Editing Proximity space

{{short description|Structure describing a notion of "nearness" between subsets}}
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 space]]s.

The concept was described by {{harvs|txt|authorlink=Frigyes Riesz|first=Frigyes |last=Riesz|year= 1909}} but ignored at the time.<ref>W. J. Thron, ''Frederic Riesz' contributions to the foundations of general topology'', in C.E. Aull and R. Lowen (eds.), ''Handbook of the History of General Topology'', Volume 1, 21-29, Kluwer 1997.</ref> It was rediscovered and axiomatized by [[Vadim Arsenyevich Efremovich|V. A. Efremovič]] in 1934 under the name of '''infinitesimal space''', but not published until 1951. In the interim, {{harvs|txt|first=A. D. |last=Wallace|authorlink=A. D. Wallace|year=1941}} discovered a version of the same concept under the name of '''separation space'''.

== 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>

== 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.

== See also ==

* {{annotated link|Cauchy space}}
* {{annotated link|Convergence space}}
* {{annotated link|Pretopological space}}
*[[Uniform space]]

== References ==

{{reflist}}
{{reflist|group=note}}

* {{citation|mr=0040748|last=Efremovič|first= V. A.|title=Infinitesimal spaces|language=Russian|journal=Doklady Akademii Nauk SSSR |series=New Series|volume=76|year=1951|pages=341–343}} 
* {{cite book|last1=Naimpally|first1=Somashekhar A.|last2=Warrack|first2=Brian D.|title=Proximity Spaces|year=1970|zbl=0206.24601|series=Cambridge Tracts in Mathematics and Mathematical Physics|volume=59|publisher=[[Cambridge University Press]]|location=Cambridge|isbn=0-521-07935-7}}
* {{citation|last=Riesz|first=F.|title=Stetigkeit und abstrakte Mengenlehre|jfm= 40.0098.07|journal=Rom. 4. Math. Kongr. 2|pages=18–24|year=1909}}
* {{citation|last=Wallace|first=A. D.|title=Separation spaces|journal=Ann. of Math.|series=2|volume=42|issue=3|year=1941|pages=687–697|mr=0004756|doi=10.2307/1969257|jstor=1969257|url=https://polipapers.upv.es/index.php/AGT/article/download/5868/8708|url-access=|url-status=|archive-url=|archive-date=}} 
* {{cite CiteSeerX|last1=Vita|first1=Luminita|last2=Bridges|first2=Douglas|title=A Constructive Theory of Point-Set Nearness|year=2001 |citeseerx=10.1.1.15.1415}}

== External links ==
* {{eom |title = Proximity space |id=Proximity_space}}

{{Topology}}

[[Category:Closure operators]]
[[Category:General topology]]