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Connectedness of coordinates with one rational point

  1. Dec 15, 2008 #1
    Hi all, i found this problem in a topology book, but it seems to be of an analysis flavour. I'm stumped.

    Show that the collection of all points in R^2 such that at least one of the coordinated is rational is connected.

    My gut says that it should be path-connected too (thus connected), but im finding the proof elusive... any thoughts?

    cheers
     
  2. jcsd
  3. Dec 15, 2008 #2
    It is path-connected. Try paths consisting of horizontal and vertical segments moving along straight lines. For instance, to move from (0, √2) to (π, 1/2), you could first move along the straight line segment from (0, √2) to (0, 1/2), and then along the straight line segment from (0, 1/2) to (π, 1/2). Now find a way to generalize that line of thought.
     
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