If $R$ is a ring, the Krull dimension (or simply dimension) of $R$ , $\dim R$ is the supremum of all integers $n$ such that there is an increasing sequence of prime ideals $\mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_n$ of length $n$ in $R$ .

If $X$ is a topological space, the Krull dimension (or simply dimension) of $X$ , $\dim X$ is the supremum of all integers $n$ such that there is a decreasing sequence of irreducible closed subsets $F_0 \supsetneq \cdots \supsetneq F_n$ of $X$ .