# Surjection from set of all lines to real projective space

• Nov 12th 2007, 07:38 AM
johnbarkwith
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 ????
• Jun 4th 2009, 06:35 PM
Rebesques
Ιt is mentioned in every definition! (Happy)
• Jun 5th 2009, 08:30 AM
HallsofIvy
There are several equivalent definitions of RP1. What definition are you using?
• Jun 5th 2009, 10:11 AM
Rebesques
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.