Definition:Compact Space/Topology/Definition 3

Definition
A topological space $T = \left({S, \tau}\right)$ is compact $\tau$ has a sub-basis $\mathcal B$ such that:
 * from every cover of $S$ by elements of $\mathcal B$, a finite subcover of $S$ can be selected.

Also see

 * Equivalence of Definitions of Compact Topological Space