The concept was described by
G. B. Halsted at the outset of his
Synthetic Projective Geometry: Given any pair of points on a projective line, they separate a third point from its
harmonic conjugate. A pair of lines in a
pencil separates another pair when a
transversal crosses the pairs in separated points. The point-pair separation of points was written AC//BD by
H. S. M. Coxeter in his textbook
The Real Projective Plane.
Application The relation may be used in showing the
real projective plane is a
complete space. The axiom of continuity used is "Every monotonic sequence of points has a limit." The point-pair separation is used to provide definitions: • {
An} is
monotonic ≡ ∀
n > 1 A_0 A_n // A_1 A_{n+1}. •
M is a
limit ≡ (∀
n > 2 A_1 A_n // A_2 M) ∧ (∀ P A_1P // A_2 M ⇒ ∃
n A_1 A_n // P M ). ==Unoriented circle==