Neighbourhood basis of connected sets but not locally connected

Question

I'm looking for a topological space in which every point has a neighbourhood basis of connected sets but is not locally connected. Note that locally connected means that every point has a neighbourhood basis of open connected sets.

Is there such a space could you define locally connected by neighbourhood basis of connected sets?

What I did: I read the first pages of the chapter about connectedness in Counterexampes in Topology but coulnd't find this discussed there.

Answer 1

Read the Wikipedia page, where you will learn the following:

$X$ is called weakly locally connected at $x$ when $x$ has a local base of connected (not necessarily open) neighbourhoods, while $x$ is locally connected at $x$ when $x$ has a local base of open connected neighbourhoods.

Theorem: if $X$ is weakly locally connected at each point, then $X$ is locally connected at each point. (proof at the linked page).

So, if you only want to define the notion for the whole space and don't care that much about indivual points then you could define it either way. So, a space such as you ask for at the start does not exist.

But the broom space has a point where it is weakly locally connected (or "connected im kleinen") but not locally connected. So at individual points we can sometimes see a difference.