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 spaces.

The concept was described by {{harvs|txt|authorlink=Frigyes Riesz|first=Frigyes |last=Riesz|year= 1909}} but ignored at the time.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. 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, {{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}} $(X, \delta)$ is a set $X$ with a relation $\delta$ between subsets of $X$ satisfying the following properties:

For all subsets $A, B, C \subseteq X$

$A \;\delta\; B$ implies $B \;\delta\; A$
$A \;\delta\; B$ implies $A \neq \varnothing$
$A \cap B \neq \varnothing$ implies $A \;\delta\; B$
$A \;\delta\; (B \cup C)$ if and only if ( $A \;\delta\; B$ or $A \;\delta\; C$ )
(For all $E,$ $A \;\delta\; E$ or $B \;\delta\; (X \setminus E)$ ) implies $A \;\delta\; B$

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 $A \;\delta\; B$ we say $A$ is near $B$ or $A$ and $B$ are {{em|proximal}}; otherwise we say $A$ and $B$ are {{em|apart}}. We say $B$ is a {{em|proximal-}} or {{em| $\delta$ -neighborhood}} 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 $A, B, C, D \subseteq X$

$X \ll X$
$A \ll B$ implies $A \subseteq B$
$A \subseteq B \ll C \subseteq D$ implies $A \ll D$
( $A \ll B$ and $A \ll C$ ) implies $A \ll B \cap C$
$A \ll B$ implies $X \setminus B \ll X \setminus A$
$A \ll B$ implies that there exists some $E$ such that $A \ll E \ll B.$

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

A {{em|proximity}} or {{em|proximal map}} 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.

References

{{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}}

Category:Closure operators

Category:General topology

proximity space

Definition

Properties

See also

References

External links