Finite geometry

Remarks

Example: the projective space $PG(2, 2)$ over the field with two elements, the Fano plane. It has $7$ points and $7$ lines. Any two points lie on exactly one line, and any two lines meet in exactly one point. The points $[1 : 0 : 0]$, $[0 : 1 : 0]$, $[0 : 0 : 1]$, $[1 : 1 : 1]$ have no three on a line, so it is a finite projective plane.

Non-example: the real projective plane satisfies the axioms of a projective plane, but it has infinitely many points, so it is not a finite geometry.

The term also names the field of mathematics that studies these structures. Finite projective spaces of dimension at least $3$ all come from finite fields, by the Veblen–Young theorem and Wedderburn's little theorem; finite planes need not.