Definition 5.11.1. Let $X$ be a topological space. Let $Y \subset X$ be an irreducible closed subset. The codimension of $Y$ in $X$ is the supremum of the lengths $e$ of chains
\[ Y = Y_0 \subset Y_1 \subset \ldots \subset Y_ e \subset X \]
of irreducible closed subsets in $X$ starting with $Y$. We will denote this $\text{codim}(Y, X)$.
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