Definition 2

The Krull dimension of the topological space X is, by definition, and is denoted by dim kr(X), or simply dim(X), the least upper bound in $\overline{\mathbf{R}}$ of the set of lengths of chains of irreducible closed subsets of X. For every point x of X, the Krull dimension of X at x is, by definition, and is denoted by $\dim_x(X)$, the greatest lower bound of the dimensions of the open neighbourhoods of x.

We have $\dim(\varnothing) = -\infty$. On the other hand, if X is nonempty, the closure of every point of X is an irreducible closed subset of X (II, § 4, No. 1, Prop. 2), and therefore the dimension of X is either $+\infty$ or a positive integer. Suppose that X is separated and nonempty; then every irreducible subset of X is reduced to a point, and X is of dimension 0. The definition of Krull dimension is therefore devoid of interest for separated spaces, but it is specially adapted to the topological spaces encountered in Commutative Algebra (spectra of rings, \* schemes *, ...). In this chapter, no confusion is to be feared with other notions of dimension of topological spaces (for example, that of Lebesgue), and we shall simply say “dimension” for “Krull dimension”.