# Definition:Local Dimension of Topological Space

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## 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 the subset relation.

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