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. The same goes for the term apartness space, which is another way to look at the same basic idea.

Notation and terminology

A proximity space is a kind of structured set: it consists of a set XX (the set of points of the space) and a proximity structure on XX. This proximity structure is given by any of various binary relations between the subsets of XX:

  • The proximity relation or nearness relation δ\delta; subsets AA and BB are proximal or near if AδBA \;\delta\; B;
  • The apartness relation \bowtie; AA and BB are apart if ABA \bowtie B.
  • The neighbourhood relation \ll; AA is a proximal neighbourhood of BB if BAB \ll A.

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 ABA \ll B doesn't necessarily mean that AInt(B)A \subseteq Int(B); sometimes it means that Cl(A)Int(B)Cl(A) \subseteq Int(B), which is stronger, or something else.)

In classical mathematics, these relations are all interdefinable:

  • AδB¬(AB)¬(AB)A \;\delta\; B \;\iff\; \neg(A \bowtie B) \;\iff\; \neg(A \ll B');
  • AB¬(AδB)ABA \bowtie B \;\iff\; \neg(A \;\delta\; B) \;\iff\; A \ll B';
  • AB¬(AδB)ABA \ll B \;\iff\; \neg(A \;\delta\; B') \;\iff\; A \bowtie B'.

(Here, ¬\neg indicates negation of truth values, while BB' is the complement of BB in XX.)

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.

Definitions

From the previous section, we have a set XX and we are discussing binary relations δ,,\delta, \bowtie, \ll on XX. These are required to satisfy the following conditions; in each row, the conditions for the various relations are all equivalent (classically). In these conditions, x,yx, y are points, while A,B,CA, B, C 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.

NameCondition for nearnessCondition for apartnessCondition for proximal neighbourhoods
Isotony (left)If ABδCA \supseteq B \;\delta\; C, then AδCA \;\delta\; CIf ABCA \subseteq B \bowtie C, then ACA \bowtie CIf ABCA \subseteq B \ll C, then ACA \ll C
Isotony (right)If BδCDB \;\delta\; C \subseteq D, then BδDB \;\delta\; DIf BCDB \bowtie C \supseteq D, then BDB \bowtie DIf BCDB \ll C \subseteq D, then BDB \ll D
Additivity (left, nullary)It is false that δA\emptyset \;\delta\; AA\emptyset \bowtie AA\emptyset \ll A
Additivity (right, nullary)It is false that AδA \;\delta\; \emptysetAA \bowtie \emptysetAXA \ll X
Additivity (left, binary)If ABδCA \cup B \;\delta\; C, then AδCA \;\delta\; C or BδCB \;\delta\; CIf ACA \bowtie C and BCB \bowtie C, then ABCA \cup B \bowtie CIf ACA \ll C and BCB \ll C, then ABCA \cup B \ll C
Additivity (right, binary)If AδBCA \;\delta\; B \cup C, then AδBA \;\delta\; B or AδCA \;\delta\; CIf ABA \bowtie B and ACA \bowtie C, then ABCA \bowtie B \cup CIf ABA \ll B and ACA \ll C, then ABCA \ll B \cap C
Reflexivity (general)If AA meets BB (their intersection is inhabited), then AδBA \;\delta\; BIf ABA \bowtie B, then AA and BB are disjointIf ABA \ll B, then ABA \subseteq B
Reflexivity (for singletons){x}δ{x}\{x\} \;\delta\; \{x\}It is false that {x}{x}\{x\} \bowtie \{x\}If {x}A\{x\} \ll A, then xAx \in A
Normality (constructive)If for every D,EXD, E \subseteq X such that DE=XD \cup E = X, either AδDA \;\delta\; D or EδBE \;\delta\; B, then AδBA \;\delta\; BIf ABA \bowtie B, then for some D,EXD, E \subseteq X such that DE=XD \cup E = X, both ADA \bowtie D and EBE \bowtie BIf ABA \ll B, then for some D,EXD, E \subseteq X such that DED \subseteq E, both ADA \ll D and EBE \ll B
Normality (simplified)If for every DXD \subseteq X, either AδDA \;\delta\; D or DδBD' \;\delta\; B, then AδBA \;\delta\; BIf ABA \bowtie B, then for some DXD \subseteq X, both ADA \bowtie D and DBD' \bowtie BIf ABA \ll B, then for some DXD \subseteq X, both ADA \ll D and DBD \ll B
Symmetry (constructive)AδBA \;\delta\; B iff BδAB \;\delta\; AABA \bowtie B iff BAB \bowtie AIf ABA \ll B, AC=XA \cup C = X, and BD=B \cap D = \empty, then DCD \ll C
Symmetry (simplified)AδBA \;\delta\; B iff BδAB \;\delta\; AABA \bowtie B iff BAB \bowtie AABA \ll B iff BAB' \ll A'
SeparationIf {x}δ{y}\{x\} \delta \{y\}, then x=yx = yUnless {x}{y}\{x\} \bowtie \{y\}, then x=yx = yx=yx = y if, for all AA, yAy \in A whenever {x}A\{x\} \ll A
Perfection (left)If AδBA \;\delta\; B, then {x}δB\{x\} \;\delta\; B for some xAx \in AABA \bowtie B if {x}B\{x\} \bowtie B for all xAx \in AABA \ll B if {x}B\{x\} \ll B for all xAx \in A
Perfection (right)If AδBA \;\delta\; B, then Aδ{y}A \;\delta\; \{y\} for some yBy \in BIf A{y}A \bowtie \{y\} for all yBy \in B, then ABA \bowtie BIf (for all yy) yBy \in B if (for all CC) yCy \in C if ACA \ll C, then ABA \ll B

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.

Warnings

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 XX and YY are (quasi)-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), equivalently if ABA \bowtie B whenever f *(A)f *(B)f_*(A) \bowtie f_*(B), equivalently if f *(C)f *(D)f^*(C) \ll f^*(D) whenever CDC \ll D. In this way we obtain categories QProxQProx and ProxProx; the forgetful functors QProxSetQProx \to Set and ProxSetProx \to Set (taking a space to its set of points) make them into topological concrete categories.

Relations to other topological structures

Preorders

Given points x,yx, y of a (quasi)-proximity space, let xyx \leq y mean that xx belongs to every proximal neighbourhood of {y}\{y\}, or equivalently (via Isotony) that {y}δ{x}\{y\} \;\delta\; \{x\}. By Reflexivity, \leq is reflexive; by Normality, \leq is transitive. (In fact, we can use these to deduce that xyx \leq y iff every proximal neighbourhood of {y}\{y\} is a proximal neighbourhood of {x}\{x\}, which is manifestly reflexive and transitive.) Therefore, \leq is a preorder.

If the quasiproximity satisfies Symmetry, then this preorder is symmetric and hence an equivalence relation.

Regardless of Symmetry, a (quasi)-proximity space is separated iff this preorder is the equality relation. That is, x=yx = y if xx belongs to every proximal neighbourhood of {y}\{y\}, or equivalently if every proximal neighbourhood of {y}\{y\} is a proximal neighbourhood of {x}\{x\}, or equivalently if {x}\{x\} is near {y}\{y\}, or equivalently if {x}\{x\} is not apart from {y}\{y\}. This may be viewed as a converse of simplified Reflexivity, which states that {x}δ{y}\{x\} \;\delta\; \{y\} whenever x=yx = y.

Conversely, given a set equipped with a preorder \leq, let AδBA \;\delta\; B if xyx \leq y for some xAx \in A and some yBy \in B, or equivalently let ABA \bowtie B if xyx \leq y for no xAx \in A and no yBy \in B, or equivalently let ABA \ll B if xyx \leq y for xAx \in A implies yBy \in B. Then we have a quasiproximity space which is symmetric iff \leq is.

In this way, we get the category ProsetProset of preordered sets as a reflexive subcategory of QProxQProx, with the category SetoidSetoid of setoids (sets equipped with equivalence relations) as a reflexive subcatgory of ProxProx.

Topological spaces

Every proximity space is a topological space; let xx belong to the closure of AXA \subseteq X iff {x}δA\{x\} \;\delta\; A, or equivalently let xx belong to the interior of AA iff {x}A\{x\} \ll 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 \subseteq X let ABA \bowtie B iff there is a continuous function f:X[0,1]f: X\to [0,1] such that f(x)=0f(x) = 0 for xAx \in A and f(x)=1f(x) = 1 for xBx \in B. 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, given above. In the case of a compact Hausdorff space (or more generally any normal regular space), we then have ABA \ll B iff Cl(A)Int(B)Cl(A) \subseteq Int(B).

Uniform spaces

If UU is a uniformity on YY (making it into a uniform space), then for all A,BYA, B \subseteq Y let AδBA \;\delta\; B iff V(A×B)V \cap (A \times B) is inhabited for every entourage (aka vicinity) VV. 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) k(A_k)_k is a list (a finite family) of sets such that there exists a same-length list of sets (B k) k(B_k)_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.

Proximities as profunctors

As a poset, the power set 𝒫X\mathcal{P}X of XX may be regarded as a category enriched over truth values. There is a notion of a bimodule over a category, also called (more specifically) a distributor or profunctor.

Then a profunctor from 𝒫X\mathcal{P}X to itself is precisely a binary relation \ll on subsets of XX 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 \ll as its underlying bimodule has the structure of an operation that, given xzx \ll z, takes this to an equivalence class of yy such that xyzx \ll y \ll z, where yy is equivalent to yy' if yyy \subseteq y' (or by any equivalence that follows). By Isotony and left binary Additivity, xyyzx \ll y \cup y' \ll z (or use right Additivity and yyy \cap y'); since y,yyyy, y' \subseteq y \cup y', 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).

As for Additivity itself, this presumably corresponds to something more general in the world of profunctors related to limits and colimits, but I haven't figured it out yet.

Symmetry probably doesn't fit into this picture very well, but who knows?

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 July 24, 2014 18:55:22 by Toby Bartels (98.23.133.202)