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. (See apartness space for a point–set version and apartness relation for a point–point version.)
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|
|Transitivity||If for every , either or , then||If , then for some , both and||If , then for some , both and|
|Symmetry (weak)||if then||if then||if then|
|Local Decomposability||If , then for some and , , , and .||If , then for some and , , , and .|
|Decomposability||If , then for some and , , , and .||If , then for some and , , , and .|
|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 , 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; see below). 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. In particular, it allows us to assert Reflexivity only for singletons, although this is often not done (to avoid mentioning points).
In classical mathematics, the weak Symmetry axioms are equivalent to the strong versions, while Decomposability is equivalent to Transitivity (and Local Decomposability is of course a special case of Decomposability). In constructive mathematics, strong Symmetry for is unreasonably strong (as is binary Additivity for ), while Local Decomposability and Decomposability appear genuinely stronger than Transitivity (but not unreasonably so); see remarks below. (We don’t bother to state a version of (Local) Decomposability for because the other axioms for seem unreasonably strong constructively, while Decomposability is automatic in classical mathematics.)
A topogeny is a relation that satisfies both forms of Isotony and all four forms of Additivity. A topogenous relation or quasipreproximity is a topogeny that also satisfies Reflexivity; a quasiproximity is a quasipreproximity that also satisfies Reflexivity and Transitivity. A topogeny is symmetric if it satisfies Symmetry; a preproximity is a symmetric quasipreproximity, and a proximity is a symmetric quasiproximity. A topogeny is separated if it satisfies Separation. A topogeny is perfect if it satisfies left Perfection, coperfect if it satisfies right Perfection, and biperfect if it satisfies both; 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). Constructively, a proximity space satisfying Regularity may be called proximally regular.
A (quasi)-(pre)-proximity space is a set equipped with a (quasi)-(pre)-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 Transitivity (so ‘pre’ is the default for them). 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 quasipreproximity. The terminology for Perfection also comes from syntopogenous spaces.
Transitivity is sometimes called normality, due to its superficial similarity with the notion of a normal space. However, the underlying topology of a proximity space need not be normal in the topological sense. If regarded as a separation axiom, the Transitivity axiom is more analogous to complete regularity than to normality, and at least in classical mathematics every proximity space is completely regular — but this is not true in constructive mathematics, or for quasi-proximity spaces, so it is debatable whether it should be attributed to the Transitivity axiom. When comparing proximities to topologies via syntopogenies, Transitivity is instead analogous to the fact that the closure of a set is closed (or the interior of a set is open), which we require of all topologies; thus arguably we should require it of all (quasi-)proximities. If this is removed from the definition of topological space, then we have a pretopological space, which is the origin of the ‘pre’ terminology above.
If and are (quasi)-(pre)-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.
As remarked above, the strong Symmetry axiom for is too strong in constructive mathematics. In fact, if it is satisfied by a nontrivial space , then excluded middle follows: for any truth value such that is true, let and ; then , so strong Symmetry gives , hence and so holds.
On the other hand, while the Symmetry axioms for and may seem stronger than the allowable weak symmetry axiom for , this is arguably an illusion. For given a proximal neighborhood space with (and weak Symmetry), if we define to mean , we obtain a proximal apartness space (with Symmetry). In fact, the Transitivity axiom for implies that if then , so that (with Symmetry) is completely determined by its restriction to subsets that are complements.
In this sense, a proximal neighborhood space actually contains more information than a proximal apartness space; it is unclear whether the latter gives rise to the former constructively. On the other side, we can of course define to mean , but with this definition we cannot even prove binary Additivity for . In fact, binary Additivity for seems too strong: it is apparently not satisfied even by a metric space.
We can, however, show the following constructively.
(Quasi-)proximal apartness spaces satisfying Decomposability are equivalent to (quasi-)proximal neighborhood spaces satisfying Decomposability.
As above, in one direction we define to mean . In the other, we define to mean that there exists a with and . Most of the axioms are easy. To prove right binary Additivity for , given and , we have and with and , whence with by distributing over .
To prove Decomposability (and hence Transitivity) for , given , we have with , whence by Decomposability for we have with , , and . By Decomposability again, we have with , , and , while , , and . Then (because of ) and (because of ), while (since ) and (because and so ) and so .
Lastly, if satisfies Symmetry and , we have with , whence by Decomposability for we have with , , and . Then by Symmetry, we have while , so .
To show that these constructions are inverses, if we start with then by a round-trip we obtain defined to mean that there exists with and . Now on one hand, implies , so . But on the other, if then Decomposability gives and with , , and , whence and .
On the other side, if we start with then by a round-trip we obtain defined to mean that there exists with and . Now on one hand, if then , so implies . But on the other, if then Decomposability gives and with and and , whence and so .
In other words, the two constructively reasonable ways to define a (quasi-)proximity space give the same result in the Decomposable case.
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 Transitivity, 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.
In constructive mathematics, a quasi-proximal nearness space (despite its general unreasonableness) or neighborhood space gives rise to a preordered set as above, while a proximal apartness space yields an inequality relation that is an apartness relation if we have Local Decomposability.
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 is 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 .
The construction of a topology works just as well for a quasi-proximity, but the result is no longer necessarily completely regular. In fact, every topology can be induced by some quasi-proximity: define if . Moreover, in this way topological spaces can be identified with (left) perfect quasi-proximity spaces.
In constructive mathematics, the three now-inequivalent kinds of (quasi-)proximity space (nearness, apartness, and neighborhood) yield three inequivalent notions of topological space: closure operators, point-set apartness spaces, and topological spaces (equivalently neighborhood spaces). Note in particular that the Transitivity axiom ensures that (in the three cases) the closure of a set is closed, that the final axiom of a point-set apartness space holds (if then ), and that the interior of a set is open. Hence, we can identify left perfect quasi-proximal apartness or neighborhood spaces with point-set apartness spaces and topological spaces respectively, by defining if as above in the second case, and if in the first.
If a quasi-proximal apartness or neighborhood space satisfies Local Decomposability, then its underlying point-set apartness space or topological space is locally decomposable (hence the name for the former axiom). Conversely, the canonical quasi-proximial apartness or neighborhood space obtained from a point-set apartness space or topological space satisfies Local Decomposability if the original space was locally decomposable.
Let be a proximal apartness or neighborhood space (with Symmetry). Then:
Suppose first that is a Locally Decomposable proximal neighborhood space and is an open set containing , which means . By Local Decomposability, we have sets and such that and and . Transitivity then gives us a set with , and Symmetry then tells us that . Let and ; then , is an open neighborhood of , and . Thus, is regular.
Now suppose is a Decomposable proximal neighborhood space and we have countable choice. Then given any opens with , essentially the same proof as above shows that is “well-inside” . Thus, applying Transitivity repeatedly (making countably many choices) we can obtain a “scale” of interpolating well-inside open subsets indexed by (dyadic) rational numbers, and thereby a function separating and , as in the classical proof of Urysohn’s lemma. (With only Transitivity instead of Decomposability, we would obtain only a function valued in the lower or upper real numbers, rather than the located real numbers.)
The arguments for proximal apartness spaces are essentially the same.
Conversely, if is a completely regular topological space, then the above definition that iff they are separated by a continuous function is also constructively valid, and produces a proximity space satisfying Symmetry and Decomposability.
If is a uniform space, then for all let iff is inhabited for every entourage (aka vicinity) . This defines a biperfect proximity structure on . In terms of the other relations, this means iff for some entourage , and iff , where .
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 between given uniform spaces 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 relation between quasi-uniformities and quasi-proximities is similar.
In constructive mathematics, all three of the relations may be defined as above from a uniformity. All the axioms of all three kinds of proximity are then provable, except Binary Additivity for , Strong Symmetry for , and (Local) Decomposability. Local Decomposability follows from uniform regularity for the uniform space, but Decomposability seems unobtainable in this way; it seems to assert locatedness of all subsets.
However, there is a different way to obtain a proximity underlying a uniformity: define if there is a uniformly continuous function such that for and for . Classically, this is equivalent to the usual definition, by the usual proof that any unform space is completely regular. Constructively, it (and a similar definition for ) yields a proximity space satisfying Decomposability, which has the same underlying topology if our original uniform space was already completely regular.
A proximity space can be identified with a syntopogenous space which is both simple and symmetric, and similarly a quasi-proximity space can be identified with one that is just simple. The relations with topological spaces and uniform spaces can then be seen as coreflections inside the category of syntopogenous spaces; see syntopogenous 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
Note that conditions (1) and (2), together with Isotony for , imply that any cluster is upwards-closed. Together with nullary Additivity for , they also imply the nullary version of (3): . However, a cluster is not generally a filter: it is not closed under binary intersections. This is most obvious when we consider how embeds in its compactification, taking each point to the “principal cluster” , i.e. the collection of sets whose closure contains , which is certainly not generally closed under intersections. (If it were a filter, of course, condition (3) would say that it is prime.)
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 Transitivity 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 Transitivity 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 Transitivity 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?
Generalized uniform structures
|binary relation||preorder||quasiuniformity||symmetric relation||equivalence relation||uniformity|
|binary function to||quasipseudometric||quasiprometric||symmetric binary function||pseudometric||prometric|
|topogeny||quasiproximity||syntopogeny||symmetric topogeny||proximity||symmetric syntopogeny|
R. Engelking, General topology, chapter 8.
S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge University Press 1970