In topology, a proximity space, also called a nearness space, is an axiomatization of notions of "nearness" that hold set-to-set, as opposed to the better known point-to-set notions that characterize topological spaces.
The concept was described by Frigyes Riesz (1909) but ignored at the time. It was rediscovered and axiomatized by V. A. Efremovič in 1934 under the name of infinitesimal space, but not published until 1951. In the interim, A. D. Wallace (1941) discovered a version of the same concept under the name of separation space.
Definition A proximity space (X, δ) is a set X with a relation δ between subsets of X satisfying the following properties:
For all subsets A, B and C of X
Proximity without the first axiom is called quasi-proximity (but then Axioms 2 and 4 must be stated in a two-sided fashion).
If A δ B we say A is near B or A and B are proximal; otherwise we say A and B are apart. We say B is a proximal or δ-neighborhood of A, written A « B, if and only if A and X−B are apart.