Editing Proximity space (section)

== Definition ==

A {{em|'''proximity space'''}} <math>(X, \delta)</math> is a set <math>X</math> with a [[Relation (mathematics)|relation]] <math>\delta</math> between subsets of <math>X</math> satisfying the following properties:

For all subsets <math>A, B, C \subseteq X</math>
# <math>A \;\delta\; B</math> implies <math>B \;\delta\; A</math>
# <math>A \;\delta\; B</math> implies <math>A \neq \varnothing</math>
# <math>A \cap B \neq \varnothing</math> implies <math>A \;\delta\; B</math>
# <math>A \;\delta\; (B \cup C)</math> if and only if (<math>A \;\delta\; B</math> or <math>A \;\delta\; C</math>)
# (For all <math>E,</math> <math>A \;\delta\; E</math> or <math>B \;\delta\; (X \setminus E)</math>) implies <math>A \;\delta\; B</math>
Proximity without the first axiom is called {{em|'''quasi-proximity'''}} (but then Axioms 2 and 4 must be stated in a two-sided fashion).

If <math>A \;\delta\; B</math> we say <math>A</math> is near <math>B</math> or <math>A</math> and <math>B</math> are {{em|proximal}}; otherwise we say <math>A</math> and <math>B</math> are {{em|apart}}. We say <math>B</math> is a {{em|proximal-}} or {{em|<math>\delta</math>-neighborhood}} of <math>A,</math> written <math>A \ll B,</math> if and only if <math>A</math> and <math>X \setminus B</math> are apart.

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

For all subsets <math>A, B, C, D \subseteq X</math>
# <math>X \ll X</math>
# <math>A \ll B</math> implies <math>A \subseteq B</math>
# <math>A \subseteq B \ll C \subseteq D</math> implies <math>A \ll D</math>
# (<math>A \ll B</math> and <math>A \ll C</math>) implies <math>A \ll B \cap C</math>
# <math>A \ll B</math> implies <math>X \setminus B \ll X \setminus A</math>
# <math>A \ll B</math> implies that there exists some <math>E</math> such that <math>A \ll E \ll B.</math>

A proximity space is called {{em|separated}} if <math>\{ x \} \;\delta\; \{ y \}</math>implies <math>x = y.</math>

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