Lemma 5.9.6 (04MF)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.9: Noetherian topological spaces
Lemma 5.9.6 (cite)

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Lemma 5.9.6. Let $X$ be a locally Noetherian topological space. Then $X$ is locally connected.

Proof. Let $x \in X$ . Let $E$ be a neighbourhood of $x$ . We have to find a connected neighbourhood of $x$ contained in $E$ . By assumption there exists a neighbourhood $E'$ of $x$ which is Noetherian. Then $E \cap E'$ is Noetherian, see Lemma 5.9.2. Let $E \cap E' = Y_1 \cup \ldots \cup Y_ n$ be the decomposition into irreducible components, see Lemma 5.9.2. Let $E'' = \bigcup _{x \in Y_ i} Y_ i$ . This is a connected subset of $E \cap E'$ containing $x$ . It contains the open $E \cap E' \setminus (\bigcup _{x \not\in Y_ i} Y_ i)$ of $E \cap E'$ and hence it is a neighbourhood of $x$ in $X$ . This proves the lemma. $\square$

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