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I've recently had the chance to learn about the weak local connectedness (connected im kleinen) property, of points in topological spaces. Also, I've been trying to construct a space which has some specific point which is weakly locally connected but not locally connected.

This sort of construction seems a bit difficult, because for some point $x \in X$ (where $X$ is our space) to be weakly locally connected but not locally connected, there must be some open $x \in U \subseteq X$ such that if $N \subseteq U$ is a nbhd of $x$ than $N$ is not open. Also, at least one such connected $N$ should exist for any arbitrary selection of $U$.

After some time I've found the infinite broom space to provide this sort of example, but actually nothing else (other than constructions of pretty similiar patholgical subspaces of $\mathbb{R}^2$).

Question:

  • Are there any other known examples of weak local connectedness without local connectedness? (I've failed to find)
  • What is the significance of this concept when studying connectedness of topological spaces? (the assumption that there is some significance, is because someone bothered to name it)
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  • $\begingroup$ Sounds like the sort of technical condition needed to do things like classifying all connected covers of a space. $\endgroup$ – Randall Aug 31 '17 at 13:52
  • $\begingroup$ Looks like there are Julia sets that are im kleinen at a point. semanticscholar.org/paper/… $\endgroup$ – Michael Lee Aug 31 '17 at 13:59
  • $\begingroup$ The broom space is an example. See the link in the link you gave for the definition of connected in the small. $\endgroup$ – William Elliot Aug 31 '17 at 20:20

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