Equivalence of Definitions of Compact Topological Subspace

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

Let $T_A = \left({A, \tau_A}\right)$ be a topological subspace of $T$, where $A \subseteq S$.

Then $T_A$ is compact iff every open cover $\mathcal C \subseteq \tau$ for $A$ has a finite subcover.

Also see

 * Compact Subspace
 * Compact Space