The Stacks project

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.