Example 5.9.5 (02HZ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.9: Noetherian topological spaces
Example 5.9.5 (cite)

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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.

Comments (2)

Comment #634 by Wei Xu on May 30, 2014 at 14:52

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.

Comment #643 by Johan on June 01, 2014 at 12:08

Yep another mistake! Thanks! Fixed here.

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