## Real Projective space.

E: set of ALL lines in R^(n+1).

The question:

Consider the natural projection p: E → RP^n sending every line to the parallel line throughthe origin. Describe the pre-image p^−1(L0) of an arbitrary point L0 ∈ RP^n and show that p^−1(L0)has a structure of a vector space (which does not depend on any choices) of dimension n.

The pre-image of every point from RP^n (which is in fact a line) is a hyper-plane of E , so a vector space of dimension n. The independence of choice is a consequence that RP^n is the quotient R^n+1\{0} / ~ , where ~ denotes x ~kx for some non negative k in R^n+1.