Compactness

Definition: Open Cover

Let $(X,\tau )$ be a topological space.

A cover of $X$ is an open cover if all of its elements are open sets.

Definition: Compactness of a Topological Space

A topological space is compact if each of its open covers has a finite subcover.

Definition: Compactness of a Subset

A subset of a topological space is compact if it is compact as a subspace.