The Stacks project

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. We suppose that $\bigcap _{\alpha \in A} Z_{\alpha } = \emptyset $. So we have $\bigcup _{\alpha \in A} (X\setminus Z_{\alpha })=X$ by complementation. As the subsets $Z_\alpha $ are closed, $\bigcup _{\alpha \in A} (X \setminus Z_{\alpha })$ is an open covering of the quasi-compact space $X$. Thus there exists a finite subset $J \subset A$ such that $X = \bigcup _{\alpha \in J} (X\setminus Z_{\alpha })$. The complementary is then empty, which means that $\bigcap _{\alpha \in J} Z_{\alpha } = \emptyset $. It proves there exists a finite subcollection of $\{ Z_{\alpha }\} _{\alpha \in J}$ verifying $\bigcap _{\alpha \in J} Z_{\alpha }=\emptyset $, which concludes by contraposition. $\square$

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