The Stacks project

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,

  4. add more here.

Proof. Omitted. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 5.27: Miscellany

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Your email address will not be published.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 0069. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 5.27.1 as pdf