Lemma 25.11.4. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology of $X$. Assume that

1. $X$ is quasi-compact,

2. each $U \in \mathcal{B}$ is quasi-compact open, and

3. the intersection of any two quasi-compact opens in $X$ is quasi-compact.

Then there exists a hypercovering $(I, \{ U_ i\} )$ of $X$ with the following properties

1. each $U_ i$ is an element of the basis $\mathcal{B}$,

2. each of the $I_ n$ is a finite set, and in particular

3. each of the coverings (25.11.0.1), (25.11.0.2), and (25.11.0.3) is finite.

Proof. This follows directly from the construction in the proof of Lemma 25.11.3 if we choose finite coverings by elements of $\mathcal{B}$ in (25.11.3.1). Details omitted. $\square$

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