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I am reading through an article by Shelah where it has a definition for a weakly locally path connected space:

Say that $X$ is weakly locally path connected (WLPC) if for every $x\in X$ and every neighborhood $u$ of $x$, there exists a neighborhood $v$ of $x$ in $u$ such that every point in $v$ can be joind to $x$ by a path through $u$.

Also I know the definition of locally path connectedness as "for every $x \in X$ and every neighborhood $u$ of $x$, there is a path connected neighborhood $v$ of $x$ contained in $u$." I wonder what is an example of a space that is not locally path connected but it is weakly locally path connected. I gave it some thought for several hours but I couldn't come up with any examples.

Kooranifar
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  • I haven't thought it through carefully, but would maybe the cone of the Hawaiian earring work? – René Bruin Jul 28 '22 at 19:35
  • @RenéBruin Is that not locally connected? – Arthur Jul 28 '22 at 19:39
  • I'm not sure to be honest. It was just a suggestion of an example to think about. Another one is $\overline{{(x,\sin(1/x);x\in]0,1]}}$ since it is not locally connected at the origin. – René Bruin Jul 28 '22 at 19:47
  • @RenéBruin I agree with Arthur about Hawaiian earring. Also I think your second example is not weakly locally path connected at $(0,0)$, since by taking $u = B_{1/2}\big( (0,0) \big)$, for any choice of $v \subset u$ there would be infinite points in $v$ that are in different $u$-path-components. – Kooranifar Jul 28 '22 at 19:51
  • Mmm, too bad. It is more difficult to come up with a example than I thought. I'm intrigued now. – René Bruin Jul 28 '22 at 19:54
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    Here's the space that is weakly locally path connected at one point but isn't locally path connected there. https://mathworld.wolfram.com/BroomSpace.html Here they say that if a space is weakly locally connected (maybe it works also for path connectedness) at ALL points then it's locally connected. https://en.wikibooks.org/wiki/Topology/Local_Connectedness – Mateo Jul 28 '22 at 20:04
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    This is similar to the difference between locally connected and connected im kleinen, examples of spaces that are cik but not locally connected will most likely work for your question too – Alessandro Codenotti Jul 28 '22 at 20:27
  • Can you give an example which will work? – René Bruin Jul 28 '22 at 20:31
  • @Kooranifar $v$ is not required to be open, but it is still required to be a neighbourhood, so it should contain an open set. Look at the "limit point" in the second example in the link above, where countably many brooms are stacked one after the other – Alessandro Codenotti Jul 28 '22 at 21:14
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    See my answer to https://math.stackexchange.com/q/2999685. – Paul Frost Jul 28 '22 at 23:18

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"Weakly locally path-connected" is actually equivalent to "locally path-connected". To prove this, suppose $X$ is weakly locally path-connected, $x\in X$, and $U$ is an open neighborhood of $x$. Let $V$ be the path-component of $x$ in $U$. I claim that $V$ is in fact open, and so is a path-connected open neighborhood of $x$ contained in $U$. To prove this, suppose $y\in V$. By weak local path-connectedness, there exists an open set $W$ such that $y\in W\subseteq U$ and every element of $W$ is connected to $y$ by a path in $U$. Then every element of $W$ is in the path-component $V$. So $V$ contains a neighborhood of each of its points, and is open.

Eric Wofsey
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    @AlessandroCodenotti: It doesn't make a difference whether they are required to be open. If they aren't open, just shrink them to smaller neighborhoods that are open. – Eric Wofsey Jul 28 '22 at 21:46
  • Ah I see your point now, you are not claiming that the local versions (WLPC at a point vs LPC at the same point) are equivalent, but that the global ones are (WLPC at all points vs LPC at all points). I agree with you then, even though I find it very surprising Shelah didn't notice – Alessandro Codenotti Jul 28 '22 at 22:00