Remark 2

If Y is a nonempty closed subset of X, one has $\operatorname{codim}(Y, X) \leq \dim(X)$. One has $\dim(X) = \sup_Y \operatorname{codim}(Y, X)$, where Y runs through the set of irreducible closed subsets of X. If Y and Y’ are two closed subsets of X such that $Y' \subset Y$, one has $\operatorname{codim}(Y, X) \leq \operatorname{codim}(Y', X)$.