Linear space (geometry)

A linear space is a basic structure in incidence geometry. A linear space consists of a set of elements called points, and a set of elements called lines. Each line is a distinct subset of the points. The points in a line are said to be incident with the line. Each two points are in a line, and any two lines may have no more than one point in common. Intuitively, this rule can be visualized as the property that two straight lines never intersect more than once.

Linear spaces can be seen as a generalization of projective and affine planes, and more broadly, of 2-${\displaystyle (v,k,1)}$ block designs, where the requirement that every block contains the same number of points is dropped and the essential structural characteristic is that 2 points are incident with exactly 1 line.

The term linear space was coined by Paul Libois in 1964, though many results about linear spaces are much older.