Lemma 5.27.1. Let $X$ be a topological space which

1. has a basis of the topology consisting of quasi-compact opens, and

2. has the property that the intersection of any two quasi-compact opens is quasi-compact.

Then

1. $X$ is locally quasi-compact,

2. a quasi-compact open $U \subset X$ is retrocompact,

3. any quasi-compact open $U \subset X$ has a cofinal system of open coverings $\mathcal{U} : U = \bigcup _{j\in J} U_ j$ with $J$ finite and all $U_ j$ and $U_ j \cap U_{j'}$ quasi-compact,

Proof. Omitted. $\square$

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