Lemma 5.11.2 (02I4)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.11: Codimension and catenary spaces
Lemma 5.11.2 (cite)

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Lemma 5.11.2. Let $X$ be a topological space. Let $Y \subset X$ be an irreducible closed subset. Let $U \subset X$ be an open subset such that $Y \cap U$ is nonempty. Then

$\text{codim}(Y, X) = \text{codim}(Y \cap U, U)$

Proof. The rule $T \mapsto \overline{T}$ defines a bijective inclusion preserving map between the closed irreducible subsets of $U$ and the closed irreducible subsets of $X$ which meet $U$ . Using this the lemma easily follows. Details omitted. $\square$

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