Basic Analysis I — Real Analysis · Problem 7.4.12

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · completeness-and-compactness

Problem

(Challenging) Let

(X, d)

be a complete metric space. Show that

K \subset X

is compact if and only if

K

is closed and such that for every

\epsilon > 0

there exists a finite set of points

x_1, x_2, \ldots, x_n

with

K \subset \bigcup_{j=1}^n B(x_j, \epsilon)

. Note: Such a set

K

is said to be totally bounded, so in a complete metric space a set is compact if and only if it is closed and totally bounded.