Definition:Compact Space/Topology/Subspace

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

A subset $Y \subseteq X$ is compact in $T$ iff the topological subspace $T_Y = \left({Y, \tau_Y}\right)$ is itself compact.

...that is, iff every open cover $\mathcal C \subseteq \tau_Y$ for $Y$ has a finite subcover.

Also see

 * Equivalent Definitions of Compact Topological Subspace