A topological space \((X, \tau)\) is compact if \(X\) is Hausdorff and quasicompact, that is, every open cover of \(X\) has a finite subcover: for every family \((U_j)_{j\in J}\) in \(\tau\) with \(X = \bigcup_{j\in J} U_j\), there is a finite subset \(J'\subseteq J\) with \(X = \bigcup_{j\in J'} U_j\).
This is the Bourbaki convention. A two-point set with the indiscrete topology satisfies the open-cover condition, so it is compact when compactness is defined by that condition alone; it is not Hausdorff, so in the Bourbaki convention it is quasicompact and not compact.