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

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#1

$Old$ November 12th, 2007, 09:38 AM

johnbarkwith $johnbarkwith is offline$

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$Default$ 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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$Old$ June 4th, 2009, 08:35 PM

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Ιt is mentioned in every definition!

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$Old$ June 5th, 2009, 10:30 AM

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There are several equivalent definitions of RP1. What definition are you using?

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$Old$ June 5th, 2009, 12:11 PM

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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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