In topology, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that hold set-to-set, as opposed to the better known point-to-set notion that characterize topological spaces.
The concept was described by Frigyes Riesz (1909) but ignored at the time.[1] 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.
A proximity space is a set with a relation between subsets of satisfying the following properties:
For all subsets
- implies
- implies
- implies
- implies ( or )
- (For all or ) implies
Proximity without the first axiom is called quasi-proximity (but then Axioms 2 and 4 must be stated in a two-sided fashion).
If we say is near or and are proximal; otherwise we say and are apart. We say is a proximal- or -neighborhood of written if and only if and are apart.
The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces.
For all subsets
- implies
- implies
- ( and ) implies
- implies
- implies that there exists some such that
A proximity space is called separated if implies
A proximity or proximal map is one that preserves nearness, that is, given if in then in Equivalently, a map is proximal if the inverse map preserves proximal neighborhoodness. In the same notation, this means if holds in then holds in