Definition:Compact Space/Topology/Subspace/Definition 1

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

Let $H \subseteq S$ be a subset of $T$.

The topological subspace $T_H = \left({H, \tau_H}\right)$ is compact in $T$ iff $T_H$ is itself compact.

That is, $T_H$ 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