Example 9.2.0.6. Let $X$ be a topological space and let $\operatorname{\mathcal{C}}$ be the category of open sets of $X$ (with morphisms given by inclusions). Then an object $U \in \operatorname{\mathcal{C}}$ is compact if and only if $U$ is quasi-compact when viewed as a topological space: that is, every open cover of $U$ admits a finite subcover.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$