Editing Proximity space (section)

== Properties ==

Given a proximity space, one can define a topology by letting <math>A \mapsto \left\{ x : \{ x \} \;\delta\; A \right\}</math> be a [[Kuratowski closure operator]]. If the proximity space is separated, the resulting topology is [[Hausdorff space|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.

Given a compact Hausdorff space, there is a unique proximity space whose corresponding topology is the given topology: <math>A</math> is near <math>B</math> if and only if their closures intersect. More generally, proximities classify the [[Compactification (mathematics)|compactifications]] of a completely regular Hausdorff space.

A [[uniform space]] <math>X</math> induces a proximity relation by declaring <math>A</math> is near <math>B</math> if and only if <math>A \times B</math> has nonempty intersection with every entourage. [[Uniformly continuous]] maps will then be proximally continuous.