A topological space $X$ is compact if, for every collection $\{U_i\}_{i \in I}$ of open sets in $X$ whose union is $X$ , there exists a finite subcollection $\{U_{i_j}\}_{j=1}^n$ whose union is also $X$ .

A subset $Y$ of a topological space $X$ is said to be compact if $Y$ with its subspace topology is a compact topological space.

Note: Some authors require that a compact topological space be Hausdorff as well, and use the term quasi-compact to refer to a non-Hausdorff compact space. The modern convention seems to be to use compact in the sense given here, but the old definition is still occasionally encountered (particularly in the French school).