Projective geometry

Remarks

Example: let $V$ be a vector space of dimension at least $3$ over a field $k$. The points of the projective space $\mathbb{P}(V)$, with the lines $\mathbb{P}(W)$ for $\dim W = 2$, form a projective geometry. Check: two distinct points span a unique $2$-dimensional $W$. If $ab$ and $cd$ meet, then $a, b, c, d$ lie in a subspace $U$ of dimension at most $3$. The planes for $ac$ and $bd$ lie in $U$, so they meet in a nonzero vector. A $2$-dimensional $W$ has $\abs{k} + 1 \ge 3$ points.

Non-example: the real affine plane $\R^2$ satisfies clauses 1 and 3 but fails Veblen's axiom. Take $a = (0,0)$, $b = (1,0)$, $c = (0,1)$ and $d = (1,2)$. The lines $ab$ (given by $y = 0$) and $cd$ (given by $y = x + 1$) meet at $(-1, 0)$, but the lines $ac$ (given by $x = 0$) and $bd$ (given by $x = 1$) are parallel.

A projective geometry of dimension $2$ is the same as a projective plane. By the Veblen–Young theorem, every projective geometry of dimension at least $3$ is some $\mathbb{P}(V)$ over a division ring.