A a A = X\A X\A X\A X\A a ε S ε (a) / X\A x S ε (a) x / X\A x n = 1, 2,... ε = 1 n a n a n S ε (a) a n / X\A Archimedes η N 1 N < η n N d(a n, a) < 1 n 1 N < η lim a n = a n a / A X\A A Q.E.D

14 3.4 X X Y X Y A Y X B A = Y B ( ) = A Y Y \A X O Y \A = O Y x A x / Y \A x / O Y x / O x (X\O) Y X\O A = (X\O) Y = Y X B A = Y B X\A = X\(Y B) = (X\Y ) (X\B) Y Y X\Y X\B X\A A Q.E.D

15 4 X x X X N x x O N O 4.1 X x X N x x N0 N x N1 N N x = x N N2 M N x N N x = M N N x N3 M N x M N = N N x N4 x N (1)x V N (2)V V X V ( ) N0 x X X X X N x N x N1 N x x O N O x N N2 M, N x U, V x U M, x V N x U V M N U V M N x N3 M x x O M O M N N x x O M N N4 4.2 Q.E.D

16 4.2 X O X O O O ( ) = O x O O x x O O = O x O x x O x O O x x O x O {x} O x O O = {x} O x O x O x O O x = O x O O x O Q.E.D

17 4.3 X, Y f : X Y f X x N N f(x) = f 1 (N) N x ( ) = f x X N f(x) f(x) O N Y O x f 1 (O) f 1 (N) f f 1 (O) x f 1 (N) x = X x N N f(x) = f 1 (N) N x O Y y f 1 (O) y f 1 (O) f(y) O O 4.2 O f(y) f 1 (O) y f 1 (O) f 1 (O) 4.2 f 1 (O) X f Q.E.D 4.4 (X, d) x X N X N x ε S ε (x) N ( ) = N x x O N O ε = ε S ε (x) O S ε (x) O N S ε (x) N S ε (x) N Q.E.D

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