can anyone think of a natural surjection from the set of all lines in R^2 to the Real projective space RP1 ????
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.