III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ B λ. 4. (X, d). X A X x. x A d(x, A) d(x, A) = inf{d(x, y) y A}. d(x, A) = 0 x A. 5. 4.. (i) d(x, A) d(x, y) + d(y, a) a A. (ii) d(x, A) d(y, A) d(x, y) d(x, A) x. 6. 4.. A X ϵ- U(A, ϵ) = {x X d(x, A) < ϵ}. U(A, ϵ). 7. (X, d). Y X y, y Y d Y (y, y ) = d(y, y ). (Y, d Y ). 1

( n ) 1/p 8. p 1. d p (x, y) = x j y j p R n j=1. 9. d(x, y) = x 1 + x y 1 + y R. 10. X. d : X X R d(x, y) = 1(x y), d(x, y) = 0(x = y),. d. 11. (X, d). X D d(x, y) 1, x, y D, D. 12. (i) d(x, y) = 1 x 1 y (0, 1]. (ii) C 1 C 2. d(x, y) C 1 x y, x y C 2 d(x, y) 13. X = (0, 1], d(x, y) = 1 x 1 y. (i) X x n (n = 1, 2, 3, ) d 1 x n. (ii) X x n (n = 1, 2, 3, ) d. 14. X = [1, ), d(x, y) = 1 x 1 y. d. 15. (X, d). ρ(x, y) =. d(x, y) 1 + d(x, y) X d(x, y) 16. (X, d), ρ(x, y) =. (X, d) 1 + d(x, y) (X, ρ). 2

d(x, y) 17. (X, d), ρ(x, y) =. X 1 + d(x, y) ad(x, y) ρ(x, y) bd(x, y), x, y X, a, b. 18. x = (x 1, x 2,, ) X. 1 x n y n d(x, y) = 2 n 1 + x n y n X. n=1 19. x = (x 1, x 2,, ) X. 1 d(x, y) = k x k y k x n = y n (n < k) 0 x n = y n n. d X. d(x, z) max{d(x, y), d(y, z)}. 20. A (X, d). A A A. 21. A (X, d). Ā A. 22. A Ā = A. 23. A A = A. 24. R n (a 1, b 1 ) (a n, b n ) R n. [a 1, b 1 ] [a n, b n ] R n. 3

III 2 25. d(x, y) R 2. ρ : R 2 R 2 R. d(0, x) + d(0, y) x 1 y 2 x 2 y 1 ρ(x, y) = d(x, y) x 1 y 2 = x 2 y 1 (a) ρ (b) p = (0, 1) R 2. {x R 2 ; ρ(p, x) < 1}. (c) (R 2, ρ) 26. R 2 X l 1, l 2 X d(l 1, l 2 ) (0 d(l 1, l 2 ) π/2) d X 27. (X, d) {x n } d(x m, x m+1 ) < m=1 {x n } 28. R 2 (a) {(x, y) R 2 ; a < x < b, c y d} (b) {(x; y) R 2 ; x 2 + y 2 < 1} \ {(0, 0)} 29. R 2 (a) {(x, y) R 2 ; x, y Q} (b) A n = {(x, 1/n) R 2 ; 0 x 1} A n. n=1 1

30. S 1 = {(x, y) R 2 ; x 2 + y 2 = 1} S 1 p, q d(p, q) p q (a) p = (cos θ p, sin θ q ), q = (cos θ q, sin θ q ) d(p, q) (b) d S 1 R 2 S 1 ( 1 ) R 2 S 1 d S 1 31. Q R 32. π : R 2 R π((x, y)) = x R 2 U π(u) R 33. U = {U R; x U ϵ > 0 st [x ϵ, x + ϵ] U} R 34. (X, U ). A, B X A \ B B \ A 35. X = {a, b} 36. R n {U(x, r); x Q n, r Q, r > 0} U(x, r) x r 37. R {(a, b); a, b R} 38. U λ, λ Λ, λ U λ U, V U V 1 U S 1 U = S 1 V, V R 2 2

39. X = {a, b, c, d, e} {{a}, {a, b, c}, {c, d}} 40. (X, U ) A λ X, λ Λ. ( λ A λ ) λ A λ 41. 40. λ A λ λ A λ 42. U = {R \ } R. 43. R 2 A = {(x, y) R 2 ; xy 0},. 44. S = {[a, b); a < b} R U 3

III 3 45. (X, U ). Y X U Y = {A Y ; A U }. (Y, U Y ). 46. (X, U ), x X. V (x) = {A U ; x A} x. B, B(x) = {A B; x A} x. 47. (X, U ) 2 1. 48. (X, U ) 2. 49. 2 1 1. 50. (X, d). (X, d). 51. X. x X X V (x)( ). (a) A V (x) x A. (b) A, B V (x) W V (x) A B W. (c) A V (x) W V (x) y W A y A, A y V (y). N(x) = {A X; B V (x), B A} x X X U. 1. 1

52. R, [a, b) [a, b]. 53. a n, n = 1, 2, 3,..,. R A = {a n ; n = 1, 2,..}. 54. R U = {(, t); t R} U {, R}. 55. R {(, t); t R} {(t, ); t R}. 56. {(a 1, b 1 ) (a n, b n ); a j, b j Q,, j = 1,..., n} R n (base). 57. l n R 2 n. A = R 2 \ ( n=1l n ). A, Ā, A. 58. ϕ : R R, A, A n R. 2 (a) ϕ( A n ) = ϕ(a n ) (b) ϕ( A n ) ϕ(a n ) (c) ϕ 1 ( A n ) = ϕ 1 (A n ) (d) ϕ 1 ( A n ) = ϕ 1 (A n ) (e) ϕ 1 (A c ) = ϕ 1 (A) c 59. f : R 2 R. A = {(x, y) R 2 ; f(x, y) 0}. A {(x, y) R 2 ; f(x, y) > 0} f. 60. θ R \ Q. A = {e 2πnθi C; n N}. C R 2. Ā = {z C; z = 1}. 2 = n=1, = n=1. 2

61. X = {(x, y) R 2 ; 0 < x + y < 1} R 2. X,, (a) A = {(x, 0) X; 0 < x < 1} (b) B = {(x, y) X; 0 < x 2 + y 2 1/2} 62. A = {(x, y) R 2 ; 0 x} B = {(x, y) R 2 ; 0 x, x 2 + y 2 < 1} (a) B R 2 (b) A B A 63. (X, d). A X. A. 64. (X, d). A X. Ā.. (a) A Ā. (b) A = U F X U X F. 65. X = C. U = {, A X; #A c < } U X. U 1, U 2 U U 1 U 2 3. 66. p. n Z p n e 0 n = n p e. v p (n) = 2 e, v p (0) = 0. d : Z Z R d(n, m) = v p (n m), d Z. 3 42. 3

67.. 0 < ϵ < 1 m = [ log 2 ϵ]. [x] x. d(0, n) ϵ n p m. {l Z; d(n, l) < ϵ} = {n + kp m ; k Z}. 68. Z H = {x 0 + ks; k Z}. (a) Z H = H. (b) Z d. p s H = Z. 69. X = R = {(x 1, x 2,...); x j R, j = 1, 2,...}. d : X X R d(x, y) = sup(min{ x n y n, 1}). d n, d (x, y) = n=1 ( 18 ). 1 x n y n 2 n 1+ x n y n 70. (X, U ) A X (a) A, A O X (b) A, A F X 4

III 4 71. (X, U ), (Y, V ). (1), (2). (1) f : X Y. (2) f x X. 72. =. 73. f : R S 1 x (cos 2πx, sin 2πx). f( 2Z) S 1. 74. X. (X, U ), (X, U 0 ). id : (X, U ) (X, U 0 ) idx = x. #X 2 id. 75. S 1 \ {(0, 1)} = R 1. 76. (0, 1) R. R = (0, 1). 77. R M = { ( 1) n (1 1 ); n = 1, 2, 3,..} n. 78. R { 1 + 1 ; n, m = 1, 2, 3,..} n m. 79. D = { m 2 ; n N, m {1, 2,, n 2n 1}} R [0, 1]. { } 1 80. A n = (x, nx ) R2 ; x > 0 A = n N A n 81. (X, U ) A B A B = A B 1.. 1

82. (X, U ), Y X X A Y, B X : (a) Int Y A = Int X (A (X \ Y )) Y 2 (b) Y A ( X A) Y 3 (c) Int X B Y Int Y (B Y ) 83. (X, U ), Y X X A Y A Y Y (Ā \ A) = 84. (a) R n (0, 1] n (b) R n M = [0, 1] n M (0, 1] n M 85. U R U 86. Q R. p < q {x Q; p x q} Q 2 Int Y Y 3 Y Y c Y. Y Y. Y Y 2

87. (a) U R n, f : U R m, d(x, y). lim x a f(x) = b inf r>0 (. sup d(f(x), b) x U,0<d(x,a)<r ) = 0 (b) f : R 2 R, (x, y) x + y,. 88. U R, a U. d, f(x) = d(a, x). f : U R 89. U R n V R m f : U R m, g : V R l. f(u) V g f : U R l ( 90. f : R 2 \ {(0, 0)} R 2, (x, y) x y R 2 R 2 x 2 +y 2, 91. R 2 R 2 (x, y), y 0 f(x, y) = (x, y), y < 0 x 2 +y 2 ), f(u) U R 2 92. f : R R, x x U = {f 1 (A); A R } R R 3

93. X = C [0, 1] [0, 1] r 4 f r = f (j) (t), U r,ϵ (f) = {g X; f g r < ϵ} j=0 sup 0 t 1. U f = {U r,ϵ (f); r = 0, 1, 2,..., ϵ > 0} (1)-(4). (1) U, (2) U U f U f, (3) U, V U f W U f W U V, (4) U U f W U, W U f, g W V U g V U 94.. X = C [0, 1] U f f X. 95. n f(z 1,..., z n ) O f = {z C n ; f(z) 0}, B = {O f ; f} O f, O g B O f O g B U = {, C n, f O f ; O f B} C n. {, C n } B U base. 4 {0, 1} ( ϵ, 1 + ϵ) 4

III 5 96. R X = [ 1, 1] f(x) = x 3 97. (X, d), (Y, ρ), (Z, σ) X f Y g f g Z 98. (X, d), (Y, ρ) f : X Y {x n } X {f(x n )} Y 99. R X = (0, ) f(x) = 1/x 100. (X, d) A X f(x) = d(x, A) 1 X 101. R A A R B = [0, 1] B B B 102. 2 2 M 2 (R) M 2 (R) 4 4. GL 2 (R) M 2 (R) GL 2 (R) = M 2 (R) 103. (X, U ) A X X A X B A A B X 1 d(x, A) = inf y A d(x, y) 1

104. (X, U ) f : X R f 1 (, a) U f 1 (a, ) U a R 105. (X, U ) (Y, V ) X = λ Λ A λ, A λ U, f : X Y, λ Λ f Aλ : A λ Y X 106. X = {a, b, c, d} U = {, X, {a}, {b}, {a, b}, {b, c, d}} f : X X f(a) = b, f(b) = d, f(c) = b, f(d) = c f c d 107. (X, U ) (X, V ). i : (X, U ) (X, V ) U V 108. (X, U ) (Y, V ) X f : X Y Y 109. (X, U ), (Y, V ). f : X Y f(a) = f(a), A X. g : X Y intf(a) f(inta), A X. 110. f : R R, x x 2, 111. (X, U ), (Y, V ) A U X f : X Y f A : A Y 2

112. f : R S 1 = {(x, y) R 2 ; x 2 +y 2 = 1}, x (cos 2πx, sin 2πx) S 1 R 2. 113. (X, U ). f, g : X R. f + g, fg, f/g(, g(x) 0 x X ). 114. (a) f : R R x sin(1/x), x 0 f(x) = 0, x = 0. f. (b) g : R R sin(1/x), x 0 g(x) = 0, x = 0. g. 115. (X, d). X A Ā = X. X. 116. R. f : R R f. 117. [a, b] = [c, d], a < b, c < d, 2. 118. (a, b) = (c, d), a < b, c < d,. 119. [0, 1] (0, 1). 2 [a, b], (a, b) R. 3

120. (0, 1) [0, 1]. 121. R = (a, b), a < b,. 122. [a, b) = (c, d], a < b,. 123. {(x, y) R 2 ; 0 < x < 1} = R 2 124. {(x, y) R 2 ; 1 < x 2 + y 2 < 2} = R S 1 3. 125. {(x, y) R 2 ; x 2 + y 2 < 1} = R 2 126. (X, U ) A X. f : X R A A X, f A. (a) A, B U X = A B f A B f X. (b) (a) A B X X = A B f X. 127. f : R R 2 f(t) = (cos t, sin t). f (0, π/2). f [0, 2π). 128. n S n ( R n+1 ). S n \{(0,..., 0, 1)} = R n. 3... 4

III 6, R n, S n = {x R n ; x = 1} R n+1. 129. X, (Y, V ), f : X Y., f 1 (V ) = {f 1 (A); A V } ( f 1 ( ) = ) X. 130. 129, f : (X, f 1 (V )) (Y, V )., f 1 (V ) f X. 131. Y, (X, U ), f : X Y., V = {A Y ; f 1 (A) U } ( f 1 ( ) = ) Y. 132. 131, f : (X, U ) (Y, V )., V f Y. 133. (X λ, U λ ), λ Λ,. (X, U ) = λ (X λ, U λ ). { λ A λ; A λ U λ λ A λ = X λ } U. 134. (R 2, U 2 ) = (R, U 1 ) (R, U 1 ). U n R n. 135. R m R n = R m+n. 136. p : R 2 R p(x, y) = x 1

137. I = [0, 1] x = y, x, y (0, 1) x y {x, y} = {0, 1}, {x, y} = {1, 0}. f : I S 1 x cos(2πx) + i sin(2πx), f : I/ S 1 f([x]) = f(x). (1)-(4). (1) f. (2) f well-defined. (3) f bijective. (4) f. 138. [0, 1] 137. S 1 = [0, 1]/. 139. z R n f : R n R n, x x + z,. 140. f : R 2 \ {0} S 1, x x/ x 141. (X, U ), (Y, V ) A X, B Y IntA IntB = Int(A B) 1 142. (X λ, U λ ), λ Λ, Int( λ A λ ) λ IntA λ 143. (X, U ), (Y, V ) A X, B Y (A B) = ( A B) (Ā B) 1 Int(A B) A B (X, U ) (Y, V ).. 2

144. (X λ, U λ ), λ Λ, λ (X λ, U λ ) 145. (X λ, U λ ), f λ : X λ Y λ, λ Λ, p λ : X = λ X λ X λ, q λ : Y = λ Y λ Y λ,. f : X Y q λ (f(x)) = f λ (p λ (x)) 2., f f λ, λ Λ,,. 146. 145, f : X Y f λ : X λ Y λ, λ Λ,. 147. X R n, R n. f : X R m f 148. 2 S 2 p = (0, 0, 1) S 2 \ {p} f : S 2 \ {p} R 2 x, (x, y, z) ( 1 z, y ), 1 z 149. R 2 \ {(0, 0)} = {(x, y) R 2 x 2 + y 2 > 1}. 150. (X n, d n ) (n = 1, 2, 3, ). n, d n (x, y) < 1, x, y X n, X = n X n d d(x, y) = n=1 1 2 n d n(x n, y n ). d X 151. C C + = {z C Imz > 0} f(z) = z i z + i f f(c +) f C + f(c + ) 2 3

152. x y x y Z. S 1 = R/ 3. 153. x y x y Q. R/. 154. R X = [0, 1] Y = [0, 1) {1.00001}. 155. X = S 1 S 1 t : X X t((x, y), (u, v)) = ((x, y), ( u, v)). X P, Q P Q P = Q P = t(q). (a),(b),(c). (a) t t = 1 X. (b). (c) P π[p ]. π[p ] 2 4. 156. (X, U ), f : X X. n f n = f f f X X f }{{} n. 3 R/ R/Z,. 4 X/. 4

III 7 157. (X, d), F X. D : X R, x d(x, F ) D(x) D(y) d(x, y). D. 158. (X, d), F X. d(x, F ) = 0 x F. 159. (X, U ), (Y, V ). X, Y X,Y, π X : X X/ X, π Y : Y Y/ Y. π X π Y : X Y X/ X Y/ Y X Y. (x, y) (x, y ) π X π Y (x, y) = π X π Y (x, y ). Q : X Y/ X/ X Y/ Y Q([x, y]) = π X π Y (x, y), [x, y] X Y/,. (a) Q well defined. (b) Q bijective 1. 160. (a) R X = [0, 1] A = (0, 1). A X/A 3. (b) X/A. 1 1

161. X = {z C 1 z } A = {z C z = 1}. (a) f(z) = ( z 1)z. f : X \ A C \ {0} bijective. (b) A X/A C f. 162. R 2 (x, y) (x, y ) (x, y) = (x, y ) (x, y) = (x, y ). R 2 / = {(x, y) R 2 ) 0 y}. 163. R 2 (x, y) (x, y ) (x, y) = (y, x ) (x, y) = (x, y ). R 2 / = {(x, y) R 2 ) x y}. 164. R x y x y Q. (a) R R/ π. R/ U W = π 1 (U). x W q Q x + q W. (b) W = R. 165. (X, U ). X T 1 x X {x} = {U; U x }. 166. T 1 (X, U ), (Y, V ) (X, U ) (Y, V ) T 1. 167. R A = (0, 1) R/A π : R R/A, π(a) 2. 2 R/A T 1. 2

168. (X, d). 169. : (T 2 ) T 1 T 0 170. (X, U ), A X. 171. (X, U ). X x X {x} = {Ū; U x }. 172. (X, U ), (Y, V ) (X, U ) (Y, V ). 173. (X, U ), (Y, V ), f : X Y. D = {(x, f(x)) X Y ; x X} X Y. 174. (X, U ), (Y, V ), f, g : X Y. A = {x X f(x) = g(x)}. 175. (X, U ), (Y, V ), f : X Y. X x y f(x) = f(y) X/. 3

176. (X, U ), (Y, V ), f : X Y. X X X, Y Y Y. (a) f f : X X Y Y, (x, y) (f(x), f(y)), X f f Y. (b) Y X. 177. N, U. (a) U. (b) (N, U ) T 1 T 2. 4

III 8 178. (X, U ) T 1 x X {x}.. 179. (X, U ) T 3 x X x A A U B U x B B A.. 180. (X, U ) T 4 F X F A A U B U F B B A.. 181.. 182. (X, U ). F, G X F G =. f : X Y (1), (2) 0 f(x) 1, (3) x F f(x) = 0, (4) x G f(x) = 1 1. 183. (X, U ) X X. W X X W = {(x, y) X X; (y, x) W } W W 184. (X, U ) (Y, V ). f : X Y,. X Y Hausdorff. 185. (X, U ), (Y, V ) Hausdorff. f, g : X Y. X D 2 f = g D f = g X. 1 Urysohn 2 D = X. 1

186. (X, U ) T 3. x, y X y {x} x {y}. 187. (X, U ) T 3. x, y X {x} = {y} {x} {y} =. 188. (X λ, U λ ) ( λ X λ, λ U λ). 189. (X, U ), A X. A X/A π : X X/A. (a) π. (b) X/A Hausdorff. 190. (X, U ) T 1 X. U. 191. (X, U ), (Y, V ). f : X Y. Y. 192. (X, d). X. 193. R U(x) = {[x, a); x < a} x R S. (a) (R, S ) T 1. (b) (R, S ) T 4 3. 194. (X, U ) T 4 T 4. 3 (R, S ) Sorgenfrey. Sorgenfrey. 2

195. n n M n (R) A = { a ij }. n n GL(n, R) M n (R). (a) D : M n (R) R (g) = detg. (b) GL + (n, R) = {g GL(n, R); detg > 0} GL + (n, R) GL(n, R). (c) f : GL(n, R) GL(n, R) f(g) = g 1. f. (d) M : GL(n, R) GL(n, R) GL(n, R) M(g 1, g 2 ) = g 1 g 2. M. 3

III 9 196. (X, U ) (Y, d), (X, U ). 197. (X, U ). f : (X, U ) (Y, V ). (Y, V ) 1. 198. (X, d) [0, 1] N. [0, 1] R. 199. (X, d).. 200.. 201. (X, d). d(x, y) = min{1, d(x, y)}. d, d d. 202. (X k, d k ), k = 1,..., n,. X = n k X d : X X R d(x, y) = n d k (x k, y k ) 2, d X k=1,. 203. 2 (X, d), (X, d ) d(x, y) d (x, y), x, y X, d U d U. 1.. 1

204. R 2 A n = {1/n} R, n = 1, 2, 3,..., X 2 T 1 Hausdorff. 205. R. A R, #A < A = A = R R. R Hausdorff. 206. (X, U ) {x n } n x, x U N N n x n U. lim x n = x. X Hausdorff n lim x n = x lim n x n = y x = y. n 207. R U = {(a, ) (a R), R, }. x n = 1 + 1, lim x n n = y y n. 208. Q. Q U (x) = {(x 1n, x + 1n } ) Q; n = 1, 2, 3,..., x 0, U (0) = {(( 1n, 1n ) Q) \ { 1m } ; m = 1, 2, 3,...}; n = 1, 2, 3,... U. (Q, U ) Hausdorff. 209. (X, U ), A U (A ) 3 (A, U A ). 210. (X, d) A. 2 R 2 x y x = y n st x, y A n. X = R 2 /. 3 U A = {A B; B U } 2

211. (X n, U n ), n = 1, 2, 3,..., X = n X n. 212. (X, U ) F F [0, 1] n X 4. 213. (X, U ) F 5 f : F S n F. 4 Tietze. 5 S n R n+1 n. 3

III 11 214. R. 215. (X, U ), (Y, V ), f : X Y. X (F (X), V F (X) ) 1. 216. R (a, b). 217. (X, U ), A X. A B Ā B. 218. R [a, b]. 219. R A [a, b], [a, b), (a, b], (a, b). 220. R 2. 221. n S n ( R n+1 ). 222. R 2. 223. (X, U ). X = F G, F G =, F, G F G. 224. X = {a, b, c, d} U = {, {a}, {a, b}, {a, b, c}, X}. (X, U ). 225. X = {a, b, c} U = {, {a}, {a, b}, {a, c}, {c}, X}. X. 1 V F (X) V F (X). 2 (X, U ) x X U x V V u. 1

226.. 227. R R 2. 228. [0, 1] [0, 1] [0, 1]. 229. R \ Q. 230. R n \ Q n (n 2). 231. (X, U ). x y, A X x, y A.. 232. (X, U ). x y, x y 3.. 233. (X, U ), {a}. {a} X = X. 234. (X, U ). 2 A, B X A B A B A, B. 235. (X, U ). 2 A, B X A B A B. 236. (X, U ). A λ, λ Λ, λ A λ λ A λ. 237. (X, U ). 2 A, B X Ā B A B. 3 ω : [0, 1] X ω(0) = x, ω(1) = y. 2

238. (X, U ). A λ, λ Λ,, 2 λ, µ Λ, Λ λ = ν 0, ν 1,, ν n = µ A νk A νk+1, k = 0,..., n 1, λ A λ. 239. 2 4. 240. (X, U ), ( =)A X. A. 241. X = {(x, sin 1x } ); x > 0 {(0, y); 1 y 1} R 2. A = { (x, sin 1); x > 0}, x B = {(0, y); 1 y 1}. (a) X = Ā. X. (b) X. (c) X. 242. R 2 X = {(0, y); 0 < y 1} {(x, 0); 0 < x 1} {( 1, y); 0 y 1, n N} n. 243. 2 2 GL(2, R) A = max{ a ij }.. GL(2, R). A B C GL(2, R). A = 1 0, B = 1 1, C = 1 0 0 1 0 1 0 1 4 3

244. (X, U ) A B A B. A B. 245. (X 1, U 1 ) (X 2, U 2 ) X 1 X 2. 246. (X, U ). f [0, 1] [0, 1] X. X. 247. (X, U ). C B A X. C A C B. 4

III 11 248. (X, U ), X x y A st A x, y. X/ π(x) x., π(x) x,. 249. (X, U ), f : X R. x, y X f(x) = a < b = f(y) a < c < b f(z) = c c X 1. 250. (X, U ) (Y, V ). (X Y, U V ). 251. (X, U ). A X, N(x) x. x A U A U N(x). 252. (X λ, U λ ), λ Λ,. λ (X λ, U λ ). 253. (X, U ), A Y X,. A X A Y. 254. (X, U ), A X. A B = U V (U, V U, U V = ) A U A V. 255. (X, U ), X x {x}... 1 1

256. Q R R. 257. (X, U ). H(X) = {f : X {0, 1}; }(, {0, 1} ) f, g (f + g)(x) = f(x) + g(x)(mod 2). f + g H(X). X H(X) = 2. 258. (X λ, U λ ), λ Λ,. λ (X λ, U λ ). 259. X = {a, b, c} U = {, {c}, {a, c}, {b, c}, X} (X, U ). 260. R n a, b. X(a, b) = {tb + (1 t)a; t [0, 1]} R n. a b X(a, b) = [0, 1]. 261. R n. 262. R n. 263. [a, b] R. f : [a, b] [a, b]. 264.. 265. (X, U ) Hausdorff A X. A. 266. (X, U ), (Y, V ). (X Y, U V ). 267. (X, U ) U. (X, U ) X. 2

268. (X, U ), A Y X,. A X A Y. 269. X X ω. U, {ω} X. (a) (X, U ). (b) (X, U ). 270. X = {0, 1}. X. 3

III 12 271. (X, U ), (Y, V ), X = Y., X Y. 272. (X, U ), (Y, V ), f : X Y. f(x). 273. (X, U ), (Y, V ), X A, Y B., A B (X Y, U V ). 274. (X, d). A X, B X. A B = d(a, B) > 0 1. 275. [a, ) R. 276. (X, U ). A j X, j = 1,..., n,. A = n j=1a j. 277. n O(n) = {T M n (R); t T T = E} R n2 2. 278. n GL(n) R n2. 279. (X, U ) Hausdorff. (X, U ). 280. (X, U ) Hausdorff. 2. 1 d(a, B) = inf x A,y B d(x, y) 2 M n (R) R n2. 1

281. Hausdorff (X, U ) x n, n = 1, 2, 3,.., x X. Y = {x, x n ; n = 1, 2, 3...} X. 282. (X, U ), A X, A,. A X/A 3. 283. (X, U ) Hausdorff. A X, A,. A X/A Hausdorff. 284. n S n R n+1 x, y x = y x y = x = y.. (a). (b) S n / Hausdorff. (c) S n /. 285. (X, U ), (Y, V ) Hausdorff., f : X Y f. f. 286. (X, U ), (Y, V ) Hausdorff., f : X Y f. 3 x y x, y A, X/ X/A. 2

287. (X, U ) Hausdorff. A, B X, A B =. A, B. i.e., A U, B V, U, V U,. 288. (X, U ) Hausdorff. X {K λ X; λ Λ} λ Λ K λ. 289. A = {[a λ, b λ ]; < a λ < b λ < +, λ Λ} R. λ Λ [a λ, b λ ]. A 2 290. (X, U ) T 1. X 4 X. 291. R R 5. 4 X, X. 5 σ. 3

292. (X, U ). X = [0, 1], U 6 : U(x) = {U x (1/n) [0, 1]; n N}(x 0) U(0) = {[0, 1/m) \ {1/n; n N}; m N}. I = [0, 1] R.. (a) U(x) x. (b) f : X I, x x. (c) X. 293. (X, U ), f : X R. f(x) > 0 x X f(x) r( x X) r > 0. 6 U x (ϵ) = {y; x y < ϵ} 4

III 13 294. R n. 295. Q R. Q. 296. X = R \ Q R. X. 297. (X, U ) U. X. 298. (X, U ) X X = X { }. X U = U {X \ K; K X }. ϕ X. (X, U ). 299. (X, U ). 298 (X, U ) 1. 300. (X, U ). (X, U ) (X, U ). (X, U ) Hausdorff (X, U ). 301. (X, U ),. 1 1

302. (X, U ). 303. (X, U ) 2 Hausdorff. y X Y = X \ {y} X. Y Y Y = X. 304. (X, U ). base B B. 305. R R R = S 1. 306. R 2 A = {(r cos θ, r sin θ) R 2 ; 0 r 1, 0 θ < π 2 }. 307. C = {z R 2 ; 1 < z 2} D = {z R 2 ; z 1}. 308. A = ( 2, 1) (0, 1). 309. (X, U ) f : X R 0, ϵ > 0 A X, f(x) < ϵ x A c. f(x), x X,. 310. R n (R n ) (R n ) = S n. 2

311. (X, U ), (Y, d), f, g : X Y. sup x X d(f(x), g(x)) = d(f(z), g(z)) z X. 312. (X, U ) Hausdorff. A U \ {, X}, A c = X \ A. A c X/A c A A = A { }. A = X/A c. π : X X/A c, f : A X/A c : π(x), x A, f(x) = π(a c ), x = (a) f. (b) f. (c) X/A c Hausdorff. (d) f 1. (e) A = X/A c. 313. [0, 1] R. A = [0, 1/3] [1/2, 1] [0, 1]/A S 1.. 314. (X, U ) Hausfdorff. U U \ {X} U = R. A = X \ U X/A = S 1.. 3

III 14 315. (X, U ) (Y, V ). Map(X, Y ) k j=1w (K j, A j ) (K j X, A j V ) B. B ( ) 1 Map(X, Y ). 316. (X, U ) Hausdorff. 317. (X, U ). C(X) = Map(X, R) d : C(X) C(X) R d(f, g) = max x X f(x) g(x). (C(X), d). 318. 317. C(X) U, d U d. U = U d. 319. 317. (C(X), d). 320. (X, U ). X R C b (X). (a) d : C b (X) C b (X) R d(f, g) = sup x X f(x) g(x). d C b (X). (b) (C b (X), d). 321. d ρ X τ = d 2 + ρ 2 X. 1. 1

322. (X, d) (Y, ρ). d U d, ρ U ρ. τ = d 2 + ρ 2. (X Y, τ) τ U τ. U d U ρ = U τ. 323. (X, d) (Y, ρ) 2.,. 324. (C([0, 1]), d). 0, 0 [0, 1] (a) f : [0, ) R f(x) = 1, x = 2 0, x [3, ). f n (x) = f(3 n x), x [0, 1],, {f n } C([0, 1]) Cauchy. (b) A = {f C([0, 1]); f([0, 1]) [0, 1]} C([0, 1]) 3. (c) A = {f C([0, 1]); f([0, 1]) [0, 1]} C([0, 1]). 325. (C([0, 1]), d). h n C([0, 1]) h 1 (t) = 0, h n+1 (t) = h n (t) + t h n(t) 2. 2 (a) h n (t). (b) h n (t) t, h n (t) h n+1 (t), lim t h n (t) = t. (c) h n (t) d t. 326. (X, d). A X. X A. 2 (X, d). 3 (X, U ), (X, U ). 2

327. (X, d). A X d A d A = d 4. : (A, d A ) A 328. a(m) n = n j m, m C,. j=1 (a) Rm > 1 {a(m) n } n Cauchy. (b) m = 1 {a(m) n } n Cauchy. 329. R n 5. 330. R n. 331. X = R, X X (u, v) d(u, v) = arctan u arctan v. (a) d X. (b) (X, d) 332. (X, d). X A, A. 333. (X, d). f : X X 6. f. 334. (X, d). f : X X. (a) X. x X x n = f(x n 1 ), x 0 = x {x, x 1, x 2,, x n } x. (b) X f. 4 d A. 5 (X, d). A X ϵ > 0 X ϵ- A. 6 d(x, y) = d(f(x), f(y)). 3

III 15 335. (X, d). (1)-(3). (a) X. (b) X. (c) X 1. 336.. 337. (X, d). f : X X 2. f(x) = x x. (a) X x F (x) = d(x, f(x)) R X. (b) F (x), x X, 0. (c) f(x) = x x. 338. (X, d). X Cauchy {x n } ˆX. (a) {x n }, {y n } ˆX d(x n, y n ) n. (b) {x n }, {y n } ˆX d(x n, y n ) 0(n ) {x n } {y n }.. (c) x = {x n } ˆX [x],x = ˆX/,. d ([x], [y]) = lim d(x n, y n ) well defined. n 1 (X, d),. 2 0 < C < 1 d(f(x), f(y)) Cd(x, y). 1

(d) (X, d ). 339. (X, d). (X, d ). 340. l 2 = {x = {x n };. x 2 n < }. l 2 l 2 n=1 l 2 l 2 (x, y) d 2 (x, y) = (x n y n ) 2 R (a) d 2 l 2. 1, k = n, (b) e k l 2 e k n =. {e k } k Cauchy 0, k n. (c) S = {x l 2 ; d 2 (0, x) = 1}. 341. RP 1 = {R 2 } 3. X = R 2 \ {0}. x, y X x y 0 λ R y = λx. (a). (b) RP 1 = X/. (c) S 1 x y y = ±x, n=1 π : S 1 S 1 /. f : S 1 / π (x) π(x) X/ well-defined. (d) S 1 / = S 1. 3 1. 2

(e) (b) RP 1 X/. RP 1 = S 1. 342. R n (x 1,...x n ) (x 1,..., x n, 0) R n+1 R n R n+1. R n d n. d n (x, y) = x y. R = n=1 Rn. (a) x, y R 2, n x, y R n d (x, y) = d n (x, y). d well-defined R. (b) (R, d ) a n = (1, 1 2,, 1 2 n, 0, ) R. {a n } Cauchy. (c) (R, d ). 343. R N = n=1 R. R. l 2 = {{x n }; x i R, n x2 n < }. R l 2 R N. 344. (X n, U n ), n N,. X 1 X 2 X 3. X = n X n. U = {U X; U X n U n n } (X, U ) 4. 345. O(2) 2. A O(2) A : R 2 R 2. S 1 A S 1: S 1 S 1 S 1. S 1 IsomS 1. ρ : O(2) A A S 1 IsomS 1. 4 X {X n }. 3

346. GL(n) = {A M(n, R); deta 0}, R n2. (a) det : GL(n) A deta R, det. (b) detgl(n). (c) GL(n). 347. O(n) = {A M(n, R); A ta = t A A = E}. O(n) R n2. 348. SO(n) = {A O(n); deta = 1} R n2. 4