Definition:Compact Space

Topology
A topological space $X$ is compact if every open cover of $X$ has a finite subcover.

See also the other equivalent definitions of compactness.

For subsets of Euclidean space, this is equivalent to being closed and bounded by the Heine-Borel Theorem.

Real Analysis
A set $X \subset \R$ is said to be compact when it is closed and bounded.

Also see

 * Equivalent Definitions of Compactness

Sources for Topology

 * : $\text{I}: \S 1$

Sources for Real Analysis

 * : $\S 2.9$