nLab
proximity space

Proximity spaces

Idea

In addition to the well-known topological spaces, many other structures can be used to found topological reasoning on sets, including uniform spaces and proximity spaces. Proximity spaces provide a level of structure in between topologies and uniformities; in fact a proximity is equivalent to an equivalence class of uniformities with the same totally bounded reflection.

Proximity spaces are often called nearness spaces, but this term has other meanings in the literature. (See for example this article.) One can clarify with the term set–set nearness space.

Definition

A proximity structure (or set–set nearness structure) on a set XX, or a proximity relation (or nearness relation) on the power set P(X)P(X) of subsets of XX, is a binary relation δ\delta on P(X)P(X) such that

  1. symmetry: AδBA\;\delta\;B iff BδAB\;\delta\;A;

  2. binary additivity: AδBCA\;\delta\;B\cup C iff either AδBA\;\delta\;B or AδCA\;\delta\;C;

  3. nullary additivity: it is never true that AδA\;\delta\;\emptyset;

  4. {x}δ{y}\{x\}\;\delta\;\{y\} if x=yx=y

  5. if for every C,DXC,D\subset X such that CD=XC\cup D=X, either AδCA\;\delta\;C or BδDB\;\delta\;D, then AδBA\;\delta\;B.

Another axiom one may require is the converse of (4):

  • separation: x=yx=y if {x}δ{y}\{x\}\;\delta\;\{y\}

In general, we say that AA and BB are proximate (or near) if AδBA\;\delta\;B, and apart otherwise. We also write ABA \ll B if not Aδ(XB)A\;\delta\;(X \setminus B).

A proximity space (or set–set nearness space) is a set XX equipped with a proximity structure δ\delta. The proximity structure or proximity space is separated if it satisfies the separation axiom (the converse of 4); note that many authors require this by default.

Variations

There are many variations possible in the list of axioms; one important consequence of the above (sometimes listed separately, allowing additivity to be weakened) is this:

  • isotony: if ACA\subset C and BDB\subset D, then CδDC\;\delta\;D if AδBA\;\delta\;B.

It is also possible to write the definition in terms of the apartness relation or the relation \ll. In particular, a (set–set) apartness space is a set XX equipped with a binary relation \bowtie on P(X)P(X) such that the negation of \bowtie is a proximity relation. This is the preferred formulation in constructive mathematics (although you'll want to rephrase the definition axiom by axiom to remove spurious double negations).

The category ProxProx

If XX and YY are proximity spaces, then a function f:XYf:X\to Y is said to be proximally continuous if AδBA\;\delta\;B implies f(A)δf(B)f(A)\;\delta\;f(B). In this way we obtain a category ProxProx, whose evident forgetful functor ProxSetProx \to Set makes it into a topological concrete category.

Relation to other topological structures

Topological spaces

Every proximity space is a topological space; let xx belong to the closure of AXA\subset X iff {x}δA\{x\}\;\delta\;A. This topology is always completely regular, and Hausdorff (hence Tychonoff) iff the proximity space is separated; see separation axiom. Proximally continuous functions are continuous for the induced topologies, so we have a functor ProxTopProx \to Top over SetSet.

Conversely, if (X,τ)(X,\tau) is a completely regular topological space, then for any A,BXA,B\subset X let AδBA\;\delta\;B iff ABA\neq \emptyset\neq B and there is no continuous function f:XI=[0,1]f:X\to I=[0,1] such that f(x)=0f(x)=0 on AA and f(x)=1f(x)=1 on BB. This defines a proximity structure on XX, which induces the topology τ\tau on XX, and which is separated iff τ\tau is a Hausdorff (hence Tychonoff) topology.

In general, a completely regular topology may be induced by more than one proximity. However, if it is moreover compact, then it has a unique compatible proximity.

Uniform spaces

If UU is a uniformity on YY (making it into a uniform space), then for all A,BYA,B\subset Y let AδBA\delta B iff V(A×B)V\cap (A\times B)\neq \emptyset for every entourage (aka vicinity) VUV\in U. This also defines a proximity structure on YY.

Uniformly continuous functions are proximally continuous for the induced proximities, so we have a functor UnifProxUnif \to Prox over SetSet. Moreover, the composite UnifProxTopUnif \to Prox \to Top is the usual “underlying topology” functor of a uniform space, i.e. the topology induced by the uniformity and the topology induced by the proximity structure are the same.

Conversely, if XX is a proximity space, consider the family of sets of the form

k=1 n(A k×A k) \bigcup_{k=1}^n (A_k \times A_k)

where (A k)(A_k) is a finite family of sets such that there exists a finite family of sets (B k)(B_k) with B kA kB_k \ll A_k and X= k=1 nB kX = \bigcup_{k=1}^n B_k. These sets form a base for a totally bounded uniformity on XX, which induces the given proximity.

In fact, this is the unique totally bounded uniformity which induces the given proximity: every proximity-class of uniformities contains a unique totally bounded member. Moreover, a proximally continuous function between uniform spaces with totally bounded codomain is automatically uniformly continuous. Therefore, the forgetful functor UnifProxUnif \to Prox is a left adjoint, whose right adjoint also lives over SetSet, is fully faithful, and has its essential image given by the totally bounded uniform spaces.

In general, proximally continuous functions need not be uniformly continuous, but in addition to total boundedness of the codomain, a different sufficient condition is that the domain be a metric space.

Syntopogenous spaces

A proximity space can be identified with a syntopogenous space which is both simple and symmetric; see syntopogenous space.

Compactifications

The (separated) proximities inducing a given (Hausdorff) completely regular topology can be identified with (Hausdorff) compactifications of that topology. The compactification corresponding to a proximity on XX is called its Smirnov compactification. The points of this compactification can be taken to be clusters in XX, which are defined to be collections σ\sigma of subsets of XX such that

  1. If AσA\in\sigma and BσB\in\sigma, then AδBA\;\delta\;B.
  2. If AδCA\;\delta\;C for all CσC\in\sigma, then AσA\in\sigma.
  3. If (AB)σ(A\cup B)\in\sigma, then AσA\in\sigma or BσB\in\sigma.

References

  • R. Engelking, General topology, chapter 8.
  • Douglas Bridges et al, Apartness, topology, and uniformity: a constructive view, pdf
  • S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge University Press 1970

Revised on March 14, 2014 12:04:49 by Victor Porton (87.68.16.14)