Lemma 81.4.5. In Situation 81.2.1 let $X/B$ be good. Let $T \subset |X|$ be a closed subset and $t \in T$. If $\dim _\delta (T) \leq k$ and $\delta (t) = k$, then $t$ is a generic point of an irreducible component of $T$.

**Proof.**
We know $t$ is contained in an irreducible component $T' \subset T$. Let $t' \in T'$ be the generic point. Then $k \geq \delta (t') \geq \delta (t)$. Since $\delta $ is a dimension function we see that $t = t'$.
$\square$

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