The Stacks project

Proof. Suppose that $T$ is stable under specialization, then for all $y\in T$ we have $\overline{\{ y\} } \subset T$. Thus $T = \bigcup _{y\in T} \overline{\{ y\} }$ which is an union of closed subsets of $X$. Reciprocally, suppose that $T = \bigcup _{i\in I}F_ i$ where $F_ i$ are closed subsets of $X$. If $y\in T$ then there exists $i\in I$ such that $y\in F_ i$. As $F_ i$ is closed, we have $\overline{\{ y\} } \subset F_ i \subset T$, which proves that $T$ is stable under specialization. $\square$