Definition:Compact Space/Topology/Subspace

Definition
Let $T = \left({X, \tau}\right)$ be a topological space

A subset $Y \subseteq X$ is compact in $T$ if the topological subspace $T\,' = \left({Y, \tau_Y}\right)$ is itself compact.

That is, if every open cover of $T\,'$ has a finite subcover.

Also see

 * Equivalent Definitions of Compact Topological Subspace