Example 5.10.4. Let $X = \{ s, \eta \}$ with open sets given by $\{ \emptyset , \{ \eta \} , \{ s, \eta \} \}$. In this case a maximal chain of irreducible closed subsets is $\{ s\} \subset \{ s, \eta \}$. Hence $\dim (X) = 1$. It is easy to generalize this example to get a $(n + 1)$-element topological space of Krull dimension $n$.

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