Definition:Krull Dimension of Topological Space

Definition
Let $T$ be a topological space.

Its Krull dimension $\operatorname {dim_{Krull}} \left({T}\right)$ is the supremum of lengths of chains of closed irreducible sets of $T$, 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 $\dim$, if there is no confusion.

Also see

 * Point is Contained in Irreducible Component