This is a follow-up on an [older question](https://mathoverflow.net/questions/297236/is-box-n-in-omega0-1-connected).


Let $\Box\_{i\in I} X\_i$ denote the [box product](https://en.wikipedia.org/wiki/Box_topology) of the spaces $X\_i$. Is there a Hausdorff space $(X,\tau)$ with $|X|>1$ such that $\Box\_{n\in\omega}X$ connected?



Connected box products of Hausdorff spaces
If $X$ is the [Irrational slope topology](https://topology.pi-base.org/spaces/S000067) then the closures of any two non-empty open sets must intersect. It easily follows that $\Box\_{n\in\omega}X$ is connected. Note that $X$ is Hausdorff but not regular. It seems (see the comments to the OP) that there are no $T\_3$ examples.


