Lemma 5.15.15. Let $X$ be a topological space. Suppose that $Z \subset X$ is irreducible. Let $E \subset X$ be a finite union of locally closed subsets (e.g. $E$ is constructible). The following are equivalent

The intersection $E \cap Z$ contains an open dense subset of $Z$.

The intersection $E \cap Z$ is dense in $Z$.

If $Z$ has a generic point $\xi $, then this is also equivalent to

We have $\xi \in E$.

**Proof.**
The implication (1) $\Rightarrow $ (2) is clear. Assume (2). Note that $E \cap Z$ is a finite union of locally closed subsets $Z_ i$ of $Z$. Since $Z$ is irreducible, one of the $Z_ i$ must be dense in $Z$. Then this $Z_ i$ is dense open in $Z$ as it is open in its closure. Hence (1) holds.

Suppose that $\xi \in Z$ is a generic point. If the equivalent conditions (1) and (2) hold, then $\xi \in E$. Conversely, if $\xi \in E$ then $\xi \in E \cap Z$ and hence $E \cap Z$ is dense in $Z$.
$\square$

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