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.
symmetry: iff ;
binary additivity: iff either or ;
nullary additivity: it is never true that ;
if for every such that , either or , then .
Another axiom one may require is the converse of (4):
In general, we say that and are proximate (or near) if , and apart otherwise. We also write if not .
A proximity space (or set–set nearness space) is a set equipped with a proximity structure . 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.
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:
It is also possible to write the definition in terms of the apartness relation or the relation . In particular, a (set–set) apartness space is a set equipped with a binary relation on such that the negation of 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).
If and are proximity spaces, then a function is said to be proximally continuous if implies . In this way we obtain a category , whose evident forgetful functor makes it into a topological concrete category.
Every proximity space is a topological space; let belong to the closure of iff . 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 over .
Conversely, if is a completely regular topological space, then for any let iff and there is no continuous function such that on and on . This defines a proximity structure on , which induces the topology on , and which is separated iff 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.
Uniformly continuous functions are proximally continuous for the induced proximities, so we have a functor over . Moreover, the composite 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 is a proximity space, consider the family of sets of the form
where is a finite family of sets such that there exists a finite family of sets with and . These sets form a base for a totally bounded uniformity on , 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 is a left adjoint, whose right adjoint also lives over , 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.
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 is called its Smirnov compactification. The points of this compactification can be taken to be clusters in , which are defined to be collections of subsets of such that