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1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeDec 30th 2020
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexAs written, I do not believe Theorem 4.1 is true. Certainly, the coreflection exists but it is unclear why the topology generated by the connected components of the open subsets of $X$ is in fact a locally connected space. It is only obvious that locally connected spaces are the fixed points of this construction. Either this case was being mistaken for the locally path-connected case or the mistake was made of assuming that connected subspaces of $X$ still need to be connected as subspaces of $R(X)$. Looking at the literature (Gleason's paper "Universally locally connected refinements") this simple refinement is used to show that the coreflection exists. However, the simple refinement and coreflection don't seem to be the same. Rather, the coreflection is only guaranteed to be the infimum (in the lattice of topologies) of locally connected topologies larger than the topology of $X$. Jeremy Brazas <a href="https://ncatlab.org/nlab/revision/diff/locally+connected+topological+space/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/locally+connected+topological+space/7">v7</a>, <a href="https://ncatlab.org/nlab/show/locally+connected+topological+space">current</a>

   As written, I do not believe Theorem 4.1 is true. Certainly, the coreflection exists but it is unclear why the topology generated by the connected components of the open subsets of $X$ is in fact a locally connected space. It is only obvious that locally connected spaces are the fixed points of this construction. Either this case was being mistaken for the locally path-connected case or the mistake was made of assuming that connected subspaces of $X$ still need to be connected as subspaces of $R(X)$. Looking at the literature (Gleason’s paper “Universally locally connected refinements”) this simple refinement is used to show that the coreflection exists. However, the simple refinement and coreflection don’t seem to be the same. Rather, the coreflection is only guaranteed to be the infimum (in the lattice of topologies) of locally connected topologies larger than the topology of $X$.

   Jeremy Brazas

   diff, v7, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 30th 2020
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   Author: Urs
   Format: MarkdownItexThanks for the alert. I forget what I was thinking there. But it looks like you already fixed it in [rev 7](https://ncatlab.org/nlab/revision/diff/locally+connected+topological+space/7)?

   Thanks for the alert. I forget what I was thinking there. But it looks like you already fixed it in rev 7?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 30th 2020
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   Author: Todd_Trimble
   Format: MarkdownItexI think I was the one who wrote that originally. Urs extracted that content from [[connected space]] into a separate article (and thus breaking some links in the process, if I recall correctly). Thanks for pointing out the (to me) subtle error, and for correcting it.

   I think I was the one who wrote that originally. Urs extracted that content from connected space into a separate article (and thus breaking some links in the process, if I recall correctly). Thanks for pointing out the (to me) subtle error, and for correcting it.