The Stacks project

Lemma 73.11.8. 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 73.11.3. 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. See Morphisms of Spaces, Lemma 66.5.5 and Morphisms of Spaces, Definitions 66.19.3, 66.5.2, 66.6.2, 66.53.2. Thus the lemma follows from Lemmas 73.11.6, 73.11.7, and 73.11.4. $\square$

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