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Math Help - Topology and Analysis

  1. #1
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    Topology and Analysis

    Can some one please assist me with this problem

    Let S=R^2\Q^2 (Points (x,y) in S have at least one irrational coordinate.) Is S connected? Prove or disprove.
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  2. #2
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    The must be some point C=\left( {\alpha ,\beta } \right) \in S that has both coordinates irrational.
    Now suppose that P=\left( {h,k} \right) \in S is any other point in S. Say that h is irrational, at least one coordinate is.
    Consider the three points \left( {h,k} \right)\,,\,\left( {h,\beta } \right)\,\& \,\left( {\alpha ,\beta } \right).
    Construct two line segments: l_1 (t) = \left( {t\left( {\alpha  - h} \right) + h,\beta } \right)\,\& \,l_2 (t) = \left( {h,t\left( {\beta  - k} \right) + k} \right)\,;\,0 \le t \le 1.
    Every point on either line segment has one of its coordinates irrational. Thus both are subsets of S.
    The union of those two line segments is connected by \left( {h,\beta } \right), thus P is connect to C by way of the path.
    Does this mean that S is connected?
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