Lemma 5.13.2. A Hausdorff space is locally quasi-compact if and only if every point has a quasi-compact neighbourhood.

Proof. Let $X$ be a Hausdorff space. Let $x \in X$ and let $x \in E \subset X$ be a quasi-compact neighbourhood. Then $E$ is closed by Lemma 5.12.4. Suppose that $x \in U \subset X$ is an open neighbourhood of $x$. Then $Z = E \setminus U$ is a closed subset of $E$ not containing $x$. Hence we can find a pair of disjoint open subsets $W, V \subset E$ of $E$ such that $x \in V$ and $Z \subset W$, see Lemma 5.12.4. It follows that $\overline{V} \subset E$ is a closed neighbourhood of $x$ contained in $E \cap U$. Also $\overline{V}$ is quasi-compact as a closed subset of $E$ (Lemma 5.12.3). In this way we obtain a fundamental system of quasi-compact neighbourhoods of $x$. $\square$

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