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Math Help - Surjection from set of all lines to real projective space

  1. #1
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    Surjection from set of all lines to real projective space

    can anyone think of a natural surjection from the set of all lines in R^2 to the Real projective space RP1 ????
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  2. #2
    Super Member Rebesques's Avatar
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    Ιt is mentioned in every definition!
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  3. #3
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    There are several equivalent definitions of RP1. What definition are you using?
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  4. #4
    Super Member Rebesques's Avatar
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    Well...
    the geometric one?
    Wiki says RP^1=(R^2-{0})/~, where ~ is the "same line through the origin" equivalence relation.


    The question goes "find a natural surjection",
    so the keyword "natural"
    demands that every line in R^2 be identified as an element of RP^1.

    Ofcourse, if that line is through the origin, the identification is obvious.
    If it is not, the slope provides an equivalence relation of all lines in the plane, and there is a unique representative of each class to pass through the origin.
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