Real projective plane

Example

The real projective plane $\mathbb{RP}^2$ is the set of 1-dimensional linear subspaces of $\R^3$. A point is written in homogeneous coordinates $[x : y : z]$ with $(x, y, z) \ne (0, 0, 0)$, where $[x : y : z] = [\lambda x : \lambda y : \lambda z]$ for every $\lambda \ne 0$. A line is the set of points of a 2-dimensional linear subspace of $\R^3$, that is a set $\set{[x : y : z] : ax + by + cz = 0}$ with $(a, b, c) \ne (0, 0, 0)$.

Check of the axioms of a projective plane:

1. Two distinct points are two distinct lines through $0$ in $\R^3$; they span exactly one plane through $0$, so they lie on exactly one line.
2. Two distinct lines are two distinct planes through $0$; their intersection is exactly one line through $0$, so they meet in exactly one point.
3. The four points $[1 : 0 : 0]$, $[0 : 1 : 0]$, $[0 : 0 : 1]$, $[1 : 1 : 1]$ have no three on a line, since any three of the vectors are linearly independent.