°ÌÁê¿ô³ØII - PDF 無料ダウンロード

July 14, 2007

Brouwer f f(x) = x x

f(z) = 0

2 f : S 2 R 2 f(x) = f( x) x S 2

3 3 2

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1. X x X U(x) U(x) x U = {U(x) x X} X 1. U(x) A U(x) x 2. A U(x), A B B U(x) 3. A, B U(x) A B U(x) 4. A U(x), B U(x) s.t. y B A U(y) U X X

1. X = R n x R n ǫ > 0 B(x, ǫ) = {y R n y x < ǫ} A U(x) ǫ > 0 s.t. B(x, ǫ) A U = {U(x) x R n }

1. U X x U, U U(x) U R n x U, ǫ s.t. B(x, ǫ) U X O 1., X O 2. U, V O U V O 3. U λ O (λ Λ) λ Λ U λ O

1. F X X F X X F 1., X F 2. F, G F F G F 3. F λ F (λ Λ) λ Λ F λ O

1. X B x B B U(x) Int B = {B }... B x B A U(x), A B B = {B }... B x B x B B = {B } = B IntB... B

1. X B 1. IntB B 2. B B 3. B

1. [U O] U O x U, U U(x) [O U] U U(x) O O s.t. x O U [O F] F F X F O [F O] U O X U F

1. U O (1), (2), (3) (4) O U U (4) U = U (4) A U(x) x B IntA A B IntA (4) A U (x) (1) (2) (3)

1. U 0 (x) U(x) x A U(x), B U 0 s.t. B A U 0 = {U 0 (x) x X} 1. U 0 (x) A U 0 (x) x 2. A, B U 0 (x) C U 0 (x) s.t. C A B 3. A U 0 (x), B U 0 (x) s.t. b B, C U 0 (b) s.t. C A U 0 U

1. {B(x, ǫ) ǫ > 0} R n X U 0 (x) = {x } U 0 = {U 0 (x)}

1. O 0 O U O, U λ O 0 (λ Λ) s.t. U = λ Λ U λ 1. X = U O 0 U ( x X, U O 0 s.t. x U) 2. U, V O 0, x U V, W O 0 s.t. x W U V O 0 O {B(x, ǫ) x R n, ǫ > 0} R n

1. S O S = { n i=1 U i U 1,, U n S} O 1. X = U S U S O (, a) (b, + ) R

1.......

2. x X U(x) = {X} X O = {, X}

2. x X {x} U(x) X O X

2. K n K n K n Zariski

2. X U 0 (x) = {x } U 0 = {U 0 (x)}

2. d: X X R X 1. d(x, y) 0 d(x, y) = 0 x = y 2. d(x, y) = d(y, x) 3. d(x, y) d(x, z) + d(z, y)... d(x, y) = y x = (y 1 x 1 ) 2 + + (y n x n ) 2 R n

2. x X ǫ > 0 B(x, ǫ) = {y X d(y, x) < ǫ} {B(x, ǫ) ǫ > 0} R n d(x, y) = y x n = 1 R

2. V K K = R C V R, v v V 1. v 0 v = 0 v = 0 2. av = a v (a K, v V ) 3. v + w v + w d(x, y) = y x d: V V R

2. R n x 2 1 + + x2 n x 1 + + x n max{ x 1,..., x n } q x 2 1 + x2 2 < ǫ x 1 + x 2 < ǫ max{ x 1, x 2 } < ǫ Figure: R 2

2. 0 1 V v V c v 0 v 1 C v 0 c C 0 1 V

2. R n V

2. J = [a, b] V C(J, V ) V C(J, V ) max f(x) a x b b f(x) dx a J R n J C(J, V ) J V sup x J f(x)

3. X Y f : X Y x X f(x) V x U f(u) V U V x X f(x) V f 1 (V ) x

3. f : R m R n x X ǫ > 0 y x < δ = f(y) f(x) < ǫ ǫ > 0

3. 1. f : X Y 2. V f(x) = f 1 (V ) x 3. V Y = f 1 (V ) X 4. F Y = f 1 (F) X 5. X A f(a) f(a)

3. X f : X Y Y f : X Y

3. C C R 2, x + yi (x, y),

3. X T T 1. T T 2. T T 3. T T 4. (X, T ) (X, T )

3. T T T T X

3. f X Y X f Y f f 1 (V ) X Y V Y f X f Y V f 1 (V )

4. A X i: A A A X U O A O X U A (x) = {B A B U(x)}, O A = {U A U O} U A {z C z = 1} S 1 S 1

4. X q: X X/ Y = X/ Y X O X Y O Y = {U X f 1 (U) O X } R x y x y Z R/ S 1

4. X X i {f i : X X i i I} f i X (initial topology) (U i ) U i X i (i I)} {fi 1 X X i (i I) i I X i X X i X = i I X i

4. I = {1,...,n} i I X i = X 1 X n {U 1 U n U i O Xi } x = (x 1,...,x n ) {U i U n U i U Xi (x i )} R n

4. I i I X i { i I U i U i O Xi i U i = X i } x = (x i ) { i I U i U i U Xi (x i ) i U i = X i } { i I U i U i O Xi } i I X i (box topology)

4. Z X = i I U i U i O Xi f : Z X i p i f : Z X i i I

5. X U O F T 1 : a, b X, U U(a) s.t. b U. T 2 : a, b X, U U(a), V U(b) s.t. U V =. T 3 : A F, b X A, U, V O s.t. A U, b V, U V =. T 4 : A, B F s.t. A B =, U, V O s.t. A U, B V, U V =. T 0 T 2+1/2 T 3+1/2 T 5 T 6

5. T 1 T 1 T 2 T 1 + T 3 T 1 + T 4

5. X 1. X 2. X a {a} 3. = {(x, x) x X} X X (a,b) U V for all U V U(a,b) U V for all U U(a), V U(b) b U U(a)

5. X = i I X i 1. X i X 2. X X i

6. Y X Y i I U i X {U i i I} Y X Y U = {U i i I} Y i I 0 U i I I 0 Y X X

6. 1. 2. X Y Y f : X Y X Y f(x)

6. X, Y X Y X Y X Y R n

6. {X i i I} i X i i X i X i

7. X Y X X U V, U V X = Y U V X U X V

7. f X Y f(x) Y X A A B A B X {A i i I} i I A i i I A i X Y X Y

7. X a a a {a} X a

7. n J 1 J n R n J i,..., J n J 1 J n X 1 S n

7. f : I I f(x) = x x I f : S 1 R f(x) = f( x) x S 1 2

7. [0, 1] X α: [0, 1] X X α(0) α(1) X

7. f : X Y X f(x) X Y X Y S 1 S 1 S 1 R 2 A B A B A = {(0, y) 0 < y 1} B = {(x,0) 0 < x 1} {(1/n, y) 0 < y 1}

7. X X X X π 0 X A B π 0 (A B) = {A, B}

8. X α: [0, 1] X α(0) = α(1) α β α: [0, 1] [0.1] X α β α β x 0 X π 1 (X, x 0 ) x 0 1 S 1 C x 0 S 1 X π 1 (X, x 0 ) X

8. π 1 (X, x 0 ) ([α], [β]) [α] [β] = [α β] α(2t), α β(t) = β(2t 1), t 1 2 t 1 2 π 1 (X, x 0 ) c x0 (t) x 0 [α] α α(t) = α(1 t)

8. ([α] [β]) [γ] = [α] ([β] [γ]) (α β) γ α (β γ) [α] [c x0 ] = [c x0 ] [α] = [α] α c x0 c x0 α α [α] [α] = [α] [α] = [c x0 ] α α α α c x0 α β α β α X x 0 x 1 l l ([α]) = [l α l] l : π 1 (X, x 0 ) π 1 (X, x 1 ) l

8. f : X Y f : π 1 (X, x 0 ) π(y, y 0 ) y 0 = f(x 0 ) (X, x 0 ) (Y, y 0 ) f g H(x,0) = f(x), H(x,1) = g(x) H : (X [0, 1], {x 0 } [0, 1] ) (Y, y 0 ) f g f g f : (X, x 0 ) (Y, y 0 ) g f 1 X f g 1 Y g: (Y, y 0 ) (X, x 0 ) f ( )

8. 1. f : (X, x 0 ) (Y, y 0 ) g: (Y, y 0 ) (X, z 0 ) (g f) = g f 2. 1 X : (X, x 0 ) (X, x 0 ) 3. f g: (X, x 0 ) (Y, y 0 ) f = g 1, 2 f : (X, x 0 ) (Y, y 0 ) f 3

8. f : (X, x 0 ) (Y, y 0 ) f : π 1 (X, x 0 ) π 1 (Y, y 0 ) X x 0 π 1 (X, x 0 ) = {1} x 0 X π 1 (X, x 0 ) 1 1 n D n 1

8. S 1 C S 1 Z π 1 (S 1, e 0 ) Z e: R S 1 e(t) = cos2πt + i sin2πt S 1 e V 0 = S 1 { 1} V 1 = S 1 {1} {V 0, V 1 } S 1 U n = (n/2 1/2, n/2 + 1/2) {U n n Z} R e U n e U n : U n V n (n = n mod 2)

8. 1. l: I S 1 p(a) = l(0) a R e l = l, l(0) = a l: I R l 2. L: I I S 1 p(a) = L(0, 0) a R p L = L, L(0, 0) = a L: I I R

8. π 1 (S 1, 1) [α] α χ([α]) = α(1) α(0) α χ: π 1 (S 1, 1) Z

8. C 0 S 1 Z π 1 (C 0, x 0 ) z z x 0 π 1 (S 1, 1) = π 1 (C 0, x 0 ) Z f : S 1 C 0 W(f,0) = f (1) π 1 (C 0, x 0 ) = Z f (winding number)

8. arg f(e(t)) = 2πθ(t) (0 t 1) θ: [0, 1] R W(f,0) = θ(1) θ(0) e(t) = cos2πt + i sin2πt x c x W(c x, 0) = 0 n W(z n, 0) = n W(f g, 0) = W(f,0) + W(g, 0) W(f/g,0) = W(f,0) W(g, 0)

8. f g: (S 1, 1) (C 0, x 0 ) W(f,0) = W(g, 0) f g C 0 (f g: S 1 C 0) W(f,0) = W(g, 0) f(z) W(f,0) = 0

8. f D 2 S 1 f S 1 W(f S 1, 0) 0 f(z) = 0 z D 2 0 f(d 2 ) f c f(0) : S 1 C 0 W(f,0) = W(c f(0), 0) = 0 c f(0) (z) f(0) (z S 1 )

8. Brouwer f : D 2 D 2 f(x) = x x D 2

8. n + 1 S n f( x) = f(x) f : S n R m f : S 1 C 0 W(f,0) f : S 2 R 2 f(s 2 )

8. f : S 2 R 2 f(x) = f( x) x S 2 S 2 3 2