Proximity space

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 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 $$(X, \delta)$$ is a set $$X$$ with a relation $$\delta$$ between subsets of $$X$$ satisfying the following properties:

For all subsets $$A, B, C \subseteq X$$ Proximity without the first axiom is called (but then Axioms 2 and 4 must be stated in a two-sided fashion).
 * 1) $$A \;\delta\; B$$ implies $$B \;\delta\; A$$
 * 2) $$A \;\delta\; B$$ implies $$A \neq \varnothing$$
 * 3) $$A \cap B \neq \varnothing$$ implies $$A \;\delta\; B$$
 * 4) $$A \;\delta\; (B \cup C)$$ implies ($$A \;\delta\; B$$ or $$A \;\delta\; C$$)
 * 5) (For all $$E,$$ $$A \;\delta\; E$$ or $$B \;\delta\; (X \setminus E)$$) implies $$A \;\delta\; B$$

If $$A \;\delta\; B$$ we say $$A$$ is near $$B$$ or $$A$$ and $$B$$ are ; otherwise we say $$A$$ and $$B$$ are. We say $$B$$ is a or  of $$A,$$ written $$A \ll B,$$ if and only if $$A$$ and $$X \setminus B$$ are apart.

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

For all subsets $$A, B, C, D \subseteq X$$
 * 1) $$X \ll X$$
 * 2) $$A \ll B$$ implies $$A \subseteq B$$
 * 3) $$A \subseteq B \ll C \subseteq D$$ implies $$A \ll D$$
 * 4) ($$A \ll B$$ and $$A \ll C$$) implies $$A \ll B \cap C$$
 * 5) $$A \ll B$$ implies $$X \setminus B \ll X \setminus A$$
 * 6) $$A \ll B$$ implies that there exists some $$E$$ such that $$A \ll E \ll B.$$

A proximity space is called if $$\{ x \} \;\delta\; \{ y \}$$implies $$x = y.$$

A or  is one that preserves nearness, that is, given $$f : (X, \delta) \to \left(X^*, \delta^*\right),$$ if $$A \;\delta\; B$$ in $$X,$$ then $$f[A] \;\delta^*\; f[B]$$ in $$X^*.$$ Equivalently, a map is proximal if the inverse map preserves proximal neighborhoodness. In the same notation, this means if $$C \ll^* D$$ holds in $$X^*,$$ then $$f^{-1}[C] \ll f^{-1}[D]$$ holds in $$X.$$

Properties
Given a proximity space, one can define a topology by letting $$A \mapsto \left\{ x : \{ x \} \;\delta\; A \right\}$$ 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.

Given a compact Hausdorff space, there is a unique proximity space whose corresponding topology is the given topology: $$A$$ is near $$B$$ if and only if their closures intersect. More generally, proximities classify the compactifications of a completely regular Hausdorff space.

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