Proposition 1.10. If $Y$ is a quasi-affine variety, then $\dim Y=\dim \overline{Y}$.
If $Z_{0}\subset Z_{1}\subset \cdots \subset Z_{n}$ is a sequence of distinct closed irreducible subsets of $Y$, then $\overline{Z_{0}}\subset \overline{Z_{1}}\subset\cdots \subset \overline{Z_{n}}$ is a sequence of distinct closed irreducible subsets of $\overline{Y}$.
Question. if $Z_i$ is irreducible in $Y$, $Z_i$ is irreducible in $\overline{Y}$? If, yes, why?
Best Answer
Yes! More generally:
Claim Let $X$ be a topological space with subspaces $A \subseteq X' \subseteq X$. If $A$ is irreducible as a subset of $X'$, then $A$ is irreducible as a subset of $X$.
Proof. Suppose $A = Z_1 \cup Z_2$ where $Z_1$ and $Z_2$ are closed in $X$. Then $Z_1, Z_2$ are also closed subspaces of $X'$, and the irreducibility of $A$ as a subset of $X'$ then implies that $A = Z_1$ or $A = Z_2$. We conclude that $A$ is irreducible as a subset of $X$.