# Definition:Local Dimension of Topological Space

## Definition

Let $X$ be a topological space.

Let $x \in X$.

The local dimension of $X$ at $x$ is the supremum of lengths of chains of closed irreducible sets of $T$ containing $x$, ordered by inclusion.

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