Definition 3

Let X be a topological space.

a) If Y is an irreducible closed subset of X, the codimension of Y in X is defined to be the least upper bound in $\overline{\mathbf{R}}$ of the lengths of chains of irreducible closed subsets of X of which Y is the smallest element.

b) If Y is a closed subset of X, the codimension of Y in X is defined to be the greatest lower bound in $\overline{\mathbf{R}}$ of the codimensions, in X, of the irreducible components of Y, and is denoted by $\operatorname{codim}(Y, X)$.