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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