Lebl-Basic-Analysis-I · Problem 7.5.10

Metric Spaces

Mathematics · continuous-functions

Problem

Let

(X, d)

be a metric space. a) If

p \in X

, show that

f \colon X \to \mathbb{R}

defined by

f(x) \colonequals d(x, p)

is continuous. b) Define a metric on

X \times X

as in Exercise 7.1.6 part b, and show that

g \colon X \times X \to \mathbb{R}

defined by

g(x, y) \colonequals d(x, y)

is continuous. c) Show that if

K_1

and

K_2

are compact subsets of

X

, then there exists a

p \in K_1

and

q \in K_2

such that

d(p, q)

is minimal, that is,

d(p, q) = \inf\{d(x, y) : x \in K_1,\, y \in K_2\}