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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- Nov 12th 2007, 08:38 AMjohnbarkwithSurjection 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 ????

- Jun 4th 2009, 07:35 PMRebesques
Ιt is mentioned in every definition! (Happy)

- Jun 5th 2009, 09:30 AMHallsofIvy
There are several equivalent definitions of RP1. What definition are you using?

- Jun 5th 2009, 11:11 AMRebesques
Well...

the geometric one?(Nerd)

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.