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Cam anyone provide me the proof of:

that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.

  • 4
    Ideally you should say what you tried. Hint: you could try showing it is path connected.2012-05-16
  • 1
    This is answered in http://math.stackexchange.com/questions/16948/pi-1-mathbb-r2-mathbb-q2-is-uncountable in both the comments (by Jacob Schlather) and as an actual answer (by Jeremy Hurwitz.) The questions, themselves, are different, so I'm not sure if this should be closed as a duplicate or not.2012-05-16
  • 0
    I would suggest this question should be left open, as it's far more likely to come up in a search related to this problem.2012-05-16
  • 0
    Michael Hardy's solution seems clear to me, but if for some reason you don't get it you might prefer to consider why $\mathbb{R}\setminus(\mathbb{Z}\times\mathbb{Z})$ is connected; it might be easier to visualize.2012-05-16
  • 0
    Another near [duplicate?](http://math.stackexchange.com/q/65705/11619)2012-05-16

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