# Neighbourhood basis of connected sets but not locally connected

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.

• If $U$ is a connected neighborhood of $x$, isn't its interior connected? – Berci Jun 1 '18 at 9:36
• @Berci No, take two tangent and closed balls in $\mathbb{R}^2$. It is a connected subset but its interior is not. – freakish Jun 1 '18 at 9:40
• If every point has a neighborhood basis of connected sets, then the space is locally connected. See en.wikipedia.org/wiki/Locally_connected_space . – Paul Frost Jun 1 '18 at 12:47

$X$ 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).