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Homemorphism between the set of circles with rational points and rectangles

  1. Aug 31, 2011 #1
    I am trying to solve a problem in which we need to prove that the set of all circles with rational points and radii is homemorphic to the set of all rectangles with vertices at rational points with the length of the diagonals as rational number. I am not able to figure out what the approach should be. Any help or hints to solve the problem would be deeply appreciated.
     
  2. jcsd
  3. Sep 1, 2011 #2
    Well, if you want to show a homeomorphism, youu need to have a topology defined
    on each of the two sets.
     
  4. Sep 1, 2011 #3
    I guess you mean each circle and each triangle is a subspace of the plane. Then I guess
    we could just show that each circle can be continuously deformed to a triangle, and viceversa. Maybe we can embed each circle in a triangle.
     
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