The idea of weakly locally connected (connected im kleinen) points, and counterexamples.

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)
• Sounds like the sort of technical condition needed to do things like classifying all connected covers of a space. – Randall Aug 31 '17 at 13:52
• Looks like there are Julia sets that are im kleinen at a point. semanticscholar.org/paper/… – Michael Lee Aug 31 '17 at 13:59
• The broom space is an example. See the link in the link you gave for the definition of connected in the small. – William Elliot Aug 31 '17 at 20:20