Space which is neither locally connected at any point nor totally disconnected

Question

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.

Answer 1

How about this space? $$\{(x,y)\in\Bbb R^2\mid x\in\{\frac1n+\frac1m\mid n,m\in\Bbb N\}, y\in[0,1] \}$$

Answer 2

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.