The Stacks project

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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