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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