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I am trying to prove that local connectedness is a topological property. Then what do I need to show? Do I need to show that if $X$ and $Y$ are homeomorphic topological spaces (let $f$ be the homeomorphism) and $X$ is locally connected at $x$ in $X$ then $Y$ is locally connected at $f(x)$ or do I need to show that if $X$ is locally connected (i.e. $X$ is locally connected at each point in $X$) then $Y$ is also locally connected (i.e. $Y$ is locally connected at each point in $Y$)?

Also provide me some hints on how to do the proof.

Thanks in advance!

ADi
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Esha
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1 Answers1

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You have to show that if $X$ is a locally connected topological space and if $Y$ is a topological space homeomorphic to $X$, then $Y$ is locally connected too. In order to do that, let $f\colon X\longrightarrow Y$ be a homeomorphism. You are assuming that, for each $x\in X$, $x$ has a neighborhood basis $\mathcal N$ consisting entirely of open, connected sets. And you want to prove that the same thing holds in $Y$. So, take $y\in Y$, take $x=f^{-1}(y)$, take a neighborhood basis $\mathcal N$ of $x$ consisting entirely of open, connected sets, and prove that $\{f(V)\mid C\in\mathcal N\}$ is a neighborhood basis $\mathcal N$ of $y$ consisting entirely of open, connected sets.

José Carlos Santos
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  • What do you mean by 'x has a neighbourhood basis N consisting entirely of open connected sets' ? I mean I know that X is locally connected iff X has a basis consisting of connected sets of X. But I do not understand what you mean by the above statement. Can you please elaborate? – Esha Jan 17 '22 at 19:04
  • I used the usual definition of locally connected topological space. – José Carlos Santos Jan 17 '22 at 19:11
  • This question is a dupe. https://math.stackexchange.com/q/1909960/121671 – Mittens Jan 17 '22 at 19:25
  • @OliverDiaz Perhaps, but you have linked to a question about local compactness. – José Carlos Santos Jan 17 '22 at 19:29
  • Willard uses the open connected definition, Engelking the connected neighbourhood definition. I'm not sure if there are spaces where this makes a difference...@Esha – Henno Brandsma Jan 17 '22 at 22:06
  • +1 to this no-nonsense post @JoséCarlosSantos. However, I am bit puzzled as to why the question was closed as dupe to that of local compactness. – User1865345 Oct 17 '23 at 10:58
  • @User1865345 I think that those who voted to close it thought that the approach to that other question can be also used here. – José Carlos Santos Oct 17 '23 at 20:43