Example 5.9.5. Any nonempty, Kolmogorov Noetherian topological space has a closed point (combine Lemmas 5.12.8 and 5.12.13). Let $X = \{ 1, 2, 3, \ldots \}$. Define a topology on $X$ with opens $\emptyset$, $\{ 1, 2, \ldots , n\}$, $n \geq 1$ and $X$. Thus $X$ is a locally Noetherian topological space, without any closed points. This space cannot be the underlying topological space of a locally Noetherian scheme, see Properties, Lemma 28.5.9.

Comment #634 by Wei Xu on

Dear project, A typo, "Any Noetherian topological space has a closed point" might possibly be "Any nonempty, Kolmogorov, Noetherian topological space has a closed point". For example, $X=\{0,1\}$ with opens $\{\emptyset, X\}$ is Noetherian but does not have a closed point.

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