Lemma 5.7.11. Let $X$ be a topological space. If $X$ is locally connected, then

1. any open subset of $X$ is locally connected, and

2. the connected components of $X$ are open.

So also the connected components of open subsets of $X$ are open. In particular, every point has a fundamental system of open connected neighbourhoods.

Proof. Omitted. $\square$

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