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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