Compact set

A set for which every open cover has a finite subcover.

Compact set

Let $(X,d)$ be a metric space and let $K\subseteq X$ . The set $K$ is compact if for every family of open sets $\{U_i\}_{i\in I}$ in $X$ such that

K\subseteq \bigcup_{i\in I} U_i,

there exist finitely many indices $i_1,\dots,i_N\in I$ such that

K\subseteq U_{i_1}\cup\cdots\cup U_{i_N}.

Compactness is a finiteness condition on the topology. In analysis it is the hypothesis behind uniform continuity (Heine–Cantor), attainment of extrema (Extreme Value Theorem), and many extraction arguments (existence of convergent subsequences ).

Examples:

In $\mathbb{R}$ with the usual metric, $[a,b]$ is compact.
In $\mathbb{R}^k$ , every closed ball $\overline{B}(0,R)$ is compact (Heine–Borel).
The open interval $(0,1)\subset\mathbb{R}$ is not compact (cover it by $(0,1-\tfrac1n)$ for $n\ge 2$ ; no finite subcover).