The Stacks project

Proof. Choose a scheme $U$, a point $u \in U$, and an étale morphism $U \to X$ sending $u$ to $\xi $. Then any sequence of nontrivial specializations $\xi _ e \leadsto \ldots \leadsto \xi _0 = \xi $ can be lifted to a sequence $u_ e \leadsto \ldots \leadsto u_0 = u$ in $U$ by Lemma 68.12.2. Conversely, any sequence of nontrivial specializations $u_ e \leadsto \ldots \leadsto u_0 = u$ in $U$ maps to a sequence of nontrivial specializations $\xi _ e \leadsto \ldots \leadsto \xi _0 = \xi $ by Lemma 68.12.1. Because $|X|$ and $U$ are sober topological spaces we conclude that the codimension of $T$ in $|X|$ and of $\overline{\{ u\} }$ in $U$ are the same. In this way the lemma reduces to the schemes case which is Properties, Lemma 28.10.3. $\square$