Basic Analysis I — Real Analysis · Problem 7.5.10

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · continuous-functions

Problem

Let

(X, d)

be a metric space. a) If

p \in X

, show that

f : X \to \mathbb{R}

defined by

f(x) \coloneq 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 : X \times X \to \mathbb{R}

defined by

g(x,y) \coloneq 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\}