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Does there exist a partition of the plane into $n=3$ (or more generally $n\ge 3$) disjoint path-connected dense subsets?

Note that the answer is yes if "path-connected" is replaced by "connected", as shown here. The linked question also shows that the answer is yes for $n=1$ (trivial) and $n=2$ (not quite trivial, but nice and explicit.)

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    I don't have an answer now nor do I have time now to give this much thought, but perhaps some of the ways that ${\mathbb Q}^{2} \cup \left({\mathbb R}\;-\;{\mathbb Q}\right)^{2}$ can be proved path-connected could be of help. See my [23 June 2009 sci.math post](http://mathforum.org/kb/message.jspa?messageID=6762982) and the earlier posts I cite in it (two from 2002 and one from 2005).2012-12-04
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    Thanks Dave, that sounds very interesting, I'll have a look at it later.2012-12-05
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    @DaveL.Renfro: Tell me if I'm wrong, but isn't there a straight-forward proof that $\mathbb{Q}^2 \cup (\mathbb{R} - \mathbb{Q})^2$ is path-connected, using the fact that any line segment with rational endpoints which is not vertical or horizontal is contained in that set? Then for any two points in your set, approximate them with points with rational coordinates, connect them with line segments, just making sure that you don't get horizontal or vertical line segments in the approximation (which is pretty easy). Or am I missing something?2012-12-05
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    This is the proof I gave in my 14 May 2002 sci.math post, where I wrote: *I asked my Ph.D. advisor whether he thought the result was worthy of publication. He said he'd think about it for a day or two and see if he could come up with a more straightforward proof. The next day he said there's a fairly easy way to do it, so it's probably not publishable.* I gave a more detailed version of this proof in my 19 October 2005 sci.math post, where I also said it can be found on p. 187 (Example 2 in Chapter 6.3) of Allan J. Sieradski, **An Introduction to Topology and Homotopy** (1992).2012-12-05
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    @DaveL.Renfro: Oh yeah, I missed that short paragraph with the easy proof when I skimmed through the post.2012-12-05

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