Lemma 5.12.5. Let $X$ be a quasi-compact Hausdorff space. Let $E \subset X$. The following are equivalent: (a) $E$ is closed in $X$, (b) $E$ is quasi-compact.

**Proof.**
The implication (a) $\Rightarrow $ (b) is Lemma 5.12.3. The implication (b) $\Rightarrow $ (a) is Lemma 5.12.4.
$\square$

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