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 neighborhood of $x$ then $N$ 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)