In a cyclic order, such as the real projective line, two pairs of points separate each other when they occur alternately in the order.
Thus the ordering a b c d of four points has (a,c) and (b,d) as separating pairs.
This point-pair separation is an invariant of projectivities of the line.
The concept was described by G. B. Halsted at the outset of his Synthetic Projective Geometry: With regard to a pair of different points of those on a straight, all remaining fall into two classes, such that every point belongs to one and only one.
If two points belong to different classes with regard to a pair of points, then also the latter two belong to different classes with regard to the first two.