Lemma 35.23.9 (0CEX)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.23: Properties of morphisms local in the fpqc topology on the target
Lemma 35.23.9 (cite)

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Lemma 35.23.9. 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 35.23.3. 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 35.23.7, 35.23.8, and 35.23.4. $\square$

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