Lebl-Basic-Analysis-I · Problem 7.4.15

Metric Spaces

Mathematics · compactness

Problem

Let

X

be a metric space and

C \subset \mathcal{P}(X)

the set of nonempty compact subsets of

X

. Using the Hausdorff metric from Exercise 7.1.8, show that

(C, d_H)

is a metric space. That is, show that if

L

and

K

are nonempty compact subsets, then

d_H(L, K) = 0

if and only if

L = K