If a metric space$E$has one of the following properties, then it has the other three:

(a) $E$is compact.

(b) Every nested sequence of nonempty closed sets in$E$has a nonempty intersection.

(c) Every infinite subset of$E$has a limit point.

(d) Every sequence in$E$has a convergent subsequence.

Proof

For the implication $(\text{a})\Rightarrow (\text{b})$, assume ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training,
learning paths, books, interactive tutorials, and more.