Lemma 5.12.6. Let $X$ be a quasi-compact topological space. If $\{ Z_\alpha \} _{\alpha \in A}$ is a collection of closed subsets such that the intersection of each finite subcollection is nonempty, then $\bigcap _{\alpha \in A} Z_\alpha $ is nonempty.

**Proof.**
Omitted.
$\square$

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