Let $(Y,\tau)$ be a [topological space](/page/Topological%20Space), let $X \subset Y$, and equip $X$ with the [subspace topology](/page/Subspace%20Topology) $\tau_X := \{X \cap U : U \in \tau\}$. Then $(X,\tau_X)$ is compact if and only if the following ambient open-cover property holds: for every index set $I$ and every family $(U_i)_{i \in I}$ of open subsets of $Y$, if $X \subset \bigcup_{i \in I} U_i$, then there exists a finite subset $J \subset I$ such that $X \subset \bigcup_{i \in J} U_i$.