Definition:Krull Dimension of Topological Space

Definition
Let $T$ be a topological space.

Its Krull dimension $\operatorname{dim_{Krull}}(T)$ is the supremum of lengths of chains of closed irreducible subsets, ordered by inclusion.

Thus, the Krull dimension is $\infty$ if there exist arbitrarily long chains.

Also denoted as
The Krull dimension can also be denoted $\operatorname{K-dim}$ or simply $\dim$, of there is no confusion.

Also see

 * Point is Contained in Irreducible Component