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

1. 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 $i = 0, \ldots , n - 1$.

2. 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 $n$.

3. 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.

4. 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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