Let $X$ be an irreducible topological space and $Y$ be closed subset of $X.$ Is $Y$ irreducible? I can prove that open subset of irreducible is irreducible but I'm unable to prove this for a closed subset. If the statement is not true can you give a counterexample?
2026-03-30 23:12:40.1774912360
Is closed subset of irreducible irreducible?
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Let $\mathbb{C}$ be the affine line ($Spec \mathbb{C}[x]$), and $V(x(x-1))$ the closed subset $\{0,1\}$. Then the former is irreducible, while the latter isn't, since it can be written as $\{0\}\cup \{1\}.$