The Stacks project

Exercise 111.18.5. Give an example of a finite, sober, catenary topological space $X$ which does not have a dimension function $\delta : X \to \mathbf{Z}$. Here $\delta : X \to \mathbf{Z}$ is a dimension function if for $x, y \in X$ we have

  1. $x \leadsto y$ and $x \not= y$ implies $\delta (x) > \delta (y)$,

  2. $x \leadsto y$ and $\delta (x) \geq \delta (y) + 2$ implies there exists a $z \in X$ with $x \leadsto z \leadsto y$ and $\delta (x) > \delta (z) > \delta (y)$.

Describe your space clearly and succinctly explain why there cannot be a dimension function.

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View Exercise 111.18.5 as pdf