Proximity space
Encyclopedia
In topology
, a proximity space is an axiomatization of notions of "nearness" that hold set-to-set, as opposed to the better known point-to-set notions that characterize topological space
s.
The concept was described by Frigyes Riesz
in 1908 and ignored at the time. It was rediscovered and axiomatized by V. A. Efremovič
in 1934, but not published until 1951. In the interim, in 1940, A. N. Wallace discovered a version of the same concept.
Definition A proximity space (X, δ) is a set X with a relation
δ between subsets of X satisfying the following properties:
For all subsets A, B and C of X
Proximity without the first axiom is called quasi-proximity.
If A δ B we say A is near B or A and B are proximal. We say B is a proximal or δ-neighborhood of A, written A « B, if and only if A δ X−B is false.
The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces.
For all subsets A, B, C, and D of X,
A proximity space is called separated if {x} δ {y} implies x=y.
A proximity or proximal map is one that preserves nearness, that is, given f:(X,δ)→(X*,δ*), if A δ B in X, then f[A] δ* f[B] in X*. Equivalently, a map is proximal if the inverse map preserves proximal neighborhoodness. In the same notation, this means if C«*D holds in X*, then f−1[C]«f−1[D] holds in X.
Given a proximity space, one can define a topology by letting A → {x : {x} δ A} 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 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×B has nonempty intersection with every entourage. Uniformly continuous maps will then be proximally continuous.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a proximity space is an axiomatization of notions of "nearness" that hold set-to-set, as opposed to the better known point-to-set notions that characterize topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s.
The concept was described by Frigyes Riesz
Frigyes Riesz
Frigyes Riesz was a mathematician who was born in Győr, Hungary and died in Budapest, Hungary. He was rector and professor at University of Szeged...
in 1908 and ignored at the time. It was rediscovered and axiomatized by V. A. Efremovič
Vadim Arsenyevich Efremovich
Vadim Arsen'evič Efremovič was a Soviet mathematician....
in 1934, but not published until 1951. In the interim, in 1940, A. N. Wallace discovered a version of the same concept.
Definition A proximity space (X, δ) is a set X with a relation
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
δ between subsets of X satisfying the following properties:
For all subsets A, B and C of X
- A δ B ⇒ B δ A
- A δ B ⇒ A ≠ ø
- A∩B ≠ ø ⇒ A δ B
- A δ (B∪C) ⇔ (A δ B or A δ C)
- (∀E, A δ E or B δ (X−E)) ⇒ A δ B
Proximity without the first axiom is called quasi-proximity.
If A δ B we say A is near B or A and B are proximal. We say B is a proximal or δ-neighborhood of A, written A « B, if and only if A δ X−B is false.
The main properties of this set neighborhood relation, listed below, provide an alternative axiomatic characterization of proximity spaces.
For all subsets A, B, C, and D of X,
- X«X
- A«B ⇒ A⊆B
- A⊆B«C⊆D ⇒ A«D
- (A«B and A«C) ⇒ A«B∩C
- A«B ⇒ X−B«X−A
- A«B ⇒ ∃E, A«E«B
A proximity space is called separated if {x} δ {y} implies x=y.
A proximity or proximal map is one that preserves nearness, that is, given f:(X,δ)→(X*,δ*), if A δ B in X, then f[A] δ* f[B] in X*. Equivalently, a map is proximal if the inverse map preserves proximal neighborhoodness. In the same notation, this means if C«*D holds in X*, then f−1[C]«f−1[D] holds in X.
Given a proximity space, one can define a topology by letting A → {x : {x} δ A} be a Kuratowski closure operator. If the proximity space is separated, the resulting topology is Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
. 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
Urysohn's lemma
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function....
, 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 whose corresponding topology is the given topology: A is near B if and only if their closures intersect. More generally, proximities classify the compactifications
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of a completely regular Hausdorff space.
A uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
X induces a proximity relation by declaring A is near B if and only if A×B has nonempty intersection with every entourage. Uniformly continuous maps will then be proximally continuous.