Definition:Compact Space/Topology

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

See also the other equivalent definitions of compactness.

A subset $Y \subseteq X$ is said to be compact (in $X$) if the topological subspace $Y$ is.

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

Also see

 * Equivalent Definitions of Compactness