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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)

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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.

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  • $\begingroup$ OK, it seems I've asked a stupid question... Do you have any idea regarding the linked one (the locally connected one)? $\endgroup$ – YuiTo Cheng May 9 at 16:06
  • $\begingroup$ @YuiToCheng: Yep, a small modification of my example works--see the answer I've added there. $\endgroup$ – Eric Wofsey May 9 at 17:14
  • $\begingroup$ @DavidMitra: Any ball in the plane intersects infinitely many of the vertical lines, so no single line contains all of the intersection with $X$. $\endgroup$ – Eric Wofsey May 9 at 17:15
  • $\begingroup$ Right. Sorry for being dense. $\endgroup$ – David Mitra May 9 at 17:18

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