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Given a locally connected space $X$ with two or more points is it always true that for any point $x$ in $X$ the subspace $X \setminus \{x\}$ is also locally connected?

I have proved it for Hausdorff spaces but I would like to know if it is true for any topological space.

balddraz
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Pepe
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  • I haven't thought much about it at the moment, but intuitively I doubt it holds. I think you probably need the extra Hausdorff condition – Riemann'sPointyNose May 22 '21 at 15:12
  • It's true not only for Hausdorff spaces but for every space where every point is closed. This last condition means that the space is T1. – Pepe May 22 '21 at 17:09

1 Answers1

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No. For instance, let $Y$ be any space that is not locally connected, and let $X=Y\cup\{x\}$ where $x$ is a new point and a subset of $X$ is open iff either it is empty or it has the form $U\cup\{x\}$ where $U\subseteq Y$ is open. Then any two nonempty open subsets of $X$ intersect, so $X$ is trivially locally connected. But $X\setminus\{x\}=Y$ is not locally connected.

It is true if $X\setminus\{x\}$ is open in $X$ (in particular, if $X$ is $T_1$), since any open subspace of a locally connected space is locally connected (you can just restrict your connected local bases at each point to the sets that are contained in the subspace).

Eric Wofsey
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