Krull dimension

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

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