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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s. The concept was described by 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, discovered a version of the same concept under the name of separation space.


Definition

A (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) implies (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 (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 ; otherwise we say A and B are . We say B is a or 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 if \ \;\delta\; \implies x = y. A or is one that preserves nearness, that is, given f : (X, \delta) \to \left(X^*, \delta^*\right), if A \;\delta\; B in X, then f \;\delta^*\; f /math> 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^ \ll f^ /math> holds in X.


Properties

Given a proximity space, one can define a topology by letting A \mapsto \left\ be a
Kuratowski closure operator In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first for ...
. 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 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 continuous function. Section 15. Urysohn's lemma is commonly used to construct continuo ...
, 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 \times B has nonempty intersection with every entourage. Uniformly continuous maps will then be proximally continuous.


See also

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References

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External links

* {{Topology Closure operators General topology