TITLE: Space which is neither locally connected at any point nor totally disconnected
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: Here's another example: $$X=(\mathbb R\times\{0\})\cup(\mathbb Q\times\mathbb Q).$$ Any neighborhood of any point is disconnected, but $\mathbb R\times\{0\}$ is a connected component, so the space is not totally disconnected.