Definition 5.20.1. Let $X$ be a topological space.

1. Let $x, y \in X$, $x \not= y$. Suppose $x \leadsto y$, that is $y$ is a specialization of $x$. We say $y$ is an immediate specialization of $x$ if there is no $z \in X \setminus \{ x, y\}$ with $x \leadsto z$ and $z \leadsto y$.

2. A map $\delta : X \to \mathbf{Z}$ is called a dimension function1 if

1. whenever $x \leadsto y$ and $x \not= y$ we have $\delta (x) > \delta (y)$, and

2. for every immediate specialization $x \leadsto y$ in $X$ we have $\delta (x) = \delta (y) + 1$.

[1] This is likely nonstandard notation. This notion is usually introduced only for (locally) Noetherian schemes, in which case condition (a) is implied by (b).

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