Proximity Structure

In General > s.a. uniformity.
* Idea: A proximity space is an intermediate concept between those of a topological space and a uniform space.
$ Def: An (Efremovich) proximity on a set X is a binary relation δ on the power set of X which obeys:
(1) If (A δ B) then (B δ A) (symmetry);
(2) (A ∪ B) δ C iff (A δ C) \(\lor\) (B δ C);
(3) If (A δ B) then A, B ≠ Ø;
(4) If ¬ (A δ B) then there exists E such that ¬ (A δ E) ∧ ¬ (X \ E) δ B;
(5) If A ∩ B ≠ Ø then (A δ B).
* Relationships: It induces a topology by A':= {x | x δ A}, for all A ⊂ X; This topology is always completely regular (and viceversa), and Tychonov if δ is separated.

Special Types and Examples
* Special case: If (6) (x δ y) implies (x = y), then the proximity is called separated (Hausdorff).
* Examples: The discrete proximity, defined by A δ B iff A ∩ B ≠ Ø; The indiscrete proximity, in which A δ B iff A, B ≠ Ø.

References
@ General: Naimpally & Warrack 70; Bridges & Vîţă 11 [using constructive logic].
> Online resources: see nLab page.

Proximity Map
$ Def: Given a set X and a group G, a map δ: X × X → G such that for all x ∈ X, δ(x, x) = 1.

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