Remark 1

The codimension of a closed subset Y of X is therefore the greatest lower bound of the codimensions of the irreducible closed subsets of Y. One has $\operatorname{codim}(\varnothing, X) = +\infty$ and, if X is not empty, $\operatorname{codim}(X, X) = 0$. Every nonempty closed subset of X contains an irreducible closed subset (II, § 4, No. 1, Prop. 5); the codimension in X of a closed subset Y is therefore always a positive integer or $+\infty$; it is zero if and only if Y contains an irreducible component of X.