Question:
Does there exist a connected metric space of first category (i.e. it can be written as a countable union of nowhere dense subsets)?
No such example is given in the book Counterexamples in Topology. Thus, it seems no easy task. I came across this paper today, which proves the existence of a countable dense homogeneous (CDH) metric space which is connected and meager-in-itself (i.e. first-category) is independent of ZFC.
However, I don't have the required mathematical maturity to finish reading it. I believe the hard part is the CDH property. Any hint? (In particular, I prefer a counterexample in the plane $\Bbb R^2$)
Related: A locally connected metric space of first category (yet unanswered)
Let $X=\mathbb{Q}\times \mathbb{R}\cup \mathbb{R}\times\{0\}\subset\mathbb{R}^2$. Then $X$ is path-connected, but it is the union of the countably many lines $\{q\}\times\mathbb{R}$ for $q\in\mathbb{Q}$ and $\mathbb{R}\times\{0\}$, each of which is closed with empty interior.