A near-linear space consists of a set of points and a set of lines satisfying the properties:

1. any line contains at least two points, and
2. any two points are on at most one line.

A near-linear space is also called a partial plane.

A linear space is a near-linear space in which every pair of distinct points are on exactly one line. (This usage of the term has no relation to its occasional appearance in linear algebra as a synonym for a vector space.)

Examples:

1. If we take all the vertices in a graph as points, and edges as lines, it is then a near-linear space in which every line contains two points.
2. Let be a finite field. Let be the elements in the Cartesian product . The solutions to a linear equation
   
   for some , where and are not both zero, form a line in . Since any two points determine a unique line, is a linear space, called the affine plane over .

Some properties:

1. In a near-linear space, if two distinct lines intersect, they intersect in one point.
2. There is no proper inclusion of lines in a near-linear space, i.e., if and are two lines such that , then .
3. In a near-linear space ,
   
   with equality holds if and only if is a linear space.
4. Let be an arbitrary point in a linear space,
   
   where the sum is taken over all lines containing . This holds because given any point, this point forms exactly one line with every other point, so counts the number of points shares in line . Summing over all lines gives all the points except .