Let $X$ be a topological space; then we say that $X$ is locally connected at $x$ if $x$ admits a neighborhood basis of open, connected sets. In this sense, a space is locally connected iff it is locally connected at every point.
Consider the following.
- It is easy to see that a space that is both locally connected and totally disconnected must be discrete.
- Cantor's leaking tent is an example of a space which is totally disconnected but not locally connected at any point.
- Several spaces are locally connected but not totally disconnected.
I've thought about spaces not locally connected at any point, since they are necessarily "much weirder" than the usual non-locally connected space. Perhaps this condition is equivalent to another one?
I suspect there must be a space which is neither locally connected at any point nor totally disconnected, but I haven't been able to produce or find an example.
Thank you in advance.