The Stacks project

Proof. Let $U$ be a scheme and $U \to X$ a surjective étale morphism. Then $Z \times _ X U \to U$ is an immersion of schemes, hence gives a homeomorphism of $|Z \times _ X U|$ with a locally closed subset $T'$ of $|U|$. By Lemma 66.4.3 the subset $T'$ is the inverse image of the image $T$ of $|Z| \to |X|$. The map $|Z| \to |X|$ is injective because the transformation of functors $Z \to X$ is injective, see Spaces, Section 65.12. By Topology, Lemma 5.6.4 we see that $T$ is locally closed in $|X|$. Moreover, the continuous map $|Z| \to T$ is a homeomorphism as the map $|Z \times _ X U| \to T'$ is a homeomorphism and $|Z \times _ Y U| \to |Z|$ is submersive. $\square$