Transcription

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2 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n = 0 0 X X O (O3) Λ = 0 O (O1) 1.1 [X, d] O(d) = {U X p U, ε > 0; N ε (p) U} O(d) (O1), (O2), (O3)

3 16 3 (O1) O(d) (O2) 2 U 1, U 2 O(d) U 1 U 2 O(d) p U 1 U 2 ε 1, ε 2 N ε1 (p) U 1, N ε2 (p) U 2 ε = min{ε 1, ε 2 } N ε (p) U 1 U 2 U 1 U 2 O(d) (O3) U λ O(d) U λ O(d) p U λ λ λ λ p U λ U λ ε > 0 N ε (p) U λ N ε (p) U λ λ 1.2 R 2 d 1, d 2, d P(x 1, x 2 ), Q(y 1, y 2 ) d 1 (P, Q) = x 1 y 1 + x 2 y 2 d 2 (P, Q) = (x 1 y 1 ) 2 + (x 2 y 2 ) 2 d (P, Q) = max { x 1 y 1, x 2 y 2 } ε O(d 1 ) = O(d 2 ) = O(d ) d (P, Q) d 2 (P, Q) d 1 (P, Q) 2d (P, Q) ε N 1 ε (P), N 2 ε (P), N ε (P) N ε 2 (P) N 1 ε (P) N 2 ε (P) N ε (P) U [R 2, d ] U [R 2, d 2 ] U [R 2, d 1 ] U [R 2, d ]

4 () O = P(X) 1.4 () O = {, X} (X, O) B U B 1 (X, O) B U p U p B U B B B U X B B λ U = B λ p U p B λ λ λ p B λ U B U X p p B p U B B p p U B p B p U p

5 B U B 1.6 (X, d) ε O(d) B(d) = {N ε (p) p X, ε > 0} B(d) 1 U ε n 1 n 1 2 n { } B 0 = N 1 (p) n p X, n Z R n 1 n p X Rn { } B 00 = N 1 (q) k q = (q1, q 2,..., q n ) R n, k Z +, q i Q, i = 1, 2,..., n U R n p U p 1 U n p 1 q 2n q 1 2n N 1 (q) p p 1 2n n U B 00 B 00 X 2 2

6 R U R 2 U U = N = {1, 2,... } k l k 0, k 1,..., k N ; k = k 0, l = k N, I ki 1 I ki (i = 1, 2,..., N) N { N 1 N 2 N M N = N 1 N 2... N i J i J i = I k J i k N i { J 1 J 2 J M U = J 1 J ( ) (X, O) B (B1) B = X B B k=1 (B2) B 1, B 2 B p B 1 B 2 B B p B B B 1 B 2 (B1) X O X B (B2) B 1 B 2 p B 1 B 2 B B (O3) (O2) (X, O) SB SB {S 1 S 2 S n S i SB, i = 1, 2,..., n, n = 0, 1, 2,...} (X, O) n = 0 0 X I k

7 R n 1 O(d) SB = { D n 1 (p) p R 2 } p R n ε > 0 δ > 0 p = p (1 δ)e 1, p = p + (1 δ)e 1 e 1 p D 1 n (p ) x D 1 n (p ) D n 1 (p ) D n 1 (p ) x p 1 1 (x p (1 δ)e 1, x p (1 δ)e 1 ) = x p 2 + (1 δ) 2 2(1 δ)(x p, e 1 ) x p 1 1 x p 2 + (1 δ) 2 + 2(1 δ)(x p, e 1 ) 2 4δ 2δ 2 2 x p 2 x p δ(2 δ) < 2δ 0 < δ ε2 2 p D 1 n (p ) D 1 n (p ) N ε (p) (O2) (O3)

8 X B (B1) (B2) B X O O B O (O1) (O2) (O3) (O3) (O1) (B1) (O2) 2 U, V O U = B λ, V = B µ (B λ, B µ B) λ Λ 1 µ Λ 2 ( ) U V = λ Λ 1 B λ µ Λ 2 B µ = B λ B µ (λ,µ) Λ 1 Λ 2 B λ B µ O (B2) 1 4 X S B B = {S 1 S 2 S n S i S, i = 1, 2,..., n, n = 0, 1, 2,...} B O B = {B λ λ Λ} { } O = Λ Λ λ Λ B λ O X S B B (B1), (B2) 0 X (B1) (B2) B X S X O S

9 X = {1, 2, 3} O = {, {1}, {2, 3}, X} O X U\V {1} {2, 3} X (i) U V {1} {2, 3} X U\V {1} {2, 3} X (ii) U V {1} {2, 3} X O 1.11 X = {1, 2, 3, 4} B = {{1}, {2, 3}, {4}} B B X O 1.12 (i) O = {(t, ) t R} {, R} (ii) O = {[t, ) t R} {, R} 1.13 X (i) X O = {X A A X } { }

10 16 10 (ii) X S = {X {a} a X} (X, O) 1.14 X (i) X O = {X A A X } { } (ii) X O = {X A A X } { }

11 ε () (X, O) X N p X p N U p U N, U O p p p 5 U X U p U p U X p U U p U X p U N N p = N N p p O p p O p N p p U p O p U U = p U ( ) (X, O) p X N (p) p U U N N (p) p N U 2.1 (X, d) p ε N (p) = {N ε (p) ε > 0} ε > 0 1 (n = 1, 2,...) n X 1 1 O p

12 (X, O) B N (p) = {B B p B} { } ( ) N (p) = N ε (p) ε > 0 N ε (p) = {q X d(p, q) ε} N (p) U p U, N N (p); N U 6 () (X, O) N (p) (N1) N (p) N N (p) p N (N2) N 1, N 2 N (p) N 3 N (p) N 3 N 1 N 2 (N3) N 1 N (p) N 2 N (p) q N 2 N 3 N (q) N 3 N 1 (N1) X N N (p) N X N (p) N N (p) p N (N2) N 1, N 2 N (p) p U 1 N 1, p U 2 N 2 U 1, U 2 p U 1 U 2 N 1 N 2 U 1 U 2 N 3 N (p) N 3 U 1 U 2 N 1 N 2 (N3) N 1 N (p) p U 1 N 1 U 1 N 2 N (p) N 2 U 1 q N 2 q U 1 N 3 N (q) N 3 U 1 N 1 (N1), (N2), (N3) N (p)

13 X p X N (p) (N1), (N2), (N3) N (p) X O X O O = {U X p U, N N (p); N U} O (O1), (O2), (O3) (O1) O, X O (O2) U 1, U 2 O p U 1 U 2 p U 1, p U 2 N 1, N 2 N (p) N 1 U 1, N 2 U 2 N 1 N 2 U 1 U 2 (N2) N 3 N (p) N 3 N 1 N 2 N 3 U 1 U 2 U 1 U 2 O (O3) U λ O p U λ λ p U λ λ U λ O N N (p) N U λ N U λ U λ O λ λ N N (p) p N (p) (X, O) p U N U O U = {q X N N (q); N N} q U q N N U N U O N N (q) (N3) N 2 N (q) q N 2 N 3 N (q ) N 3 N N q U N 2 U q U q N 2 U U O 2.4 R 2 N (p) = { ABC p ABC } N (p) R N (p) (N1), (N2), (N3)

14 ( ) (X, O) F F c = X F 8 () 3 (C1) X (C2) (C3) (O1), (O2), (O3) () X A A A A A A A (C3) A A = {F F A } A A = A 9 X A A = {x X x A } x / A A F x U = F c = X F U A x x / A A U x x 3.1 ( (Sorgenfrey) ) X = R O Sor t R N Sor (t) = {[t, t + ε) ε > 0} R Sor = (R, O Sor )

15 16 15 (i) (, α), [α, β), [β, ) R Sor (ii) (α, β), (β, ) R Sor (iii) (, α], [α, β] R Sor (iv) (α, β] R Sor (v) R Sor 1 (vi) R Sor 2 (vi) B t R [t, t + 1) t B t B t B t [t, t + 1) B t t {B t } t R 10 ( ) 4 (CL1) = (CL2) A A (CL3) A B = A B (CL4) A = A (CL2), A (CL1), (CL4) (CL3) (C2) A B A B A B A B A A B, B A B A B A B 11 X A X A X (CL1), (CL2), (CL3), (CL4) A X O X (C1), (C2), (C3) (C1) (CL1) (CL2) X X = X (C2) F 1, F 2 F 1 F 2 = F 1 F 2 = F 1 F 2 F 1 F 2 (C3) A B A B A B A B = B A B = A B = B A B F λ λ F λ F λ λ F λ F λ = F λ λ λ F λ λ F λ (CL2) X

16 16 16 A (CL4) A F A A F A F A F F = F 12 (X, O) {N (p)} A X p A N N (p); A N p / A F ; A F, p / F, F U; A U =, p U, U N N (p); A N = (X, O) A X A = X 3.2 R (i) Q (ii) R Q (iii) 2 Z [ ] { 1 2 = n 2 n Z, m N } m (iv) Z + 2 Z = { n + 2 m n, m Z } 3.3 ε > 0 U = (b n a n ) < ε n=1 (a n, b n ) Q n=1 (X, O) 1 p X A X (i) p A p N N A ( p A) (ii) p A p N N (A {p}) ( p A {p}) (iii) p A p N N A N (X A) ( p A X A)

17 16 17 (iv) p A p N N A ( p / X A) (v) p A p N N A = {p} ( p / A {p}) (X, O) A X (i) A A d A A (ii) A A A (iii) A Int A A (iv) A = A d A A d 3.4 X = R A R A = A 1 A 2 A 3 A 4, A 1 = { 1 n n = 2, 3, 4,... } A 2 = {t Q 1 < t < 2} A = {0} A 1 [1, 4] A d = {0} [1, 4] A 3 = (2, 3) A 4 = (3, 4) A = {0} A 1 [1, 2] {3} {4} Int A = (2, 3) (3, 4) A A R { A = t = 1 } n n = 1, 2, 3,... A d = {0}, (A d ) d = { B = t = 1 n + 1 } m n, m 1, 2, 3,... B d = {0} A

18 16 18 C = { t = 1 n + 1 m + 1 l } n, m, l 1, 2, 3,... C d = {0} B 13 (i) A = A X A (ii) Int A = X X A (iii) A = Int A A, Int A A = (iv) Int A A (i),(ii) (iii) (i),(ii) (iv) X A X A 14 A = A A d p A A p A d p / A A {p} = A N N (p) N (A {p}) = N A 3.6 (A B) d = A d B d (i) (A B) A B (ii) A B = (A B) = A B (i) p (A B) p A B = A B p / Int(A B) p A p / Int(A B) p / Int A p A p B (ii) p A p A p A B p Int(A B) p N N A B p A p / B N N B = N A p Int A p / Int(A B) (A B) (A B) ( A B) p (A B) p A B p A B A B = A B (X Int B) p Int B p / Int A p Int A p Int(A B) A B = A B = A B Int(A B) = Int A Int B

19 16 19 p A B p / A B p N N A B = p Int A N A N A = N N B = N A B = p B p / Int A p A p A p B A B = A, B 3.7 H = n=1 { ( (x, y) x 1 ) } 2 + y 2 = 1 n n 2 R X R 2 { ( X = (x, y) x ( + y n) 2 = ) } 2 n n=1 3.9 Y R 2 Y = {(x, y) (x n) 2 + y 2 = n 2 } n= Z R 2 Z = r (1,2) Q 3.11 W R 2 W = n= 3.12 V R 2 V = n= {(x, y) (x r) 2 + y 2 = r 2 } {( ) 1 x, cos x + n π 2 < x < π } 2 { (x, tan x + n) π 2 < x < π } 2

20 C R 2 C = {( x, sin x) 1 } 0 < x X X X O 3.15 α Z + α Z = {n + α m n, m Z} 3.16 D X E X D E X 3.17 D X X G D G = G 3.18 SB X D X S SB D X 3.19 G X A X G A = G A 3.20 A X G A = G A G X 3.21 Int(A) Int(B) Int(A B) Int(A) Int(B) (A B) A X α(a) = Int(A), β(a) = Int(A) 3.22 A A α(a) A β(a) A 3.23 A = α(a) B = α(b) A B = α(a B) 3.24 A X α(α(a)) = α(a) β(β(a)) = β(a) 3.25 U, V X α(u) α(v ) = 3.26 U, V X U V = X α(u) = X V 3.27 R A 7 A, Int A, A, α(a), β(a), α(int A), β(a)

21 X A A X A = A c A A = A a 2 (i) A 14 (ii) R A (i) (A) A, (Int A) A (ii) R A A, (A), (Int A), ( A) 3.30 A = (X A) 3.31 A Int( A) = 3.32 A B = A B = (A B) = A B 3.33 A, B (A B) (B A) (A B) 3.34 A, B (A B) (A B) (B A) ( A B) 3.35 R A, B ( ( A)) = ( A) 3.37 A B = (A B) = (A B) ( A B) A R A A (α, β) 1 2 ( (Baire) ) R 2 A n (n = 1, 2....) R n A n I 1 = (0, 1) I 1 A 1 t 1 I 1 A 1 t 1 / A 1 0 < ε < 1 2 I 2 = (t 1 ε, t 1 + ε), I 2 I 1 A 1 I 1, I 2,..., I k I k+1 I k A k t k I k A k t k / A k 0 < ε < 1 2 k I k+1 = (t k ε, t k + ε), I k+1 I k A k (t k ) t k+1 t k < 1 2 k t t n I n t R n A n

22 16 22 Q R Q 2 A R R A R G δ F σ G δ F σ F σ G δ 3.38 Q R F σ R Q R G δ Q R G δ R Q R F σ Q R G δ U i R Q = i=1 U i U i Q Q 1 Q = {r 1, r 2, r 3,... } V i = R {r i } V i U i V j = Q (R Q) = i=1 j=1

23 () (X, O) Y X X Y O Y = {U Y U O} Y O Y (Y, O Y ) O Y (O1), (O2), (O3) 15 (X, O) Y X (i) B (X, O) B Y (Y, O Y ) = {B Y B B} (ii) SB (X, O) SB Y (Y, O Y ) = {S Y S SB} (i) Y U Y U X B U = B λ ( ) U Y = Bλ Y = (B λ Y ) Y B Y (ii) SB X {S 1 S 2 S n Y S i SB, i = 1, 2,..., n, n Z + } Y S 1 S 2 S n Y = (S 1 Y ) (S 2 Y ) (S n Y ) SB Y 16 (i) (X, O) Y X q Y N (X, O) q N Y (Y, O Y ) q (Y, O Y ) q N Y (X, O) q N N Y = N Y (ii) {N (p) p X} (X, O) q Y N Y (q) = {N Y N N (q)} {N Y (q) q Y } (Y, O Y )

24 16 24 (i) N (X, O) q q U N U O q U Y N Y N Y Y q N Y Y q q U Y N Y U Y O Y U Y = U Y U O N = U N Y N X q N Y = N Y (ii) q Y N Y = N Y N X q N 1 N (q) N 1 N N 1 Y Y q N 1 Y N Y {N Y (q) q Y } (Y, O Y ) 4.1 (X, d) Y X Y d Y d Y Y p, q Y d Y (p, q) = d(p, q) (Y, d Y ) (X, d) Y (Y, d Y ) ε Nε Y (p) Nε Y (p) = {q Y d Y (p, q) < ε} = {q X d(p, q) < ε} Y = N ε (p) Y 4.2 (R n ) S n 1 = {(x 1, x 2,..., x n ) R n x x x 2 n = 1} D n = {(x 1, x 2,..., x n ) R n x x x 2 n 1} S n 1 (n 1) D n n S n 1 D n R n (X, O) A Y X (i) A X A X Y A Y A Y = A X Y (ii) A X Int X A Y Int Y A Int X A = Int Y A Int X Y (i) p A Y N N (p) N Y N Y (p) (N Y ) A N A p A X

25 16 25 p A X Y N Y N Y (p) N N (p) N Y = N Y A Y N Y A = (N Y ) A = N A p A Y (ii) p Int X A N N (p) N A N Y A p Int Y A p Int X Y p Int Y A Int X Y N N (p) N Y A N Y N = N Y A p Int X A 4.3 (i) X = R Y = (0, 2), A = (1, 2) A X = [1, 2] A Y = [1, 2) (ii) Y = [0, 2], A = [1, 2] Int X A = (1, 2) Int Y A = (1, 2] (i) A Y Y X A X (ii) A Y Y X A X 4.4 (Cantor ) [0, 1] C C i (i = 0, 1, 2,...) (i) C 0 = [0, 1] (ii) C i 2 i [a, d] b c a < b < a + d 2 < c < d C i [a, d] [a, b] [c, d] = [a, d] (b, c) C i+1 C i+1 2 i+1 C = i=1 3 1 (b, c) [a, d] 3 1 b, c [a, d] (ternary set) C i

26 S 0 1 φ : C S x C i x C i x C i 1 C i 2 x s i = 1 s i = 0 x C φ(x) = (s 1, s 2, s 3,...) S φ C S 4.5 ( Sierpinski ) T n T n 4 n 1 ( ) n T n T 0 4 T = n=1 T n

27 X Y A X (i) (Int X A) Y Int Y (A Y ) (ii) A X Y A Y Y 4.7 A X U A V X U V V (V A) X 4.8 A B X A B A B 4.9 X Y A X U Y A Y Y A Y U Y = 4.10 X X A, B A B = X A B (B A) = (A B) B A = (i) A, B A, B C X (ii) C = C A A C B B (iii) C A A C B B C X (iv) C A A C B B C X 4.11 X X A, B, C A B = X C A B (i) C A C B C X (ii) C A C B C X 4.12 X X A, B A B = X S A U B S B B S Int(A U B )

28 ( ) 2 (X, O X ) (Y, O Y ) f : X Y Y V O Y f 1 (V ) X V O Y f 1 (V ) O X 5.1 (X, d X ) (Y, d Y ) f : X Y ε-δ x X ε > 0 δ > 0 d X (x, y) < δ d Y (f(x), f(y)) < ϵ f(x) ε N ε (x) Y U = f 1 (N ε (f(x))) X x N δ (x) f 1 (N ε (f(x))) δ > 0 d X (x, y) < δ y N δ (x) f 1 (N ε (f(x))) f(y) N ε (f(x)) d Y (f(x), f(y)) < ϵ V Y U = f 1 (V ) X x U V f(x) V ε > 0 N ε (f(x)) V δ > 0 d X (x, y) < δ d Y (f(x), f(y)) < ϵ f(n δ (x)) N ε (f(x)) V N δ (x) f 1 (N ε (f(x))) f 1 (V ) = U 18 (X, O X ), (Y, O Y ) {N X (p)}, {N Y (q)} f : X Y 3 (a) f : X Y (b) p X f(p) Y N Y f 1 (N Y ) p (c) p X V Y N Y (f(p)) V X N X (p) f(v X ) V Y

29 16 29 (a) (b) N Y f(p) U Y O Y f(p) U Y N Y p f 1 (U Y ) f 1 (N Y ) f f 1 (U Y ) f 1 (N Y ) p (b) (c) V Y N Y (f(p)) (b) f 1 (V Y ) p V X N X (p) V X f 1 (V Y ) f(v X ) V Y (c) (a) U Y O Y p f 1 (U Y ) f(p) U Y V Y N Y (f(p)) V Y U Y (c) V X N X (p) f(v X ) V Y p V X f 1 (V Y ) f 1 (U Y ) f 1 (U Y ) 19 (X, O X ), (Y, O Y ) f : X Y A X f ( A ) f(a) U = Y f(a) U f 1 (U) X A A f 1 (U) = f ( A ) U = f ( A ) f(a) U Y A = X f 1 (U) A A A f(a) U = f(a) U = f ( A ) f(a) f ( A ) U = A X f 1 (U) = A 5.2 f : X Y B Y f 1 (B) f 1 (B) ι : Y X, ι(p) = p ( p Y ) 20 (X, O) Y X ι : Y X (Y, O Y ) (X, O) O Y Y ι : Y X (Z, O Z ) f : Z Y f : Z Y ι f : Z X U X ι 1 (U) = U Y Y ι : Y X U Y Y

30 16 30 ι f : Z X Y U Y U Y = ι 1 (U) f 1 (U Y ) = f 1 (ι 1 (U)) = (ι f) 1 (U) Z 21 X, Y {U λ } X U λ X X = λ U λ f : X Y λ f U λ : U λ Y U λ i λ : U λ X i λ f U λ = f i λ f U λ W Y (f U λ ) 1 (W ) U λ U λ X (f U λ ) 1 (W ) X (f U λ ) 1 (W ) = f 1 (W ) U λ ( ) f 1 (W ) U λ = f 1 (W ) U λ = f 1 (W ) λ λ 22 X, Y K 1, K 2 X X = K 1 K 2 f : X Y f K 1 : K 1 Y, f K 2 : K 2 Y i = 1, 2 f K i W Y (f K i ) 1 (W ) K i (f K i ) 1 (Y W ) K i K i X (f K i ) 1 (Y W ) X (f K i ) 1 (Y W ) = f 1 (Y W ) K i (f 1 (Y W ) K 1 ) (f 1 (Y W ) K 2 ) = f 1 (Y W ) (K 1 K 2 ) = f 1 (Y W ) f 1 (W ) 2 () X X = λ K λ x X U x U x K λ U x K λ λ = λ 1, λ 2,..., λ N

31 X, Y X = λ K λ f : X Y f K λ : K λ Y x X U x x 23 f U x U x = (U x K λ1 ) (U x K λ2 ) (U x K λn ) f (U x K λi ) f U x X I = [0, 1] X γ : I X X (path) x = γ(0) γ y = γ(1) γ γ x y ω 5.3 ( ) I = [0, 1] I 2 = { (x, y) R 2 0 x 1, 0 y 1 } f Peano t f(t) 4 q 0, q 1, q 2, q 3 : I 2 I 2 ( t q 0 (s, t) = 2, s ) ( 2 s q 1 (s, t) = 2, 1 + t ) ( s q 2 (s, t) = 2, 1 + t ) ( 2 2 t q 3 (s, t) = 2, 1 s ) 2 I 2 2 (0, 0), (1, 0) q 0 ((0, 0)) = (0, 0), q 0 ((1, 0)) = q 1 ((0, 0)), q 1 ((1, 0)) = q 2 ((0, 0)), q 2 ((1, 0)) = q 3 ((0, 0)), q 3 ((1, 0)) = (1, 0)

32 16 32 (1) S = I 2 4 S 0, S 1, S 2, S i = 0, 1, 2, 3 S i = q i (S) I = [0, 1] 4 I 0, I 1, I 2, I 3 I i S i f 1 : I S f 1 (0) = (0, 0), f 1 (1) = (1, 0) (2) 0 n 15 n = 4i + j (0 i, j 3) 4 S ij = q i (S ) ij 4 n = 4i + j n n + 1 I = [0, 1] 16 4 I 00, I 01,..., I 33 f 2 : I S q 0 (f 1 (4s)) q 1 (f 1 (4s 1)) f 2 (s) = q 2 (f 1 (4s 2)) q 3 (f 1 (4s 3)) ( 0 s 1 ) ( s 1 ) ( s 3 ) ( 4) 3 4 s 1 f 2 f 2 (0) = (0, 0), f 2 (1) = (1, 0) f 2 (I ij ) S ij

33 16 33 (k) 4 n = i 1 4 k 1 + i 2 4 k i k i k (0 i 1, i 2,..., i k 3) S i1 i 2...i k = q i1 (S i2...i k ) n = i 1 i 2... i k 4 n n + 1 k = I 4 k I i1i 2...i k I i1i 2...i k S i1 i 2...i k f k : I S q 0 (f k 1 (4s)) q 1 (f k 1 (4s 1)) f k (s) = q 2 (f k 1 (4s 2)) q 3 (f k 1 (4s 3)) ( 0 s 1 ) ( s 1 ) ( s 3 ) ( 4) 3 4 s 1

34 16 34 I i1 i 2...i k I i1 i 2...i k 1, S i1 i 2...i k S i1 i 2...i k 1 f k (I i1 i 2...i k 1 ) S i1i 2...i k 1 t I 1 f k (t) f k 1 (t) 2 k 2 2 {f k (t)} f(t) = lim k f k(t) f(t) t I 4 t = i i i k 4 k + (0 i k 3) = 0.i 1 i 2 i 3... {f(t)} = k=1 S i1i 2...i k ( ) f : I I 2

35 16 35 ( ) (X, O X ), (Y, O Y ) f : X Y f 1 : 1 f 1 : Y X f (X, O X ) (Y, O Y ) 5.4 () S n x (0,..., 0, ±1) x n+1 = 0 φ ± (x) R n φ + : S n {(0,..., 0, 1)} R n, φ : S n {(0, 0,..., 0, 1)} R n ( ) x 1 x 2 x n φ ± (x 1, x 2,..., x n+1 ) =,,..., 1 x n+1 1 x n+1 1 x n+1 φ 1 ± (u 1, u 2,..., u n ) = ( 2u1 u 2 + 1,..., φ ± ) 2u n u 2 + 1, 1 ± u 2 u X 2 p, q X h : X X h(p) = q 5.5 R n S n 5.6 () R 3 A A i (i = 1, 2, 3,...) (i) A 1 4

36 16 36 (ii) A i 4 i A i+1 4 A i+1 4 i+1 A 1 A 2 A n A = i=1 A i 24 S 0 1 ψ : A S x A i x A i x A i 1 A i 4 x t i = 00 t i = 01 t i = 11 t i = 10 x A ψ(x) = (t 1, t 2, t 3,...) S 0, 1 2 ψ A S 25 ψ 1 φ : C A

37 () X R 2 A E B, C, D 3 X f f A B x y C (x, y) (r, θ) D 1 B E 1 f X f(x) f(x) X f 2 x f(x) f(f(x))... f n (x) f 1 (x) = f(x), f 2 (x) = f(f(x)), f n (x) = f(f (n 1) (x)) f n f 2 f 3 f f : X f(x) f : R 2 R 2 X 1 = X X 2, X 3, X 4,...

38 16 38 i 2 X i X i 1 A i = { (x, y) i 1 } x 2 + y 2 i Y 1 = f(x) Y 2, Y 3, Y 4,... Y 4 = X 1 i 2 Y i Y i 1 A i f i : X i X i 1 Y i Y i 1 X i 1 f i 1 i = 2 f f : R 2 R 2 f(x) = f i (x), (x X i X i 1 ) f Smale X Y f : X Y

39 16 39 ( ) X 2 O 1, O 2 O 1 O 2 O 2 O 1 O 1 O () X O = P(X) 5.9 () X O = {, X} 5.10 (i) id : R Sor R id : R R Sor O Sor (ii) f : R R f(t) = { 0 (t < 0), 1 (t 0) f : R Sor R (iii) g(t) = f( t) g : R Sor R ( ) (X, O X ) f : X Y f Y O Y (f) O Y (f) = { V Y f 1 } (V ) O X O Y (f) f : X Y O X O Y (f) (O1) (O2) (O3)

40 16 40 () X {O λ λ Λ} 1 λ : (X, O λ ) (X, O) X O = O λ {O λ λ Λ} λ Λ O = λ Λ O λ O = {U X λ Λ, U O λ } O (O1) (O2) (O3) 2 X, Y X Y 2 {0, 1} X Y = X {0} Y {1} (X Y ) {0, 1} { i 1 : X X Y ; i 1 (x) = (x, 0) i 2 : Y X Y ; i 2 (y) = (y, 1) X Y i 1, i 2 O X Y (X Y, O X Y ) X Y 26 X Y i 1 (X) = X {0}, i 2 (Y ) = Y {1} i 1, i 2 O X Y = {i 1 (U) i 2 (V ) U O X, V O Y } 27 f : X Z, g : Y Z f g : X Y Z f g i 1 = f, f g i 2 = g f g(x, 0) = f g i 1 (x) = f(x), f g(y, 1) = f g i 2 (y) = g(y) f g ( ) (Y, O Y ) f : X Y f X O X (f 1 ) O X (f 1 ) = { f 1 } (V ) X V O Y O X (f 1 ) f : X Y O Y O X (f 1 ) (O1) (O2) (O3)

41 16 41 () X {O λ λ Λ} 1 λ : (X, O) (X, O λ ) X O = sup O λ {O λ λ Λ} λ Λ X SB = λ Λ O λ SB = {U X λ Λ, U O λ } SB O = sup O λ λ Λ 5.11 X, Y f : X Y (a) (b) (a) B Y f 1 (Int B) Int ( f 1 (B) ) (b) B Y f 1 (B) f ( 1 B ) 5.12 X, Y f : X Y (a) (b) (a) A X f(a d ) f(a) (b) B Y ( f 1 (B) ) f 1 ( B) 5.13 (X, O X ) A X O[A] = {U (V A) U, V O} A O 5.14 X, Y f : X Y Y f 5.15 X, Y X A, B A B = X A B (B A) = (A B) B A = f : X Y f A : A Y f B : B Y f

42 X n = Y n = ( {(x, y) R 2 x 1 ) 2 + y 2 = 1 n n 2 {(x, y) R 2 ( x 1 1 n) 2 + y 2 = }, X = ( n n=1 ) 2 } X n, Y = f : Y X f(y n ) = X n f f Z n = {(x, y) R 2 (x n) 2 + y 2 = n 2 }, Z = g : Z Y g(z n ) = Y n g g [W, d] h : Z W n h Z n n=1 Z n n=1 Y n

43 ( ) (X, O X ), (Y, O Y ) X Y B X Y = {U V U O X, V O Y } O X Y O X Y (X Y, O X Y ) B X Y B X Y (U 1 V 1 ) (U k V k ) = (U 1 U k ) (V 1 V k ) B X Y 6.1 R 2 R R R 2 ε N ε (p) δ S δ (p) = {(x, y) x x 0 < δ, y y 0 < δ} p = (x 0, y 0 ) N ε (p) S ε (p) N 2 ε (p) 2 28 (X, O X ), (Y, O Y ) p 1 : X Y X, p 2 : X Y Y p 1 (p, q) = p, p 2 (p, q) = q O X Y (i) p 1 : X Y X, p 2 : X Y Y (ii) O X Y p 1 : X Y X, p 2 : X Y Y (iii) (Z, O Z ) g : Z X Y g p 1 g, p 2 g (i) U X p 1 1 (U) = U Y O X Y V Y p 1 2 (V ) = X V O X Y (ii) O X Y p 1 : X Y X, p 2 : X Y Y p 1 1 (U) = U Y O

44 16 44 p 1 2 (V ) = X V O U V O B X Y O O X Y O (iii) p 1 g (p 1 g) 1 (U) = g 1 (U Y ) O Z p 2 g (p 2 g) 1 (V ) = g 1 (X V ) O Z g 1 (U Y ) g 1 (X V ) = g 1 (U V ) O Z X Y B X Y g Z X Y g Z 29 X, Y {N X (p)}, {N Y (q)} (p, q) X Y N X Y (p, q) = {N X N Y X Y N X N X (p), N Y N Y (q)} {N X Y (p, q)} X Y (p, q) X Y N X Y (p, q) N X N Y (p, q) W X Y (p, q) U X, V Y (p, q) U V W p U, q V N X N X (p), N Y N X (p) N X U, N Y V N X N Y W 30 (i) f : X 1 X 2, g : Y 1 Y 2 (f g)(p, q) = (f(p), g(q)) f g : X 1 Y 1 X 2 Y 2 (ii) (p) = (p, p) : X X X p 1, p 2 p 1 (f g)(p, q) = f(p) = f p 1 (p, q), p 1 (p) = p p 2 g 1 : Z X, g 2 : Z Y g : Z X Y g(x) = (g 1 (p), g 2 (x)) g g 1 = p 1 g, g 2 = p 2 g 31 () A X, B Y A B 2 X A Y O A O B X Y O A B 2 (A B, O A O B ) (A B, O A B ) 1 A B

45 A B : (A B, O A O B ) (A B, O A B ) 20 X Y (A B, O A O B ) X Y 27 1 A B : (A B, O A B ) (A B, O A O B ) p 1, p 2 p 1 : (A B, O A B ) A, p 2 : (A B, O A B ) B p 1 : (A B, O A B ) A X A B i A B X Y p 1 p 1 A i A X i A p 1 = p 1 i A B A B f : X Y (i) X U f(u) Y f (ii) X F f(f ) Y f p 1, p 2 W X Y p p 1 (W ) q Y (p, q) W W U V B X Y (p, q) U V W p U p 1 (W ) p 1 (U V ) = U p 1 (W ) X = Y = R Z R 2 Z = {(x, y) x > 0, y > 0, xy = 1} Z R 2 p 1 (Z) = (0, ) R R 2 R 32 y Y i (y) 1 : X X Y i (y) 1 (x) = (x, y) p 1 2 (y) = X {y} i (y) 1 : X X {y} X Y i (y) 1 p 1 X Y X X {y}

46 S n 1 R R n {0} h h((x 1, x 2,..., x n ), t) = (e t x 1, e t x 2,..., e t x n ) 6.3 T 2 R 3 2 (R + r cos u) cos v (R + r cos u) sin v ((cos u, sin u), (cos v, sin v)) r sin u T 2 S 1 S 1 2 n X 1 X n = (X 1 X n 1 ) X n X i O i X 1 X n B X1 X n = {U 1 U n U i O i, i = 1, 2,... n} {X λ λ Λ} λ Λ X λ { X λ = f : Λ } X λ f(λ) X λ λ Λ λ {X λ λ Λ} Λ f λ f(λ) X λ f(λ) = x λ f (x λ ) X λ = {(x λ ) x λ X λ } λ Λ

47 16 47 p λ : λ Λ X λ X λ p λ ((x λ )) = x λ λ Λ X λ {(X λ, O λ ) λ Λ} p λ : λ Λ X λ X λ λ Λ X λ SB X λ = {p 1 λ (U λ) λ Λ, U λ O λ } U λ1 U λk λ λ 1,...,λ k X λ = p 1 λ 1 (U λ1 ) p 1 λ k (U λk ) Λ Λ λ U λ X λ λ Λ U λ λ Λ X λ 33 (X λ, O λ ) (i) p λ : λ Λ X λ X λ (ii) p λ : λ Λ X λ X λ (iii) (Z, O Z ) g : Z λ Λ X λ g p λ g : Z X λ (i) U λ X λ p 1 λ (U λ) SB X λ (ii) O λ Λ X λ p λ : λ Λ X λ X λ λ p 1 λ (U λ) O SB X λ O O (iii) p λ g (p λ g) 1 (U λ ) = g 1 (p 1 λ (U λ)) O Z λ Λ X λ B X λ g Z λ Λ X λ g Z

48 16 48 Z {g λ : Z X λ } g : Z λ Λ X λ g(x) = (g λ (x)) g g λ = p λ g λ Λ X λ = X λ Λ X λ X Λ X Λ (x λ ) f(λ) = x λ f : Λ X X Λ Λ X X Λ = {f : Λ X} S i = 1, 2, 3,... N X i = {0, 1} = 2 S = X 1 X 2 X 3 = 2 N s = (s 1, s 2, s 3,...) p i (s) = s i X i S = 2 N φ : C 2 N 2 N p i i s i φ p i φ C i 1 2 i C i 2 i 0, 1, 0, 1,... C i i p i φ : C {0, 1} = 2 φ : C 2 N φ 1 : 2 N C S = 2 N s = (s 1, s 2, s 3,...) N i (s) = {S i s } N i (s) φ 1 C i 1 i φ S = 2 N 6.6 R I I = [0, 1] f : I R f(t) t R I (x t ) = (f(t))

49 16 49 R I R I f 1, f 2, f 3,... f R I f V (f) i N f i V (f) t 1,..., t k ε 1 > 0,..., ε k > 0 i N f i (t 1 ) f(t 1 ) < ε 1,..., f i (t k ) f(t k ) < ε k 6.7 N N N s = {n i } S N i (s) = {N N i s } s = {n i } 1 φ(s) = n n n 3 + n 4 + φ N N 1 (1, ) Q φ : N N (1, ) Q X Y 2 f, g : X Y H : X I Y x X { H(x, 0) = f(x) H(x, 1) = g(x) f g : X Y H

50 16 50 t I f t (x) = H(x, t) f = f 0, g = f 1 f t : X Y f g t X Y x X γ x (t) = H(x, t) γ x : I Y Y f(x) g(x) 6.8 S 1 = {(x, y) R 2 x 2 + y 2 = 1} R 2 i c 0 (x, y) = 0 i c 0 : S 1 R 2 H(x, y, t) = ((1 t)x, (1 t)y) 6.9 i : S 1 R 2 (1, 0) c 1 i c 1 : S 1 R 2 R 2 {0} i c 1 : S 1 R 2 {0}

51 X Y (i) f f (ii) f g g f (iii) f g, g h f h (i) F (x, t) = f(x) F f f (ii) F f g G(x, t) = F (x, 1 t) G g f (iii) F, G f g, g h { ( ) F (x, 2t) 0 t 1 2 H(x, t) = ( G(x, 2t 1) 1 2 t 1) H f h X Y { f f : X Y 35 g f g f : X Z g g : Y Z F, G f f, g g H(x, t) = G(F (x, t), t)

52 16 52 { H(x, 0) = G(F (x, 0), 0) = G(f(x), 0) = g(f(x)) = g f(x) H(x, 1) = G(F (x, 1), 1) = G(f (x), 1) = g (f (x)) = g f (x) H G(F (x, t), t) = G((F 1)(x, t, t)) = G((F 1)((1 )(x, t))) H = G (F 1) (1 ) H : X I 1 X I I F 1 Y I G Z S 1 S 1 p ± = (±1, 0) S 1 p : R S 1 p(t) = (cos t, sin t) { p ( π,π) : ( π, π) S 1 {p } p (0,2π) : (0, 2π) S 1 {p + } { s + : S 1 {p } ( π, π) s : S 1 {p + } (0, 2π) f : S 1 S 1 p(s 0 ) = f(p + ) s 0 R p f = f p f : [0, 2π] R f(0) = s 0 [0, 2π] p f R p S 1 f S 1 [0, 2π] f p : [0, 2π] S 1 ε > 0 t s < ε d(f p(t), f p(s)) < 2 [0, 2π] N 0 = t 0, t 1,..., t N 1, t N = 2π t k t k 1 < ε f p([t k 1, t k ]) π { f p([t k 1, t k ]) S 1 {p } f p([t k 1, t k ]) S 1 {p + } f k : [t k 1, t k ] R { s + f p [tk 1,t f k = k ] s f p [tk 1,t k ]

53 16 53 f k p f k = f p [tk 1,t k ] d 0 = s 0 f 1 (0) k = 1, 2,..., N 1 d k = f k+1 (t k ) f k (t k ) d k 2π, 0, 2π t [t k 1, t k ] f(t) = f k (t) + d 0 + d 1 + d d k 1 f : [0, 2π] R f(0) = s 0 p f = f p f : S 1 S 1 p f = f p, p f = f p 2 f, f : [0, 2π] R t 0 [0, 2π] f(t 0 ) = f (t 0 ) f = f t [0, 2π] p( f(t)) = p( f (t)) = f p(t) n Z f(t) f (t) = 2nπ n Z t [0, 2π] t 0 [0, 2π] f(t 0 ) f (t 0 ) = 0 n = 0 f : S 1 S 1 p f = f p f : [0, 2π] R f deg(f) Z deg(f) = f(2π) f(0) 2π 6.10 p w = z p f : S 1 S 1 deg(f) = p 36 S 1 S 1 f g : S 1 S 1 deg(f) = deg(g) f g H : S 1 I S 1 F : [0, 2π] I R p F = H (p 1) [0, 2π] I p 1 S 1 I F R p H S 1 [0, 2π] I H (p 1) : [0, 2π] I S 1 ε > 0 (t t ) 2 + (s s ) 2 < ε d(h(p(t), s), H(p(t ), s )) < 2

54 16 54 [0, 2π] I N 0 = t 0, t 1,..., t N 1, t N = 2π; 0 = s 0, s 1,..., s N 1, s N = 1 (t k t k 1 ) 2 + (s j s j 1 ) 2 < ε H (p 1) ([t k 1, t k ] [s j 1, s j ]) π { H (p 1) ([t k 1, t k ] [s j 1, s j ]) S 1 {p } H (p 1) ([t k 1, t k ] [s j 1, s j ]) S 1 {p + } F k,j : [t k 1, t k ] [s j 1, s j ] R { s + H (p 1) [tk 1,t F k,j = k ] [s j 1,s j] s H (p 1) [tk 1,t k ] [s j 1,s j ] F k,j p F k,j = H (p 1) [tk 1,t k ] [s j 1,s j] k, j = 1, 2,..., N 1 d k,j (s) = F k+1,j (t k, s) F k,j (t k, s) (s [s j 1, s j ]) d k,j [s j 1, s j ] d k,j (s) 2π, 0, 2π d k,j = d k,j (s) s [s j 1, s j ] d k,j(t) = F k,j+1 (t, s j ) F k,j (t, s j ) (t [t k 1, t k ]) d k,j [t k 1, t k ] d k,j (t) 2π, 0, 2π d k,j = d k,j (t) t [t k 1, t k ] d k,j + d k+1,j = F k+1,j (t k, s j ) F k,j (t k, s j ) + F k+1,j+1 (t k, s j ) F k+1,j (t k, s j ) = F k+1,j+1 (t k, s j ) F k,j (t k, s j ) = F k,j+1 (t k, s j ) F k,j (t k, s j ) + F k+1,j+1 (t k, s j ) F k,j+1 (t k, s j ) = d k,j + d k,j+1 (t, s) [t k 1, t k ] [s j 1, s j ] F (t, s) = F k,j (t, s) + d 1,1 + + d k 1,1 + d k,1 + + d k,j 1

55 16 55 F : [0, 2π] I R p F = H (p 1) { f(t) = F (t, 0) { g(t) = F (t, 1) p f(t) = p F (t, 0) = H(p(t), 0) = f(p(t)) = f p(t) p g(t) = p F (t, 1) = H(p(t), 1) = g(p(t)) = g p(t) F (2π, 0) F (0, 0) deg(f) = 2π F (2π, 1) F (0, 1) deg(g) = 2π s [0, 1] p(f (0, s)) = p F (0, s) = H (p 1)(0, s) = H(p(0), s) = H(p(2π), s) = H (p 1)(2π, s) = p F (2π, s) = p(f (2π, s)) n Z F (2π, s) F (0, s) = 2nπ F n Z s [0, 1] n deg(f) = deg(g) X, Y A X B Y (A B) = ( A B) (A B) A X Y x X A(x) Y A(x) = {y Y (x, y) A} V X A(V ) Y A(V ) = A(x) = {y Y x V ; (x, y) A} x V

56 A(x) f(x) 6.14 A X Y x X U X x A(x) = A(U) x U:open 6.15 B X Y x X U X x B(x) = B(U) x U:open A X Y B Y Z B A X Z B A = {(x, z) X Z y Y ; (x, y) A, (y, z) B} 6.16 B A g f(x) 6.17 A X Y B Y Z B A X Z 6.18 X, Y, P p : P X, q : P Y A f : A X, g : A Y h : A P p h = f q h = g P X Y 6.19 {p λ : P X λ λ Λ} A {f λ : A X λ λ Λ} h : A P p λ h = f λ ( λ Λ) P λ Λ X λ X 1 x 0 X c x0 1 X H : X I X H(x 0, t) = x 0 X 6.20 C I Cone(C) Cone(C) = {(tx, 1 t) x C, 0 t 1}

57 16 57 n Z y n τ y n X = n Z τ y n (Cone(C)) (i) X (ii) X y s f s : X X (iii) X

58 ( ) (X, O) X X/ π : X X/ X/ O X/ O X/ = {U X/ π 1 (U) O} X/ X/ (X/, O X/ ) (O X/ ) (O1), (O2), (O3) X A p A, p q q A π X X/ 1 : 1 37 X/ O X/ (i) π : X X/ (ii) π : X X/ (iii) (Z, O Z ) g : X/ Z g g π : X Z (i) O X/ (ii) O X/ π : X X/ U O π 1 (U ) O O X/ (iii) W O Z (g π) 1 (W ) (g π) 1 (W ) = π 1 (g 1 (W )) g 1 (W ) O X/ 7.1 (X, O) R s t s t Z [t] S 1 (cos 2πt, sin 2πt) R/Z S 1 R/Z R R n Z [t] R/Z R n Z (t + n ε, t + n + ε) S 1 (cos 2πt, sin 2πt) {(cos 2πx, sin 2πx) t ε < x < t + ε}

59 π : R R/Z 7.3 (X, O) [ 1, 1] [0, 2π] (u, 0) ( u, 2π), u [ 1, 1] X/ 7.4 X/ X/ φ : X/ R 3 (( φ(u, v) = 3 + u cos v ) ( cos v, 3 + u cos v ) sin v, u sin v ) φ R 3 φ [ 1, 1] [0, 2π] φ π R 3 2 y 0 y 0 φ 1 [ 1, 1] [0, 2π] 7.5 X = S 1 [0, 1] (cos u, sin u, 0) ( cos u, sin u, 0), u X/ X, Y A X, B Y h : A B X Y h X h Y X h Y = X Y/ ; x h(x) ( x A) 38 π i 1 : X X Y X h Y, π i 2 : Y X Y X h Y X, Y X h Y

60 16 60 π i 1 π i 1 X U π i 1 (U) π i 1 (X) X h Y W π i 1 (U) = π i 1 (X) W π 1 W X Y = X {0} Y {1} π 1 W = U {0} V {1} U A A h h(u A) B Y V V B = h(u A) V π i n D n 2 D± n n 1 S± n 1 id : Sn 1 + S n 1 D+ n id D n n Sn 7.7 n inv : R n {0} R n {0} inv(x) = 1 x 2 x R n 2 R n ± inv Rn + inv R n n Sn X A X A p A q p = q p, q A A 1 X/ A X/ A X/A R/Z S 1 R/ Z A X/ A A 7.8 X/ A A {π(u) X U A X } 7.9 [0, 1]/ {0,1} S 1 R/Z 7.10 S 1 D 2 1 D 2 / S 1 2 S X X [0, 1] X {1} 1 X Cone(X) Cone(S 1 ) D 2

61 16 61 P n (R), P n (C) R n+1 0 P n (R) = {L L R n+1 1 } n C n+1 1 P n (C) = {L L C n+1 1 } n 0 x x 1 { p : R n+1 {0} P n (R) p C : C n+1 {0} P n (C) { p(x) = p(x ) x = rx ( r R {0}) p C (x) = p C (x ) x = cx ( c C {0}), C { x x x = rx ( r R {0}) 1 : 1 { x C x x = cx ( c C {0}) P n (R) = (R n+1 {0})/ P n (C) = (C n+1 {0})/ C 0 x = (x 1, x 2,..., x n+1 ) L (x 1 : x 2 : : x n+1 ) L { p(x 1, x 2,..., x n+1 ) = (x 1 : x 2 : : x n+1 ) p C (x 1, x 2,..., x n+1 ) = (x 1 : x 2 : : x n+1 ) (R n+1 {0})/, (C n+1 {0})/ C P n (R), P n (C) 7.12 P 1 (R) S 1 P 1 (R) S 1 p : R 2 {0} P 1 (R) (0 θ π) 1 : 1 2 : 1 P 1 (R) [0, π] S 1

62 P 2 (R) p : R 3 {0} P 2 (R) S 2 p S 2 : S 2 P 2 (R) 2 : 1 p(x) = p(y) y = ±x S 2 45 A + 45 A A E B P 2 (R) B = p S 2(A + ) = p S 2(A ) A ± p S 2 B B A + A C P 2 (R) C = p S 2(A E ) P 2 (R) = B C p S 2 A E 2 : 1 A E+ A E C = p S 2(A E+ ) A E+ p S C A E+ 2 C P 2 (R)

63 () P 1 (C) S 2 P 1 (C) S 2 C 2 C z C w φ : C z P 1 (C) ψ : C w P 1 (C) φ(z) = (z : 1), ψ(w) = (1 : w) P 1 (C) P 1 (C) = φ(c z ) ψ(c w ) φ(z) = ψ(w) (z : 1) = (1 : w) z 0, w 0 w = 1 z P 1 (C) C 2 C z C w z w w = 1 z P 1 (C) = (C z C w )/ C z R 2 C z z = x + i y R 2 (x, y) 5.7 φ 1 + S2

64 16 64 φ 1 + C z = R 2 S 2 {(0, 0, 1)} 2x 1 x + i y (x, y) x 2 + y 2 2y + 1 x 2 + y 2 1 C z S 2 {(0, 0, 1)} C z P 1 (C) S 2 {(0, 0, 1)} f P 1 (C) C z 1 (0 : 1) f(0 : 1) = (0, 0, 1) 1 : 1 f : P 1 (C) S 2 f(1 : 0) = (0, 0, 1) f C w S 2 {(0, 0, 1)} 1 : 1 C w {0} φ : S 2 {(0, 0, 1)} R 2 C w {0} = C z {0} = R 2 {0} φ 1 + S 2 {(0, 0, ±1)} φ R 2 {0} w = u + i v z = 1 w = u i v u 2 + v ( 2 ) u u 2 + v 2, v u 2 + v 2 φ 1 + (u 2 + v 2 ) 2 u 2 + v 2 + (u 2 + v 2 ) 2 = 1 u 2 + v 2 + (u 2 + v 2 ) 2 φ 1 2(u 2 + v 2 ) = (u, v) 2u u 2 +v 2 2v u 2 +v 2 u 2 +v 2 (u 2 +v 2 ) 2 (u 2 +v 2 ) 2 2u(u 2 + v 2 ) 2v(u 2 + v 2 ) u 2 + v 2 (u 2 + v 2 ) 2 ( 2u(u 2 + v 2 ), 2v(u 2 + v 2 ) ) C w R 2 f C w S 2 {(0, 0, 1)} 7.15 p C : C 2 {0} P 1 (C) C 2 {0} = R 4 {0} R 4 {0} S 2 3 S 3 Hopf : S 3 R 4 {0} p C S 2

65 16 65 x x x x 2 4 = 1 2(x 1 x 3 x 2 x 4 ) Hopf(x 1, x 2, x 3, x 4 ) = 2(x 1 x 4 + x 2 x 3 ) x x 2 2 x 3 2 x 4 2 S 3 S P 3 (R) SO(3) p : R 4 {0} P 3 (R) S 3 p S 3 : S 3 P 3 (R) : 1 p(x) = p(y) y = ±x SO(3) S ( 3 2) SU(2) 1 : 1 SU(2) α β γ δ αᾱ + γ γ = 1 β + δ δ = 1 α β + γ δ = 0 = 1 αδ βγ = 1 δ 3 β α(β β + δ δ) = δ δ = ᾱ β = γ SU(2) ( α γ γ ᾱ ), αᾱ + γ γ = 1 (x, y, u, v) S 3 α = x + i y, γ = u + i v S 3 ( SU(2) (x, y, u, v) x + i y u + i v u + i v x i y ) SU(2) C 2 P 1 (C) (z : w) α β γ δ (αz + βw : γz + δw)

66 16 66 P 1 (C) S 2 S 2 S 2 P 1 (C) S 2 (x + i y : 1) 1 x 2 + y x 2 + y 2 1 ( ) e i θ 0 A(θ) = 0 e i θ xy 2θ R(2θ) P 1 (C) (x + i y : 1) A(θ) (e i θ (x + i y) : e i θ ) = (e 2i θ (x + i y) : 1) 2x 2y = (x cos 2θ y sin 2θ + i (x sin 2θ + y cos 2θ) : 1) S 2 xy 2θ R(2θ) 2x 2x cos 2θ 2y sin 2θ 1 x 2 + y 2 2y R(2θ) x 2 + y 2 2x sin 2θ + 2y cos 2θ + 1 x 2 + y 2 1 x 2 + y 2 1 ( S 2 ) B = P 1 (C) ( ) ( (x + i y : 1) B x + i y 1 x 2 x + i y + 1 : 1 + y 2 ) 1 + i 2y = x 2 + y 2 + 2x + 1 : 1 S 2 x 2 + y 2 = r 2 ( r 2 ) 1 + i 2y r 2 + 2x + 1 : 1 1 = (r 2 1) 2 +4y 2 +(r 2 +2x+1) 2 (r 2 +2x+1) 2 1 2r r 2 +4x(r 2 +1) (r 2 +2x+1) 2 = 1 r r x 2 + y y 2x = 2x 2y 2r 2 2 r 2 +2x+1 4y r 2 +2x+1 (r 2 1) 2 +4y 2 (r 2 +2x+1) 2 (r 2 +2x+1) 2 2r 2 2 r 2 +2x+1 4y r 2 +2x+1 8x 2 4x(r 2 +1) (r 2 +2x+1) 2 1 x 2 + y x 2 + y 2 1 2y 2x xz x 2 + y 2 1 π 2 B S2 xz

67 16 67 π A(θ) B 2 ( ) BA(θ)B 1 cos θ i sin θ = i sin θ cos θ ( ) A(φ)BA(θ)B 1 e i(φ+η) cos θ e i(φ η) i sin θ A(η) = e i(φ η) i sin θ e i(φ+η) cos θ SU(2) S 2 A A A A SO(3) SU(2)/{±1} P 3 (R) 7.17 X X Y X Y Y Y/ Y X/ i X = [ 1, 2] x x ( 1 x 1) 1 < t 1 Y = [ 1, t) (1, 2] i : Y/ Y X/ 7.18 B X/ Y = π 1 (B) i : Y/ Y X/ 7.19 π : X X/ B X/ Y = π 1 (B) i : Y/ Y X/ 7.20 X A r : X A a A r(a) = a r X r x y r(x) = r(y)r X/ r A 7.21 (i) A X X A X/ A π(a) (ii) A X X A X/ A π(a) X = R A = Q

68 λ Λ f λ : X λ Y λ λ f λ : λ X λ λ Y λ ( λ X) / λ f λ λ Y λ , 2 X x 1 y x 2 y 2 X/ 1 (X/ 1 )/ 2 X/ R 2 H 2 = {(x, y) y < 0} R 2 2 R 2 1, R 2 2 H2 1, H 2 2 h : H2 1 H 2 2 h(x, y) = (x 1y ), y (y < 0) R 2 1 h R 2 2 R h id : H 2 1 H 2 2 R 2 1 id R 2 2 R π : R R/ Z π 1 : R Q R/ Z Q R Q π 1 π 1 R Q/ π 1 R/ Z Q

69 X (i) X T 1 2 p, q X p U, q / U U X (ii) X Hausdorff 2 p, q X p U, q V, U V = U, V X (iii) X X T 1 p X p F X p U, F V, U V = U, V X (iv) X X T 1 2 F 1, F 2 X F 1 U, F 2 V, U V = U, V X

70 T T 1 X T 1 p X;1 {p} X T 1 X {p} q X {p} q U U X {p} X T 1 p X; N N (p) N = {p} X T 1 p q p U q q / N N (p) N q p q / N N (p) N N N (p) N = {p} p q q / N N (p) N p N q / N p U N q / U T 1 X A d p A d p A d p / A d p U U U (A {p}) = U p A d q U A d q p U {p} q q A d (U {p}) (A {q}) U (A {p}) = 8.1 X 2 X T X p X; N N (p) N = {p} X p q U, V p U X V N = X V p q / N = N q / N q / N N (p) N q p q / N N (p) N N N (p) N = {p}

71 16 71 p q q / N N (p) N p N q / N p U N V = X N q V U V = T 1 X = {(p, p) p X} X X p q p U, q V, U V = U, V U V = X X X X X A r : X A p A r(p) = p X A X A p / A r(p) A p r(p) p U, r(p) V, U V = U, V r p U r(u ) V U U p X A q (U U ) A r(q) = q q U U, r(q) V 8.2 X O T X = R {0, 1} (t, 0) (t, 1) (t < 0) X = X/ T 1

72 T 1 X 4 (a) X (b) p X N N (p) p V V N (c) p X p F X p U, F V, U V = U, V X (d) X F F = U F U, U X (a) (b) N p p U N U F = X U p V, F W, V W = V, W V X W V X W V W = V F = V U N (b) (c) X F p p W W X F W W p U U W U V = X W V X W V W = V U = X V = W X F F V (c) (d) F U U p / F p / p U, F V, U V = U, V X p / V (d) (a) p / F p / U F U, p / U p / U p V U V = 41 X, Y X Y (p, q) X Y N U X, V Y p U, q V, U V N X, Y U 0 X, V 0 Y p U 0, U 0 U, q V 0, V 0 V U 0 V 0 (p, q) U 0 V 0 U 0 V 0 U V N (b) X Y

73 R 0 { { } } 1 N (0) = ( ε, ε) n n = 1, 2,... ε > 0 { O X } = (R, O ) 1 n n = 1, 2, (X, d) F 1, F 2 F 1 p ε p N(p, ε p ) F 2 = U = ( p, ε ) p 2 p F 1 N F 2 q δ q N(q, δ q ) F 1 = V = ( q, δ ) q U, V F 1 U, F 2 V 2 q F 2 N U V = U V p F 1, q F 2 N ( p, ε ) ( ) p 2 N q, δ q 2 d(p, q) < ε p 2 + δ q 2 d(p, q) < ε p d(p, q) < δ q 1 X F X U X F U V X F V, V U F 1 = F, F 2 = X U F 1, F 2 V, W X F 1 V, F 2 W, V W = F V V X W X F 2 = U 2 n 2 2 X F 0 X U 1 X F 0 U 1 0 < q < 1 2 q U q X F 0 U q, U q U 1 q < q U q U q

74 q d n 2n U q X n 2 q U q q = d d ± 1 2n U, U d 1 2 n d+1 2 n U U d 1 2 n d+1 2 n 1 U q 3 2 f : X R { 1, p U q f(p) = inf{q q 2 p U q } f [0, 1] f(f 0 ) = {0}, f(x U 1 ) = {1} f p U q f(p) q f(p) < q p U q q < q p U q U q q f(p) q f(p) = 0 q > 0 p U q ε > 0 q > 0 q < ε p U q f(p ) q < ε f(p) = 1 q < q < 1 p / U q p / U q ε > 0 1 q < q < 1 1 ε < q < q p X U q p / U q f(p ) q > 1 ε 0 < f(p) < 1 0 < q < q < f(p) < q < 1 p U q U q ε > 0 f(p) ε < q < f(p) < q < f(p) + ε q, q p U q U q f(p) ε < q f(p ) q < f(p) + ε f F 0 = A, U 1 = X B 43 ( (Uryson) ) X A, B X f : X [0, 1] f(a) = {0}, f(b) = {1}

75 ( (Tietze) ) X A X A f : A R X F : X R f(a) [ 1, 1] [ 1 1] 3 { } { A 0 = p A f(p) 1, B 0 = p A 3 f(p) 1 } 3 A 0, B 0 A X [ F 0 : X 1 3, 1 ] { 3 F 0 (A 0 ) = 1 } { } 1, F 0 (B 0 ) = 3 3 A f(p) F 0 (p) 2 [ 3 2 ] A 1 = { p A f(p) F 0(p) 2 9 } {, B 1 = p A f(p) F 0(p) 2 } 9 A 1, B 1 [ X F 1 : X 2 9, 2 ] { F 1 (A 1 ) = 2 }, F 1 (B 1 ) = { } A f(p) F 0 (p) 9 F 1 (p) 4 9 [ F n+1 : X 1 ( ) n+1 2, 1 ( ) ] n [ ( ) n+1 ( ) ] n f F 0 F 1 F n ( ) n ( 2 ) n ( ) n ( 2 3 ) n F n+1 ( ) n+2 2 A f(p) F 0 (p) F n+1 (p) 3 n F k F = F n A f k=0 n=0 F f(a) [ 1, 1] g(p) = 2 arctan f(p) π g : A ( 1, 1) G : X [ 1, 1] B = G 1 (±1)

76 16 76 A, B X A 1 B 0 h : X [0, 1] F (p) = tan π h(p) G(p) 2 X 45 ( ) 2 2 X 2 B = {B n } F 1, F 2 F 1 p F 2 p B n, B n F 2 = B n B B {U n } F 1 U n F 1, F 2 {V n } F 2 V n n n U n = U n (V 1 V n ), V n = V n (U 1 U n ), U = n U n, V = n V n F 1 U, F 2 V, U V = B (B n, B m ) B n B m {(C n, D n )} n f n : X [0, 1] C n 0 X D n 1 d : X X R d(p, q) = n 1 2 n f n(p) f n (q) X X d X p q d(p, q) > 0 p N 1 N 2 N 2 N 1 p C n, q / D n n X O O d p q d(p, q) X ε O d O W O, p W p C n, D n W n p 1 ( ) ( ) 2 n N 1 1 p, 2 n q N p, 2 n d(p, q) < 1 2 n f n (p) = 0 f n (q) < 1 q D n W W O O d

77 ( ) (i) R Sor (ii) R Sor R Sor R Sor R Sor R Sor 2 (i) F 1, F 2 R Sor F 1 t ε t > 0 [t, ε t ) F 2 = U = t F 1 [t, ε t ) F 1 U U 1, 2 s F 2 [s, δ s ) F 1 = V = s F 2 [s, δ s ) t F 1, s F 2 [t, ε t ) [s, δ s ) = U V = (ii) F = {(t, t) t R} R Sor R Sor {(t, t)} = F [t, t +ε) [ t, t+ε) 1 {(t, t)} F F R Sor R Sor : R F (t) = (t, t) R F F 1 = (Q), F 2 = (R Q) R Sor R Sor U, V F 1 U, F 2 V U V t R ε t > 0 { t Q [t, t + ε t ) [ t, t + ε t ) U t R Q [t, t + ε t ) [ t, t + ε t ) V R Q { 2 S n = t R Q ε t > 1 } n n S n = R Q S n (α, β) S n n (α, β) ε t > 1 t (α, β) n q [q, q + ε q ) [ q, q + ε q ) U q t S n [q, q +ε q ) [ q, q +ε q ) [t, t+ε t ) [ t, t+ε t ) U V 8.6 T X T 1 (a) (b) (a) U = {p} p U, U X; (b) F = {p} p F, F X;

78 T 1 X (A d ) d A d 8.9 T 1 X ( A ) d = A d 8.10 X/ [x] X 8.11 n {x 1, x 2,..., x n } N(x 1 ), N(x 2 ),..., N(x n ) 8.12 f : X Y, b : Y X g f = 1 X Y X f(x) Y T X D X f : X Y f D : D Y f(x D) Y f(d) 8.15 X, Y D X, E Y h : D E f : X Y h 1 : E D g : Y X f, g g = f X Cone(X) 8.17 X f : X X {x f(x) = x} X X Cone(X) 8.20 X x V V X 8.21 X X = R x n N { U n (x) = {x} y Q x y < 1 } n {U n (x) n N} x X X 2

79 X A X A = {U X U A } 8.24 (R. H. Bing) θ > 0 1 X = {(x, y) Q Q y 0} (x, y) X ε > 0 B ε (x, y) = {(x, y)} {(z, 0) z (x + θy) < ε z (x θy) < ε} (i) {B ε (x, y) ε > 0} (x, y) X (ii) X (iii) B ε1 (x 1, y 1 ) B ε2 (x 2, y 2 ) X 8.25 (M. Brown) X = N a, b U(a, b) = {an + b N n Z} (i) {U(a, b)} X (ii) p {kp k N} (iii) P (iv) X (v) k ka U(c, d) U(a, b) U(c, d) (vi) X 1 2N 8.26 X π : X X/ X/ 8.27 X {(x, y) R 2 y }0 y > 0 (x, 0) {(x, 0)} (x, 0) x X 8.28 X I = [0, 1] O 1 O P(R Q) O 1 = O P(R Q) X

80 ( ) X X A 9.1 ( ) [a, b] 9.2 R {( n, n) n = 1, 2, 3,...} R 9.3 R n A A A X A = {A λ } A λ1 A λ2 A λn 46 X X F = {F λ } Fλ = U = {X F λ } X X = (X F λ1 ) (X F λ2 ) (X F λn ) = X (F λ1 F λ2 F λn ) F λ1 F λ2 F λn X U = {U λ } F λ = X U λ F = {F λ } F λ1 F λ2 F λn = X (U λ1 U λ2 U λn ) F U X 9.4 Cantor 47 X F X U = {U λ } F F U λ U 0 = X F X U 0 = U {U 0 } X X X = U 0 U λ1 U λ2 U λn F U λ1 U λ2 U λn 48 X Y f : X Y f(x) Y

81 16 81 Y U = {U λ } f(x) f(x) U λ f f 1 (U λ ) X f 1 (U) = {f 1 (U λ )} X X X = f 1 (U λ1 ) f 1 (U λ2 ) f 1 (U λn ) f(x) U λ1 U λ2 U λn 49 X f : X R f(x) R t 0 f(x) t 1 f(x) t 0 = f(x 0 ) f t 1 = f(x 1 ) f 50 X K p / K p / K q K p q U q, V q p U q, q V q, U q V q = q K V = {V q } K K V q1 V q2 V qn U = U q1 U q2 U qn p V qi K p / K V = V q1 V q2 V qn p U, K V, U V = 1 X K 51 X F 1, F 2 X p F 1 p / F 2 U p, V p p U p, F 2 V p, U p V p = p F 1 U = {U p } F 1 F 1 U p1 U p2 U pn U = U p1 U p2 U pn F 1 U V = V p1 V p2 V pn F 2 U pi U F 1 U, F 2 V, U V = 52

82 16 82 X Y f : X Y F X F f(f ) Y f(f ) 53 X, Y X Y W = {W λ } X Y (p, q) X Y (p, q) W λ W λ U (p,q) X, V (p,q) Y (p, q) U (p,q) V (p,q) W λ p q Y V p = {V (p,q) } Y Y = V (p,q1) V (p,q2) V (p,qn ) U p = U (p,q1 ) U (p,q2 ) U (p,qn ) U p Y = U p ( ) V (p,q1) V (p,q2) V (p,qn ) = U p V (p,q1 ) U p V (p,q2 ) U p V (p,qn ) U (p,q1 ) V (p,q1 ) U (p,q2 ) V (p,q2 ) U (p,qn ) V (p,qn ) W λ1 W λ2 W λn U (p,qi ) V (p,qi ) W λi U p Y W p X p U p U = {U p } X X = U p1 U p2 U pn X Y = (U p1 U p2 U pn ) Y = U p1 Y U p2 Y U pn Y U pi Y W 9.5 X {C λ λ Λ} λ C λ U λ C λ C λ0 λ 1,..., λ N Λ C λ0 C λ1 C λn U 9.6 I 2 (0, y) (1, y) I 2 / S 1 I

83 I 2 (0, y) (1, 1 y) I 2 / 9.8 I 2 I 2 (0, 0) I 2 / S I 2 (0, y) (1, y), (x, 0) (x, 1) I 2 / S 1 S X Y f : X Y X x y f(x) = f(y) X/ Y 9.11 X, Y, Z f : X Y Z A X, B Y W Z f(a B) W U X, V Y A U, B V, f(u V ) W 9.12 X, Y X f : X Y R y Y M(y) = sup{f(x, y) x X} M(y) Y , 8.25 X 9.14 X {f n } X x n f n (x) f n+1 (x) {f n } g 9.15 X, Y f : X Y {F n } X F n F n+1 f ( n=1 F n) = n=1 f(f n) 9.16 X f : X X A X f(a) = A 9.17 π : X X/ Y π 1 : X Y X/ Y X Y π 1 π 1 X Y/ π 1 X/ Y 9.18 X = [ 1, 1] x x, ( x < 1) Y = X/ A = π([0, 1]), B = π([ 1, 0]) A Y, B Y A B Y 1 π( 1) / A A 9.19 X = [0, 1] Z (x, n) (x, m), (0 x < 1) Y = X/ A = π([0, 1] {0}) A Y A = Y

84 ( (Tikhonov) ) {X λ λ Λ} λ Λ X λ () X X (Ord1) x x ( ) (Ord2) x y y x x = y (Ord3) x y y z x z () ( ) X [X, ] (Ord4) x y y x X A b X A x A x b ( ) [X, ] [X, ] Y X [Y, ] Y a Y X 9.20 Z + Z Q R (, b] 9.21 X Y = P(X) Y y, y Y y y y y [Y, ] 9.22 [Y, ] Z Y X [Z, ]

85 16 85 [X, ] X X ( (Zorn) ) ( ) F Z = λ Λ X λ F Z F = {F } F F F F Z (i) F F Z (ii) F F 1, F 2,..., F n (iii) F Z F F {F α } 2 F α, F β F α F β F β F α α F α F α1 F α1, F α2 F α2,..., F αn F αn i F α1, F α2,..., F αn F αi F F max F max (i),(ii),(iii) F max λ Λ p λ : Z = λ Λ X λ X λ p λ F max {p λ (F ) F F max } X λ X λ {p λ (F ) F F max } q λ F F max p λ (F ) q λ U λ F F max U λ p λ (F ) p 1 λ (U λ) F

86 16 86 p 1 λ (U λ) F F max p 1 λ (U λ) F max q λ q = (q λ ) λ Λ X λ = Z p λ (q) = q λ ( λ Λ) q p 1 λ 1 (U λ1 ) p 1 λ 2 (U λ2 ) p 1 λ n (U λn ) F F max F max p 1 λ 1 (U λ1 ) p 1 λ 2 (U λ2 ) p 1 λ n (U λn ) F q F F q F F F F q F F F F {X λ λ Λ} X λ λ Λ x λ X λ (x λ ) λ Λ X λ X λ ( λ Λ) λ Λ X λ () {X λ λ Λ} φ : Λ X λ λ Λ λ Λ φ(λ) X λ λ Λ x λ X λ φ(λ) = x λ

87 16 87 Z Z, Q, R { 9.24 (i) n 1 } m n, m Z+ (ii) n 1 m n, m, l Z + l [X, ] α = [X, ] 2 [X, ] [Y, ] X Y (x, y) (x, y ) y < y { y = y x x α = [X, ], β = [Y, ] X Y α β 2 [X, ], [Y, ] X, Y X Y X {0} Y {1} (X Y ) {0, 1} (x, ε) (x, ε ) ε < ε { ε = ε x x α = [X, ], β = [Y, ] X Y

88 16 88 α + β 2 [X, ] [Y, ] f : X Y f x y f(x) f(y) f [X, ] [Y, ] 0 = [, ] 1 = [{0}, ] 2 = [{0, 1}, ] n = [{0, 1, 2,..., n 1}, ] ω = [Z +, ] ω + 1, ω + 2,..., ω + n,... ω + ω = ω 2, ω 2 + 1,..., ω 2 + ω = ω 3, ω 3 + 1,... ω 4,..., ω 5,..., ω ω,... ω ω ω,..., ω ω ω ω,... ωωωω.... ω ω, ω ωω,..., ω,... ε 2 α, β α β α = [X, ], β = [Y, ] X Y = Y X = {f : Y X}

89 16 89 X Y X x 0 X0 Y = {f X Y f(y) x 0 y Y } f g f(y) g(y) y f(y) < g(y) α = [X, ] a a a 1 = min{x X x > a} a a X a = {x X x < a} α a = [X a, ] α a a α [X, ] α a f : X X a A = {x X f(x) < x} f(a) X a f(a) < aa A A a α = [X, ] β = [Y, ] 2 f : X Y, g : X Y f = g f g X 1 = {x X f(x) g(x)} x 0 f(x 0 ) < g(x 0 ) f(x 0 ) > g(x 0 ) f(x 0 ) < g(x 0 ) g g 1 f(x 0 ) < x 0 x 0 X 1 f g 1 f(x 0 ) = g g 1 f(x 0 ) = f(x 0 ) f g 1 f(x 0 ) = x 0 f(x 0 ) = g(x 0 ) f(x 0 ) > g(x 0 )

90 α = [X, ] β = [Y, ] α α a β b X 1 = {a X b Y α a = β b } Y 1 = {b Y a X α a = β b } b b β b β b a X 1 α a = β b b 1 b Y 1 a X 1 b Y 1 f : X 1 Y 1 X 1 X, Y 1 Y α β X X 1, Y Y 1 X X 1 a 0 a X 1 a < a a X 1 a 0 X 1 [X 1, ] = α a 0 Y Y 1 b 0 [Y 1, ] = β b 0 α a 0 = β b 0 a 0 / X 1 X X 1 Y 1 = Y β α a 0 Y Y 1 α = β b 0 α β α < β α β α = β α < β α = β α β α = [X, ], β = [Y, ]Y X β α X [X, ] R ()

91 16 91 X X a 0 a 1 (i) X X φ : P(X) { } X X W φ(w ) W (ii) X W W [W, W ] φ x W φ(x W x ) = x 1 (i) a 0, a 1, a 2,... a 0 = φ(x) a 1 = φ(x {a 0 }) a 2 = φ(x {a 0, a 1 })... a 0 < a 1 < a 2 < [{a 0, a 1, a 2,...}, ] φ (ii) φ a 0 = φ(x) 2 2 φ α = [W, W ], β = [V, V ] α β β α α β f : W V y V f : W V y f : W V y V f : W V W W W = {x W f(x) x} W = W [W, W ] x...

92 16 92 X φ X φ X X φ 3 X φ W φ W 2 1 φ W φ... 4 W = X W X a = φ(x W ) W +1 = W {a } a W +1 W +1 φ... X (i) Y Y X X Y Y (ii) X φ : P(X) { } X X 2 x X x x {y X y > x} g(x) = φ({y X y > x}) g X X g(x) > x

93 16 93 (iii) X A {y X a A y a} A h(a) = φ({y X a A y a}) h X X h(a) A A h(a) A A = h( ) h( ) = φ(x) (iv) Y y y Y y y 0 y y 0 y y y 0 ω + 1 (v) f : Y X Y Y y f y : Y y X { a Y b f y (a) = g(f y (b)) ( ) a Y f y (a) = h(f y (Y a )) 1 Y = ω + 1 = {0, 1, 2,..., ω} x 0 = φ(x) f ω (0) = x 0 f ω (1) = g(x 0 ) f ω (2) = g(g(x 0 )) f ω ( ) 2 y < y Y ( ) f y, f y Y y f y f y 3 a Y a + f a f a+ f a+ ( ) { a Y a f a+ (a) = g(f a (a )) a Y f a+ (a) = h(f a (Y a )) 4 a Y b < a b f b f a f a ( ) f a (b) = f b+ (b)

94 a Y ( ) f a y f y Y y max Y + 1 Y = (Y + 1) y max f : Y X

95 () X X 1, X 2 X 1, X 2 X X = X 1 X 2, X 1 X 2 = X 1, X 2 X X X A 10.1 R {t} 55 X 3 (a) X (b) X X (c) {0, 1} 2 X {0, 1} (a) (b) X U V = X U X = U V (b) (c) f X {0, 1} f 1 (0) X {0} {0, 1} f 1 (0) X (c) (a) X X = X 1 X 2 f : X {0, 1} f(x 1 ) = {0}, f(x 2 ) = {1} X {0, 1} 10.2 R R 56 X Y f : X Y f(x) Y f(x) f(x) {0, 1} g g f : X f(x) {0, 1} ( ) X f : X R 2 p 0, p 1 X f(p 0 ) < c < f(p 1 ) f(q) = c q X f(x) R

96 X A X B X A B A B f B {0, 1} B f A : A {0, 1} f(a) = {0} p B f(p) = 1 B p U B f(u B ) = {1} X p U U B = U B p A U A q U A q U B f(q) = 1 f(a) = {0} {( 10.3 R 2 A = t, sin 1 ) } t > 0 A = A t {(0, s) 1 s 1} {(0, s) 1 s 1} B A B 10.4 {( A n = t, sin 1 ) 0 < t < 1 } B 0 = {(0, ±1)} A n B 0 t n n (A n B 0 ) = B 0 58 X K 1 K 2 K 0 = n K n K 0 K 0 = K 0 K 0 K 1 X U, V K 0 U, K 0 V, U V K 1 = K 0 = n K n U V N K N U V W n = X K n n W n = X n K n = X K 0 U, V K 1 W 1 W 2 K 1 U V W N K N U V K N U V K N = U K N = V K N = K 0 = U K 0 = K 0 = V K 0 = 10.5 C 1 C 2 C 3 i C i 59 X {A λ } X λ A λ λ A λ

97 16 97 f λ A λ {0, 1} λ A λ p λ A λ f(p) = 0 f A λ f(p) ( ) = 0 f(a λ ) = {0} f A λ = {0} λ λ 60 2 X, Y q 0 Y X {q 0 } X Y p X {p} Y X Y 2 (p, q 0 ) A p = X {q 0 } {p} Y A p = X {q 0 } = {A p } p X A p = X Y p X 10.6 R n R n S n {(0,..., 0, ±1)} n 1 (S n {(0,..., 0, 1)}) (S n {(0,..., 0, 1)}) S n R n+1 {0} S n R R 1 {0} R n 2 R n {X λ λ Λ} λ Λ X λ λ Λ p λ : λ Λ X λ X λ X λ q λ X λ 1 q = (q λ ) λ Λ X λ Λ Λ i (q) Λ : λ Λ X λ λ Λ X λ (x λ ) λ Λ X λ i (q) Λ ((x λ)) = (y λ ), y λ = { x λ (λ Λ ) q λ (λ Λ Λ ) i (q) Λ λ Λ X λ X (q) Λ = ( ) i(q) (q) Λ λ Λ X λ X Λ λ Λ X λ q X (q) Λ X (q) Λ = λ Λ Λ p 1 λ (q λ) = { (x λ ) } X λ x λ = q λ, λ Λ Λ λ Λ

98 16 98 Λ Λ X (q) Λ X (q) X (q) = Λ Λ X(q) Λ Λ Λ X (q) Λ λ Λ X λ λ Λ X λ = {q} = X (q) = λ Λ X λ (x λ ) λ Λ X λ N Λ = {λ 1, λ 2,..., λ N } x λ1 U λ1, x λ2 U λ2,..., x λn U λn (x λ ) p 1 λ 1 (U λ1 ) p 1 λ 2 (U λ2 ) p 1 λ N (U λn ) N i (q) Λ ((x λ 1, x λ2,..., x λn )) ( p 1 λ 1 (U λ1 ) p 1 λ N (U λn ) ) X (q) Λ N X (q) 62 X (i) p X p C(p) (ii) C(p) X (iii) p, q X C(p) C(q) = C(p) = C(q) (i) p X C(p) C(p) (ii) C(p) p (iii) C(p) C(q) C(p) C(q) p C(p) C(q) C(p) q C(p) C(q) C(q) C(p) = C(q) X C(p) X p X C(p) = {p} X { } X = R n n = 1, 2, 3,... {( 1 (, 0], n + 1, 1 ) n = 1, 2, 3,...}, (1, ) n 10.8 C Q, R Q 10.9 R Sor

99 X R 2 X = {(± π ) } 2, y y R ( π ) 2, y { (x, n + sec x) {( π ) } 2, y y R π 2 < x < π 2, n Z } {(± π 2, y ) y R } 63 X 1 x {Z λ } Z 0 = λ Z λ Z 0 Z 0 Z 0 2 Z 0 = Z 0 Z 0, x Z 0 Z 0, Z 0 X X U, V Z 0 U, Z 0 V, U V = Z 0 U V W λ = X Z λ λ W λ = X Z 0 U, V X X = U V W λ1 W λ2 W λn Z λ1 Z λ2 Z λn U V Z 0 = Z λ1 Z λ2 Z λn U, Z 0 = Z λ1 Z λ2 Z λn V x Z 0 U, Z 0 V Z 0 Z λ Z 0 U Z 0 U = Z i Z A i X A i A i+1 i A i A X C X A X A C A X X U U U α, U β U U α0, U α1,..., U αn α = α 0, β = α N, U αi 1 U αi, (i = 1, 2,..., N)

100 U R 2 {(a n, b n )} U = n (a n, b n ) X R 2 { X = (x, y) x = ± π } 2 n= { (x, y) π 2 < x < π 2, y = 1 } cos x + n ( π ) 2, 0 X n N R n R 2 { R n = (x, y) x n } n + 1, y n X R 2 X = {(x, y) x = ±1} R n (1, 0) X S n I S R [0, ) n= n 2 R n R n 2 S n S n 2 (i) {x R n x } (ii) {x R n x } R n R R A X X A B X A A B A X X A C X A X C

101 A X, B X A B A B A B A X, B X A B A B X, Y A X, B Y X Y A B R n (R. H. Bing ) 8.24 X (M. Brown ) 8.25 X