Definition:Compact Space/Topology/Subspace

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

A subset $H \subseteq S$ is compact in $T$ iff the topological subspace $T_H = \left({H, \tau_H}\right)$ is itself compact.

That is, $H \subseteq S$ is compact in $T$ iff every open cover $\mathcal C \subseteq \tau_H$ for $H$ has a finite subcover.

Also see

 * Equivalent Definitions of Compact Topological Subspace