In topology, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that applies set-to-set, as opposed to the better-known point-to-set notion that characterizes topological spaces.

The concept was described by 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, discovered a version of the same concept under the name of separation space.

Definition

A with a relation satisfying the following properties:

1. )

Proximity without the first axiom is called (but then Axioms 2 and 4 must be stated in a two-sided fashion).

are are is a or are apart.

The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces.

A proximity space is called

A or

Properties

be a Kuratowski closure operator. If the proximity space is separated, the resulting topology is Hausdorff. Proximity maps will be continuous between the induced topologies.

The resulting topology is always completely regular. This can be proven by imitating the usual proofs of Urysohn's lemma, using the last property of proximal neighborhoods to create the infinite increasing chain used in proving the lemma.

if and only if their closures intersect. More generally, proximities classify the compactifications of a completely regular Hausdorff space.

A uniform space has nonempty intersection with every entourage. Uniformly continuous maps will then be proximally continuous.

See also

• Uniform space

References

External links