Topology/Compactness/Covering/Definition

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

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

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

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

holds.