# Connected metric space of first category

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.
• @DavidMitra: Any ball in the plane intersects infinitely many of the vertical lines, so no single line contains all of the intersection with $X$. – Eric Wofsey May 9 at 17:15