The problem with which this paper is concerned is that of investigating a class of topological properties commonly called separation properties. A topological space which satisfies only the definition may be very limited in open sets. By use of the separation properties, specific families of open sets can be guaranteed.
The purpose of this thesis is to present an axiomatic foundation for the development of the natural numbers from two points of view. It makes no claim at originality other than at the point of organization and presentation of previously developed works.