Quasi-compact object and topos

An object \(X \in T\) is called quasi-compact if any cover \(\bigsqcup_{i} U_{i} \twoheadrightarrow X\) can be refined to a finite cover, i.e. there is a finite subset \(J \subseteq I\) such that the map \(\bigsqcup_{j \in J} U_j \to X\) is still an effective epimorphism.

We say a topos \(T\) is quasi-compact if \(* \in T\) is quasi-compact.