1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1.2 R A 1.3 X : (1)X (2)X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f

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1 1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1. R A 1.3 X : (1)X ()X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f 1 (A) f X X f 1 (A) = X f 1 (A) = A a A f f(x) = a x X f(x) = a A x f 1 (A) f 1 (A) = X y Y f f(x) = y x X X = f 1 (A) x f 1 (A) y = f(x) A A = Y Q.E.D

2 1.5 f : X Y X f(x) ( ) f : X Y g : X f(x) g(x) = f(x) (x X) g O f(x) O = U f(x) Y U x g 1 (O) g(x) O = U f(x) f(x) O = U f(x) f(x) U x f 1 (U) g 1 (O) = f 1 (U) f : X Y f 1 (U) X g : f(x) 1.4 f(x) Q.E.D 1.6 f : X Y X Y ( ) f f f X Y Q.E.D 1.7 Y X Y = X Y X ( ) A X A = A = Y A A a A A a Y = X a X a Y A Y a Y a N N Y A X A Y Y Y A Y = A Y = Y A Y A Y = Y = Y A = Y A A A = A X = Y A = A X A = X Q.E.D

3 1.8 X Y, Z Z Y Z Z Y Z Z ( ) Z Y Z X Z X Z Y Z Z Y Z Y = Z Y Y Z Z Y = Z Y = Y 1.7 Y Q.E.D A = (x, sin( π )) R 0 < x < 1}, B = (0, y) 1 < y < 1}, X = A B x X ϕ : (0, 1) A ϕ(x) = (x, sin( π )) ϕ (0, 1) x A B [ 1, 1] B A = X X X A, B A B = A B X = R, A = (0, 1), B = (1, ) A B = 1} A, B

4 1.9 X X F = F λ λ Λ} (1)X = λ Λ F λ () F λ (3) F X ( ) X A λ Λ F λ A F λ () F λ A = F λ A = F λ (i) λ Λ F λ A = A = A X = A ( λ Λ F λ ) = (A F λ ) λ Λ = = λ Λ (ii)f λ0 A = F λ λ 0 Λ F λ0 A (3) λ Λ F λ F λ0 F λ A = F λ A = F λ F λ A = F λ X\A F λ0 A F λ F λ0 (X\A) A A X\A X\A = X\A A A = A = F λ F λ0 (X\A) A = (X\A) A = λ Λ F λ A = F λ F λ A (1) X = λ Λ F λ A X A = X Q.E.D

5 1.10 X Y X Y X Y ( ) = P X : X Y X X X Y 1.4 X Y = y Y x X X y} X X} Y Y X y} x} Y X Y (x, y) Z(x, y) = (X y}) (x} Y ) (x, y) (X y}) (x} Y ) (X y}) (x} Y ) X y} x} Y 1.9 Z(x, y) X Y = (x,y) X Y Z(x, y) Z(x, y) Z(x, y ) X Y (x, y ), (x, y)} Z(x, y) Z(x, y ) Z(x, y) Z(x, y ) Q.E.D X X X C X (1)C ()C C C C = C 1 X X X R\S 0 = (, 1) ( 1, 1) (1, ) 3 n 1 R n+1 \S n = x R n+1 ; x > 1} x R n+1 ; x > 1}

6 1.11 X ( ) C C C C C C = C C, D C D = C D C D 1.9 C C D C = C D D C D C C = D Q.E.D

7 X (path) [0, 1] X γ : [0, 1] X γ(0) γ (beginning point) γ(1) γ (end point) γ γ(0) γ(1) [0, 1] I X X x, y x y X (path-connected space).1 ( ) X A X A A = X A X A A a A X b X b / A b X γ : [0, 1] X γ(0) = a, γ(1) = b γ 1 (A) [0, 1] γ(0) = a A 0 γ 1 (A) γ 1 (A) γ(1) = b / A 1 / γ 1 (A) γ 1 (A) [0, 1] [0, 1] Q.E.D R n C (convex set) C x, y x y C x, y C (1 t)x + ty C (0 t 1) C 1 S r (x) R n 3 R n

8 S r (x) R n S r (x) = y R n ni=1 (x i y i ) < r} u, v S r (x) 0 t 1 p = (1 t)u + tv u, v S r (x) d(u, x) = u x < r, d(v, x) = v x < r p S r (x) d(p, x) = n (p i x i ) i=1 = p x = (1 t)u + tv x = (1 t)(u x) + t(v x) (1 t)(u x) + t(v x) = (1 t) u x +t v x < (1 t)r + tr = r. X I = [0, 1] γ : I X γ 1 : I X γ 1 (t) = γ(1 t) (0 t 1) γ 1 ( ) ϕ : I I ϕ(t) = 1 t (0 t 1) ϕ γ 1 = γϕ γ 1 I X Q.E.D.3 α : I X, β : I X α(1) = β(0) αβ : I X α(t) (0 t 1 (αβ)(t) = ) β(t 1) ( 1 t 1) αβ

9 .4 X X x, y x y x y ( ) (1) γ : I X γ(t) = x (t I) γ x x path x x () x y x y pathγ : I X γ(0) = x, γ(1) = y γ 1 : I X γ 1 (t) = γ(1 t) (t I) γ 1 (0) = γ(1 0) = γ(1) = y γ 1 (1) = γ(1 1) = γ(0) = x γ 1 y x path y x (3) x y y z path α : I X, β : I X α(0) = x, α(1) = y β(0) = y, β(1) = z α(1) = y = β(0).3 αβ : I X path (αβ)(t) = α(t) (0 t 1 ) β(t 1) ( 1 t 1) (αβ)(0) = α(0) = x (αβ)(1) = β(1) = z αβ x z path x z Q.E.D

10 .5(Glueing Lemma) X, Y A, B X f : A X, g : B X f g : A B X (f g)(x) = f(x) (x A) g(x) (x B) (1)A, B A B f, g f g ()A, B A B f, g f g ( ) (1) () Y O (f g) 1 (O) A B x A x B (f g)(x) = f(x) O (f g)(x) = g(x) O x (f g) 1 (O) (f g)(x) O x f 1 (O) g 1 (O) (f g) 1 (O) = f 1 (O) g 1 (O) f f 1 (O) A A A B f 1 (O) A B g 1 (O) A B (f g) 1 (O) A B Q.E.D.6 f : X Y X f(x) ( ) x, y f(x) x = f(x ), y = f(y ) X x, y X γ(0) = x, γ(1) = y pathγ : I X fγ : I Y (fγ)(0) = f(γ(0)) = f(x ) = x (fγ)(1) = f(γ(1)) = f(y ) = y fγ x y path Q.E.D

11 .7 X( ), Y ( ) X Y X Y ( ) = (x, y), (x, y ) X Y Y y, y Y α(0) = y, α(1) = y α : I Y α : I X Y α(t) = (x, α(t)) (t I) α ( ) X α(0) = (x, α(0)) = (x, y) α(1) = (x, α(1)) = (x, y ) β(0) = x, β(1) = x β : I X β : I X Y β(t) = (β(t), y ) (t I) β ( ) β(0) = (β(0), y ) = (x, y ) β(1) = (β(1), y ) = (x, y ) (x, y) (x, y ), (x, y ) (x, y ) (x, y) (x, y ) = P X : X Y X P Y : X Y Y.6 Q.E.D ( ) P X α : I X (P X α)(t) = P X ( α(t)) = P X (x, α(t)) = x P X α P Y α : I Y (P Y α)(t) = P Y ( α(t)) = P Y (x, α(t)) = α(t) P Y α = α α f : W X Y P X f P Y α Q.E.D

12 .8 R n X X ( ) x X U(x) = y X x y} U(x) = X (1) U(x) z U(x) X x X ε S ε (x) X S ε (x) w α : I S ε (x) α(t) = (1 t)w + tz (0 t 1) α(0) = w, α(1) = z α(t) S ε (x) d(α(t), z) = α(t) z = (1 t)w + tz z = (1 t)(w z) = (1 t) w z = (1 t)d(w, z) z U(x) β : I X β(0) = x, β(1) = z βα 1 x w w U(x) S ε (x) U(x) U(x) () U(x) X\U(x) = U(w) w X\U(x) U(w) X\U(x) U(x) (3) X = U(x) x U(x) U(x) U(x) X X U(x) = X Q.E.D

13 3 X, Y f : X Y, g : X Y (homotopic) F : X I Y F (x, 0) = f(x), F (x, 1) = g(x) (x X) f g F f g (homotopy) F : f g X, Y A X f : X Y, g : X Y A (relative homotopic) f(x) = g(x) (x A) F : X I Y F (x, 0) = f(x), F (x, 1) = g(x) (x X) F (x, t) = f(x) = g(x) (x A, t I) f g rel A F f g A F : f g rel A A = f g f g rel A 1 f : I R f(x) = (cos (πx), sin (πx)) (x I) g : I R g(t) = (0, 0) (t I) F : I I R F (x, t) = ((1 t) cos πx, (1 t) sin πx) (x, t I) F f g ( ) X C R n X C f : X C, g : X C f g F : X I C F (x, t) = (1 t)f(x) + tg(x) (x X, t I) f(x), g(x) C C F (x, t) = (1 t)f(x) + tg(x) C F (x, 0) = f(x) F (x, 1) = g(x) (x X) F ( ) F : f g

14 3.1 X, Y Map(X, Y ) X Y Map(X, Y ) Map(X, Y ) f, g Map(X, Y ) f g ( ) (1) F : X I Y F (x, t) = f(x) (x X, t I) O Y F 1 (O) = (x, t) X I F (x, t) O} = (x, t) X I f(x) O} = (x, t) X I x f 1 (O)} F 1 (O) = f 1 (O) I f 1 (O) X F 1 (0) X I () f g F : X I Y F (x, 0) = f(x), F (x, 1) = g(x) G : X I Y G(x, t) = F (x, 1 t) (x X, t I) G(x, 0) = F (x, 1 0) = g(x) G(x, 1) = F (x, 1 1) = f(x) ϕ : I I ϕ(t) = 1 t (t I) ϕ 1 X X F (1 X ϕ)(x, t) = F (1 X ϕ(x, t)) = F (1 X (x), ϕ(t)) = F (x, 1 t) = G(x, t) G g f (3) f g, g h F, G : X I Y F (x, 0) = f(x), F (x, 1) = g(x) G(x, 0) = g(x), G(x, 1) = h(x) H : X I Y H(x, t) = F (x, t) (0 t 1 ) G(x, t 1) ( 1 t 1) t = F (x, ) = g(x) = G(x, 1) H well-defined H(x, 0) = f(x), H(x, 1) = g(x) (x X) Glueing Lemma H f h Q.E.D

15 3.

16 4 X α, β : I X α(1) = β(0) α β α β : I X (α β)(s) = α(s) (0 s 1 ) β(s 1) ( 1 s 1) X p X X α : I X α(0) = α(1) = p α p (basic point) X loop p X loop Ω(p) = α : I X α α(0) = α(1) = p} 4.1 Ω(p) 0, 1} Ω(p) α, β Ω(p) α β rel 0, 1} ( ) ( ) α Ω(p) x α α = β Ω(p) α β rel 0, 1}} p X loop Π 1 (X, p) = α α Ω(p)} 4. Π 1 (X, p) ( )

17 4.3 α, β Π 1 (X, p) α β = α β welldefined ( ) α = α β = β = α β = α β α α rel 0, 1} β β rel 0, 1} F : I I X, G : I I X H : I I X F (s, 0) = α(s) F (s, 1) = α (s) F (0, t) = α(0) = α (0) = p F (1, t) = α(1) = α (1) = p (s I, t I) G(s, 0) = β(s) G(s, 1) = β (s) G(0, t) = β(0) = β (0) = p G(1, t) = β(1) = β (1) = p (s I, t I) H(s, t) = F (s, t) (0 s 1 ) G(s 1, t) ( 1 s 1) s = F (, t) = p = G( 1, t) H well-defined φ : [0, 1 ] [0, 1], φ(s) = s G ψ H(s, t) = F (s, t) = F (φ(s), 1 x (t)) = F ((φ 1 X )(s, t)) = F (φ 1 X )(s, t) F (φ 1X )(s, t) (0 s 1 ) G (ψ 1 X )(s, t) ( 1 s 1) F (s, 0) = α(s) (0 s 1 H(s, 0) = ) } G(s 1, 0) = α(s 1) ( 1 s 1) = (α β)(s) F (s, 1) = α H(s, 1) = (s) (0 s 1) } G(s 1, 1) = α (s 1) ( 1 s 1) = (α β )(s) H(0, t) = F (0, t) = α(0) = α (0) = p H(1, t) = G(1, t) = β(1) = β (1) = p 1 } I I I H Glueing Lemma H : α β α β rel 0, 1} α β = α β Q.E.D

18 4.4 ( ) X α, β, γ α(1) = β(0), β(1) = γ(0) (α β) γ α(β γ) rel 0, 1} α(4s) (0 s 1) 4 ((α β) γ)(s) = β(4s 1) ( 1 4 s 1) γ(s 1) ( 1 s 1) α(s) (0 s 1) (α (β γ))(s) = β(4s ) ( 1 s 3) 4 γ(4s 3) ( 3 s 1) 4 ( ) F : I I X F (s, t) = α( s β( γ( t+1 4 t+1 s s t+ 4 1 t+ 4 ) = α( 4s t+1 ) (0 s t+1 4 ) ) = β(4s t 1) ( t+1 4 s t+ 4 ) ) = γ( 4s t t ) ( t+ 4 s 1) F t I F (0, t) = α(0), F (1, t) = β(1) (α β) γ α(β γ) rel 0, 1} Q.E.D ( ) α, β, γ Π 1 (X, p) ( α β ) γ = α β γ = (α β) γ = α (β γ) = α β γ = α ( β γ ) ( α β ) γ = α ( β γ Q.E.D

19 4.5 ( ) X α : I X α(0) = x, α(1) = y e x α α rel 0, 1}, α e y α rel 0, 1} e z : I X, e z (s) = z (s I) e p : I X e p (s) = p e p Π 1 (X, p) ( ) F : I I X x (0 s t F (s, t) = ) α( s t ) = α( s t) 1 t t ( t s 1) F ( ) F (s, 0) = α(s) F (s, 1) = (e x α)(s) F (0, t) = x F (1, t) = α(1) (s I, t I) α e x α rel 0, 1} Q.E.D 4.6 ( ) X α : I X α(0) = x, α(1) = y α α 1 e x rel 0, 1}, α 1 α e y rel 0, 1} α 1 : I X, α 1 (s) = α(1 s) (s I) α 1 = α 1 α ( ) F : I I X α(s) (0 s t ) F (s, t) = α(t) ( t s 1 t ) α 1 (s 1) (1 t s 1) F ( ) F (s, 0) = α(0) = x = e x (s) F (s, 1) = (α α 1 )(s) F (0, t) = α(0) = x F (1, t) = α 1 (1) = x (s I, t I) e x α α 1 rel 0, 1} Q.E.D

1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)

1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1) X α α 1 : I X α 1 (s) = α(1 s) ( )α 1 1.1 X p X Ω(p)