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 (, (, and ( 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 ( Details omitted. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01H7. Beware of the difference between the letter 'O' and the digit '0'.