位相入門演習 演習問題 4 (連続写像と同相写像) 実数値関数と同様に, ε ...

4(), ε-δ . , . .. (X, d X ) (Y,d Y ) f : X → Y , f a ∈ X .∀ε >0, ∃δ >0, s.t. “ ∀x ∈ X, d X (x, a) 0, a δ f ε . 4-1. f : X → Y (i)(iii) ( 2.13).(i) f .(ii) Y U , f −1 (U) X .(iii) Y F , f −1 (F ) X .Proof. a ∈ X .“∀x ∈ X, d X (x, a) 0 . ,N dX (a; δ) ⊂ f −1 (N dY (f(a); ε)) ⊂ f −1 (U). f −1 (U) X .(ii)⇒(i) a ∈ X ε>0 , N dY (f(a); ε) Y (2-2). (ii) f −1 (N dY (f(a); ε)) X . a ∈ f −1 (N dY (f(a); ε)) , N dX (a; δ) ⊂ f −1 (N dY (f(a); ε)) δ>0 . f a ∈ X . a , f .(ii)⇔(iii) () (, f −1 (Y \U) =X \ f −1 (U)). (ii) (iii) .. , 4-1 . , , 2-2 (R n ,d i )(i =1, 2, ∞). . ( 3-3 3-7 , ). . 4-1. , ε-δ .13

4-2. y 0 ∈ Y , f : X ∋ x ↦→ f(x) =y 0 ∈ Y . 4-3. () X Y f : X → Y . 4-4. f,g : X → V X V .h : X → V ( 2.16).(1) h(x) :=f(x)+g(x). (2) h(x) :=r · f(x). (, r ∈ R .) 4-5. (X, d) f : X → R , d ′ (x, y) :=d(x, y)+|f(x) − f(y)| f (X, d ′ ) . . 1. A ⊂ X f : X → Y , f A (restriction) f| A . f| A , ∀x ∈ A, f| A (x) :=f(x) A Y . 4-2. f : X → Y , A ⊂ X f| A A Y ( 2.4).Proof. Y U , f| −1A (U) =f −1 (U) ∩ A . f −1 (U) 4-1(ii) X , f −1 (U) ∩ A 2-11(2) A . 4-1(ii) f| A . 4-6. X = A 1 ∪ A 2 f : X → Y , f| A1 f| A2 .(1) A 1 , A 2 , f ( 2.5).(2) A 1 , A 2 , f ( 2.15).(3) f . .. , 4-6 X = ⋃ ni=1 A i . 4-7. X = ⋃ λ∈Λ A λ f : X → Y , λ ∈ Λ f| Aλ . A λ , f .. , A λ , f . f , {A λ | λ ∈ Λ} X 1 X = ⋃ λ∈Λ A λ , f| Aλ .* 4-8. , .{0 if x

. X Y , X Y 1 1 . , , X Y .. (topological invariant) . . . 4-10. R ( 3.13).(1) N ≈ Z (2) [0, 1) ≈ (0, 1] (3) (−1, 1) ≈ R (4) N ≉ Q **(5) [0, 1] ≉ (0, 1) 4-11. V h : V → V .(1) h(x) :=x + x 0 .( x 0 ∈ V .) (2) h(x) :=r · x. ( r ∈ R \{0}.) 4-12. (R n ,d i )(i =1, 2, ∞) , . .. 4-12 , 1-3 3 2-3 . , 3 . 4-13. (X × Y ) × Z ≈ X × (Y × Z) . 4-14. pr i : X 1 × X 2 → X i (i =1, 2) (2.5). 4-15. X, Y 1 ,Y 2 , f : X → Y 1 × Y 2 , pr i ◦f (i =1, 2) . ( 2.7). 4-16. y 0 ∈ Y , f : X → X × Y f(x) :=(x, y 0 ) . f : X → f(X) .. 4-134-16 . , ( 1-12 ) ( 4-4)... X C ∗ (X) . 4-17. ‖f‖ := sup x∈X |f(x)| C ∗ (X) :. (X, d X ) (Y,d Y ) f : X → Y (isometry):∀ x, x ′ ∈ X, d X (x, x ′ )=d Y (f(x),f(x ′ )).. f : X → Y , X Y (isometric) . X Y , (X, d X ) Y (f(X),d Y | f(X) 2) . , . 4-18. . 4-19. C ∗ (N) l ∞ .15

4-3. X C ∗ (X) .Proof. x 0 ∈ X , i : X → C ∗ (X) x ↦→ d(x, ·) − d(x 0 , ·) ∈ C ∗ (X) , .* 4-20. i . 4-4. (X n ,d n ) X = ∏ n∈Æ X n , X d ( 1-12).d(x, y) :=∞∑min{2 −n ,d n (x n ,y n )}. (, x =(x n ) n∈Æ , y =(y n ) n∈Æ .)n=1(1) i ∈ N pr i : X → X i .Proof. d ′ i (x i,y i ) := min{2 −i ,d i (x i ,y i )} d ′ i X i , 2-6(2) id i :(X i ,d ′ i ) → (X i,d i ) . , pr i : X → (X i ,d ′ i ) , pr i : X → (X i ,d i ) . pr i : X → (X i ,d ′ i ) . a =(a n ) n∈Æ ∈ X ε>0 , δ := ε>0 . x =(x n ) n∈Æ ∈ X d(x, a) 0 d(z,a) 0 , d(z,a)