Basic Analysis I — Real Analysis · Problem 7.1.8

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · metric-spaces

Problem

Let

(X, d)

be a metric space. For nonempty bounded subsets

A

and

B

let

d(x,B) \coloneq \inf\{d(x,b) : b \in B\} \quad \text{and} \quad d(A,B) \coloneq \sup\{d(a,B) : a \in A\}.

Now define the Hausdorff metric as

d_H(A,B) \coloneq \max\{d(A,B), d(B,A)\}.

Note:

d_H

can be defined for arbitrary nonempty subsets if we allow the extended reals. a) Let

Y \subset \mathcal{P}(X)

be the set of bounded nonempty subsets. Prove that

(Y, d_H)

is a so-called pseudometric space:

d_H

satisfies the metric properties (i), (iii), (iv), and further

d_H(A,A) = 0

for all

A \in Y

. b) Show by example that

d

itself is not symmetric, that is

d(A,B) \neq d(B,A)

. c) Find a metric space

X

and two different nonempty bounded subsets

A

and

B

such that

d_H(A,B) = 0