Lemma 5.19.3. Let $T \subset X$ be a subset of a topological space $X$. The following are equivalent

$T$ is stable under specialization, and

$T$ is a (directed) union of closed subsets of $X$.

Lemma 5.19.3. Let $T \subset X$ be a subset of a topological space $X$. The following are equivalent

$T$ is stable under specialization, and

$T$ is a (directed) union of closed subsets of $X$.

**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.
$\square$

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