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Show set is a polygon connected set?

  1. Sep 24, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the complement of the set S = {all (x, y) with x and y rational numbers} is a polygon connected set. Is it an open set?

    2. Relevant equations



    3. The attempt at a solution

    The complement of S = {all (x, y) with x not rational or y not rational}
    Let the points (x, y) and (z, w) exist in the complement of S.

    Case 1:

    Suppose x is not rational and z is not rational.

    Case 2:

    Suppose x is not rational and w is not rational.

    Case 3:

    Suppose y is not rational and z is not rational.

    Case 4:

    Suppose y is not rational and w is not rational.

    I don't really know how to actually prove my four cases.
     
  2. jcsd
  3. Sep 24, 2009 #2
    Example: (pi,2) to (e,4)

    Line segment from (pi,2) to (pi,pi), and another line segment from (pi,pi) to (e,pi), and finally another segment from (e,pi) to (e,4).
     
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