The Stacks project

Lemma 35.20.9. The property $\mathcal{P}(f) =$“$f$ is a universal homeomorphism” is fpqc local on the base.

Proof. This can be proved in exactly the same manner as Lemma 35.20.3. Alternatively, one can use that a map of topological spaces is a homeomorphism if and only if it is injective, surjective, and open. Thus a universal homeomorphism is the same thing as a surjective, universally injective, and universally open morphism. Thus the lemma follows from Lemmas 35.20.7, 35.20.8, and 35.20.4. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 35.20: Properties of morphisms local in the fpqc topology on the target

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Your email address will not be published.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