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Math Help - Metric Geometry Proof

  1. #1
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    Metric Geometry Proof

    How would one go about proving that a line in a metric geometry has infinitely many points? I assume it has something to do with the fact that the ruler applied to the line maps that line to the set of all reals and thus since the reals are infinite the line must contain that same number of points. Am I on the right track?

    Thanks,
    Ultros
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  2. #2
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    Quote Originally Posted by Ultros88 View Post
    How would one go about proving that a line in a metric geometry has infinitely many points? I assume it has something to do with the fact that the ruler applied to the line maps that line to the set of all reals and thus since the reals are infinite the line must contain that same number of points. Am I on the right track?
    All proofs is geometry depend on the particular set of definitions and axioms in use on the textbook. From the wording of this question it seems as if you have what Edwin Moise calls the Ruler Axiom: Every line has a coordinate system.
    So yes you are correct that this means there is a one-to-one correspondence between the points on a line and the real numbers.
    Note that between any two numbers that is a third number: a < b\quad  \Rightarrow \quad a < \frac{{a + b}}{2} < b.
    This means that between any two points on a line there is a third point.
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