Projective plane

Remarks

Example: for a field $k$, the projective plane $\mathbb{P}^2(k)$ has as points the one-dimensional subspaces of $k^3$ and as lines the two-dimensional subspaces, with incidence given by inclusion. Check: two distinct one-dimensional subspaces span exactly one plane. Two distinct planes in $k^3$ meet in a line through $0$, by the dimension formula $2 + 2 - 3 = 1$. The points $[1:0:0]$, $[0:1:0]$, $[0:0:1]$, $[1:1:1]$ have no three collinear. For $k = \F_2$ this is the Fano plane, with $7$ points and $7$ lines.

Non-example: an affine plane satisfies clauses 1 and 3 but not clause 2, since it has parallel lines.

Clause 3 excludes degenerate cases, such as a single line with all points on it. The axioms are self-dual: exchanging points and lines turns each clause into a statement that also holds, so the dual of a projective plane is a projective plane. In a finite projective plane every line has $n + 1$ points for some $n \ge 2$, the order of the plane.