A topological space \(X\), not necessarily Hausdorff, is quasicompact if every open cover of \(X\) has a finite subcover. A quasicompact space is compact if it is also Hausdorff.
Requiring compact spaces to be Hausdorff is the convention of Bourbaki and of algebraic geometry, where the Zariski topology on the spectrum \(\Spec A\) of a commutative ring \(A\) is always quasicompact and is Hausdorff if and only if every prime ideal of \(A\) is maximal. In [@Mun00] a space is compact under the open-cover condition alone, so compact in that sense means quasicompact in the Bourbaki convention.