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. The same goes for the term apartness space, which is another way to look at the same basic idea.
A proximity space is a kind of structured set: it consists of a set (the set of points of the space) and a proximity structure on . This proximity structure is given by any of various binary relations between the subsets of :
The conditions required of these relations are given below in the Definitions. (We say ‘proximal neighbourhood’ instead of simply ‘neighbourhood’ to avoid misapplying intuition from general topology. That doesn't necessarily mean that ; sometimes it means that , which is stronger, or something else.)
In classical mathematics, these relations are all interdefinable:
In constructive mathematics, any one of these relations may be taken as primary and the others defined using it; thus we distinguish, constructively, between a set–set nearness space, a set–set apartness space, and a set–set neighbourhood space.
From the previous section, we have a set and we are discussing binary relations on . These are required to satisfy the following conditions; in each row, the conditions for the various relations are all equivalent (classically). In these conditions, are points, while are subsets, and we require them for all points or subsets. We list the conditions roughly in order of increasing optional-ness, then define terminology for relations satisfying them.
|Name||Condition for nearness||Condition for apartness||Condition for proximal neighbourhoods|
|Isotony (left)||If , then||If , then||If , then|
|Isotony (right)||If , then||If , then||If , then|
|Additivity (left, nullary)||It is false that|
|Additivity (right, nullary)||It is false that|
|Additivity (left, binary)||If , then or||If and , then||If and , then|
|Additivity (right, binary)||If , then or||If and , then||If and , then|
|Reflexivity (general)||If meets (their intersection is inhabited), then||If , then and are disjoint||If , then|
|Reflexivity (for singletons)||It is false that||If , then|
|Normality (constructive)||If for every such that , either or , then||If , then for some such that , both and||If , then for some such that , both and|
|Normality (simplified)||If for every , either or , then||If , then for some , both and||If , then for some , both and|
|Symmetry (constructive)||iff||iff||If , , and , then|
|Separation||If , then||Unless , then||if, for all , whenever|
|Perfection (left)||If , then for some||if for all||if for all|
|Perfection (right)||If , then for some||If for all , then||If (for all ) if (for all ) if , then|
When both left and right rules are shown, we only need one of them if we have Symmetry, but we need both if we lack Symmetry (or if we are using proximal neighbourhoods in constructive mathematics). Even so, Isotony is usually given on both sides, since it is convenient to combine both directions into a single statement. On the other hand, Isotony is equivalent to the converse of binary Additivity, so sometimes these are combined instead (so Isotony does not explicitly appear), usually on only one side when Symmetry is used.
Whether made explicit or not, Isotony is very fundamental, and it is what allows the axioms after Additivity to be written in different forms. In particular, we need Reflexivity only for singletons, although this is often not done (to avoid mentioning points). Similarly, we usually simplify Normality as shown (although this is appropriate for constructive mathematics only when defining neighbourhood spaces). In the same vein, Symmetry for proximal neighbourhoods is usually given in the simplified form (although now that is not appropriate for constructive mathematics).
A topogeny is a relation that satisfies both forms of Isotony and all four forms of Additivity. A quasiproximity is a topogeny that also satisfies Reflexivity and Normality. A topogeny (or quasiproximity) is symmetric if it satisfies Symmetry; a proximity is a symmetric quasiproximity. A topogeny or (quasi)-proximity is separated if it satisfies Separation. A topogeny or quasiproximity is perfect if it satisfies left Perfection, coperfect if it satisfies right Perfection, and biperfect if it satisfies both; a proximity (or a symmetric topogeny) is usually called simply perfect if it satisfies any form of Perfection, because then it must satisfy both (except in constructive mathematics using proximal neighbourhoods).
A (quasi)-proximity space is a set equipped with a (quasi)-proximity. All of these terms may be used with nearness, apartness, or proximal neighbourhoods, as explained in the previous section; nearness is usually the default.
Some authors require a (quasi)-proximity to be separated; conversely, some authors do not require a (quasi)-proximity to satisfy Normality. The term ‘topogeny’ is also not found in the literature (except here, in a generalization of nearness spaces); it is derived from ‘topogenous relation’, a term used in the theory of syntopogenous spaces for a nearness topogeny satisfying Reflexivity. (Thus, quasiproximities and topogenous relations are the same thing for authors who use nearness and do not require Normality.) The terminology for Perfection also comes from syntopogenous spaces.
If and are (quasi)-proximity spaces, then a function is said to be proximally continuous if implies , equivalently if whenever , equivalently if whenever . In this way we obtain categories and ; the forgetful functors and (taking a space to its set of points) make them into topological concrete categories.
Given points of a (quasi)-proximity space, let mean that belongs to every proximal neighbourhood of , or equivalently (via Isotony) that . By Reflexivity, is reflexive; by Normality, is transitive. (In fact, we can use these to deduce that iff every proximal neighbourhood of is a proximal neighbourhood of , which is manifestly reflexive and transitive.) Therefore, is a preorder.
Regardless of Symmetry, a (quasi)-proximity space is separated iff this preorder is the equality relation. That is, if belongs to every proximal neighbourhood of , or equivalently if every proximal neighbourhood of is a proximal neighbourhood of , or equivalently if is near , or equivalently if is not apart from . This may be viewed as a converse of simplified Reflexivity, which states that whenever .
Conversely, given a set equipped with a preorder , let if for some and some , or equivalently let if for no and no , or equivalently let if for implies . Then we have a quasiproximity space which is symmetric iff is.
Every proximity space is a topological space; let belong to the closure of iff , or equivalently let belong to the interior 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 there is a continuous function such that for and for . 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, given above. In the case of a compact Hausdorff space (or more generally any normal regular space), we then have iff .
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
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
Then a profunctor from to itself is precisely a binary relation on subsets of that satisfies Isotony. Adding Reflexivity makes it a co-pointed profunctor, and Normality morally makes it a coassociative coalgebra with Reflexivity as counit. (Actually, coassociativity is trivial when enriched over truth values, as is the claim that Reflexivity, once it exists, is a counit, but we say ‘coassociative’ to clarify which sense of ‘coalgebra’ we mean.)
The sense in which Normality makes this a coalgebra is actually a bit involved, and it only quite works because of Additivity. A coalgebra with a given profunctor as its underlying bimodule has the structure of an operation that, given , takes this to an equivalence class of such that , where is equivalent to if (or by any equivalence that follows). By Isotony and left binary Additivity, (or use right Additivity and ); since , we have the desired equivalence.
This suggests that if we want a notion of proximity without Additivity, then Normality must become more complicated, being a structure rather than just a property (and a structure that should be preserved by proximal maps).
Symmetry probably doesn't fit into this picture very well, but who knows?
R. Engelking, General topology, chapter 8.
S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge University Press 1970