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1. (a) ( R n , d (n) ) の部分集合 A を A = { (x 1 , . . . , x n ) ∈ R n | ∑ n

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位相入門演習 No.6問題 extra

2013/1/25

1. (a) ( R n , d (n) ) の部分集合 A を A = { (x 1 , . . . , x n ) ∈ R n | ∑ n

i=1 x 2 i = 1 } で定める。

x ∈ R n に対して、 d(x, A) を求めよ。

x ∈ A のとき、 x ∈ / A のときで異なる表式をもつことに注意。

(b) ∀ a ∈ A に対して、 d(x, A) < d(x, a) が成り立つような (X, d) と A の例を挙げよ。

2. (問 13.8) (X, d) を距離空間、 A, B ⊂ X とする。次を示せ。

(a) (A ∪ B) = A ∪ B を示せ。

(b) (A ∪ B) i = A i ∪ B i を示せ。

(c) (A ∪ B) d = A d ∪ B d を示せ。

3. (X, d) を距離空間とする。

(a) (問 13.9) d ′ (x, y) = 1+d(x,y) d(x,y) もまた X 上の距離関数であることを示し、

(X, d) の開集合系 O d = (X, d ′ ) の開集合系 O d

′

を示せ。

(b) ρ(x, y) = min (1, d(x, y)) もまた X 上の距離関数であることを示し、

(X, d) の開集合系 O d = (X, ρ) の開集合系 O ρ

を示せ。

4. C[0, 1] を [0, 1] 上の連続関数の全体とする。２つの距離関数 d ∞ (f, g) = sup

0 ≤ t ≤ 1 | f (t) − g(t) | d 2 (f, g) =

(∫ 1

0

(f(t) − g(t)) 2 dt ) 1/2

に対し (C[0, 1], d ∞ ) の開集合系と (C[0, 1], d 2 ) の開集合系は一致するか？

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