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Math Help - Rationals, Pathwise connected

  1. #1
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    Rationals, Pathwise connected

    Problem:
    Show that the set  S = \left\{ (x,y) \in \mathbb{R}^2 | x or y \in Q \right\} is pathwise connected.
    =========================
    I can only think of a generalized path.

    Since  0 \in Q , then there is a parametric path consisting of two lines conneting the x-axis and y-axis. So the parametric path is \gamma (t) = (0,0) and the parameteric path is dependent on the values of x, y.

    If x > 0, y > 0, then \gamma: [0, x] \rightarrow [0, y]
    If x < 0, y < 0, then \gamma: [x, 0] \rightarrow [y, 0]

    and vice versa. However, I do not know how to prove this for all points.

    Thank you for reading. Any help is greatly appreciated.
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  2. #2
    MHF Contributor

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    You are almost done...

    Let us show that every point of S is connected to (0,0). You proved that this is true for the points lying on either of the two axes.

    Now, if (x,y) is in S, either x or y is rational. If x is rational, I guess you know how to connect (x,y) to the x-axis, and hence to (0,0). And you can do the same with the y-axis if y is rational.

    At that point, we are done: if M and N are in S, you can connect M to (0,0) and then (0,0) to N.

    Laurent.
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