Definition:Krull Dimension of Topological Space

Definition
Let $T$ be a topological space.

Its Krull dimension $\map {\dim_{\mathrm {Krull} } } T$ is the supremum of lengths of chains of closed irreducible sets of $T$, ordered by the subset relation.

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