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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