Definition 5.10.1. Let $X$ be a topological space.

A

*chain of irreducible closed subsets*of $X$ is a sequence $Z_0 \subset Z_1 \subset \ldots \subset Z_ n \subset X$ with $Z_ i$ closed irreducible and $Z_ i \not= Z_{i + 1}$ for $i = 0, \ldots , n - 1$.The

*length*of a chain $Z_0 \subset Z_1 \subset \ldots \subset Z_ n \subset X$ of irreducible closed subsets of $X$ is the integer $n$.The

*dimension*or more precisely the*Krull dimension*$\dim (X)$ of $X$ is the element of $\{ -\infty , 0, 1, 2, 3, \ldots , \infty \} $ defined by the formula:\[ \dim (X) = \sup \{ \text{lengths of chains of irreducible closed subsets}\} \]Thus $\dim (X) = -\infty $ if and only if $X$ is the empty space.

Let $x \in X$. The

*Krull dimension of $X$ at $x$*is defined as\[ \dim _ x(X) = \min \{ \dim (U), x\in U\subset X\text{ open}\} \]the minimum of $\dim (U)$ where $U$ runs over the open neighbourhoods of $x$ in $X$.

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