Proposition 7

Let $U$ be an open subset of a topological space $X$. The mapping $V \mapsto \overline{V}$ (closure in $X$) is a bijection of the set & irreducible subsets of $U$ which are closed in $U$ onto the set & irreducible subsets & $X$ which are closed in $X$ and meet $U$; the inverse bijection is $Z \mapsto Z \cap U$. In particular, this bijection maps the set & irreducible components & $U$ onto the set & irreducible components & $X$ which meet $U$.

If $V$ is closed in $U$ and irreducible, $\overline{V}$ is irreducible (Proposition 2) and $V = \overline{V} \cap U$. Conversely, if $Z$ is irreducible and closed in $X$ and meets $U$, $Z \cap U$ is a non-empty open subset of $Z$, hence irreducible (Proposition 3) and dense in $Z$ and, as $Z$ is closed, $Z = \overline{Z} \cap U$. This proves the proposition.