Corollary

Let $X$ be a topological space and $E$ a subspace with only ajinite number of distinct irreducible components $Q_i$ ($1 \leq i \leq n$); then the irreducible components of the closure $\overline{E}$ in $X$ are the closures $\overline{Q}_i$ of the $Q_i$ ($1 \leq i \leq n$) and $\overline{Q}_i \# \overline{Q}_j$ for $i \neq j$.

$\overline{E}$ is the union of the $\overline{Q}_i$, which are irreducible (Proposition 2); as $Q_i$ is closed in $E$, $\overline{Q}_i \cap E = Q_i$; as $Q_i \notin Q_j$ for $i \neq j$, $\overline{Q}_i \notin \overline{Q}_j$, whence the corollary by virtue of Proposition 6.