Topology/Compactness/Covering/Definition

Compact (covering property)

A topological space ${}X$ is called compact if for every open covering

$X=\bigcup _{i\in I}U_{i}\,\,\,{\text{ with }}U_{i}{\text{ open and }}I{\text{ an arbitrary index set}},$

there exists a finite subset ${}J\subseteq I$ such that

${}X=\bigcup _{i\in J}U_{i}\,$

holds.