Let $(X,\tau)$ be a [topological space](/page/Topological%20Space). For a family $\mathcal F$ of closed subsets of $X$, say that $\mathcal F$ has the finite intersection property if, for every finite subfamily $\mathcal E \subset \mathcal F$,
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\begin{align*}
\bigcap_{F \in \mathcal E} F \ne \varnothing,
\end{align*}
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where the empty intersection is interpreted as $X$. Then $X$ is compact if and only if every family $\mathcal F$ of closed subsets of $X$ with the finite intersection property satisfies
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\begin{align*}
\bigcap_{F \in \mathcal F} F \ne \varnothing.
\end{align*}