Compactness in Metric Spaces

Compactness generalizes the Heine-Borel property to arbitrary metric spaces. A space is compact if every open cover has a finite subcover (or equivalently, every sequence has a convergent subsequence). Compact spaces share many properties with finite sets: continuous functions attain extrema, and spaces are complete and totally bounded.

Definition

Definition8.1Compact metric space

A metric space $(X, d)$ is compact if every open cover has a finite subcover. Equivalently, $X$ is sequentially compact: every sequence in $X$ has a subsequence converging to a point in $X$ .

RemarkEquivalence in metric spaces

In metric spaces (unlike general topological spaces), compactness and sequential compactness are equivalent. This is not true in arbitrary topological spaces.

Example[a, b] is compact

By the Heine-Borel theorem, $[a, b] \subseteq \mathbb{R}$ is compact. More generally, closed bounded subsets of $\mathbb{R}^n$ are compact.

Example(0, 1) is not compact

$(0, 1)$ is not compact. The sequence $x_n = 1/n$ has no convergent subsequence in $(0, 1)$ (the limit $0 \notin (0, 1)$ ).

ExampleUnit sphere in R^n

The unit sphere $S^{n-1} = \{x \in \mathbb{R}^n \mid \|x\| = 1\}$ is compact (closed and bounded in $\mathbb{R}^n$ ).

Properties of compact spaces

Theorem8.1Compact implies complete and totally bounded

Every compact metric space is complete and totally bounded.

Theorem8.2Continuous images of compact sets are compact

If $K$ is compact and $f : K \to Y$ is continuous, then $f(K)$ is compact.

RemarkExtreme Value Theorem

As in $\mathbb{R}$ , continuous functions on compact spaces attain their maximum and minimum.

Compactness in function spaces

ExampleArzelà-Ascoli

In $C([a, b])$ with the sup norm, a set is compact iff it is closed, bounded, and equicontinuous (Arzelà-Ascoli theorem). This is not the same as "closed and bounded" — boundedness alone is insufficient.

Summary

Compactness in metric spaces:

Equivalent to sequential compactness (every sequence has convergent subsequence).
Compact spaces are complete and totally bounded.
Continuous images are compact (EVT holds).
In $\mathbb{R}^n$ : compact $\Leftrightarrow$ closed and bounded (Heine-Borel).

See Heine-Borel and Arzelà-Ascoli.