Non-Desarguesian planes give examples of 2-dimensional projective spaces that do not arise from vector spaces over division rings, showing that the restriction to dimension at least 3 is necessary.
A projective spaceS can be defined abstractly as a set P (the set of points), together with a set L of subsets of P (the set of lines), satisfying these axioms :
Each two distinct points p and q are in exactly one line.
Veblen's axiom: If a, b, c, d are distinct points and the lines through ab and cd meet, then so do the lines through ac and bd.
Any line has at least 3 points on it.
The Veblen–Young theorem states that if the dimension of a projective space is at least 3 (meaning that there are two non-intersecting lines) then the projective space is isomorphic with the projective space of lines in a vector space over some division ringK.