nLab
proximity space

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Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

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Extra stuff, structure, properties

Examples

Basic statements

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Basic homotopy theory

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. (See apartness space for a point–set version and apartness relation for a point–point version.)

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 c)A \;\delta\; B \;\iff\; \neg(A \bowtie B) \;\iff\; \neg(A \ll B^{\mathsf{c}});
  • AB¬(AδB)AB cA \bowtie B \;\iff\; \neg(A \;\delta\; B) \;\iff\; A \ll B^{\mathsf{c}};
  • AB¬(AδB c)AB cA \ll B \;\iff\; \neg(A \;\delta\; B^{\mathsf{c}}) \;\iff\; A \bowtie B^{\mathsf{c}}.

(Here, ¬\neg indicates negation of truth values, while B cB^{\mathsf{c}} is the complement of BB in XX, i.e. XBX\setminus B.)

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
TransitivityIf for every DXD \subseteq X, either AδD cA \;\delta\; D^{\mathsf{c}} or DδBD \;\delta\; B, then AδBA \;\delta\; BIf ABA \bowtie B, then for some DXD \subseteq X, both AD cA \bowtie D^{\mathsf{c}} and DBD \bowtie BIf ABA \ll B, then for some DXD \subseteq X, both ADA \ll D and DBD \ll B
Symmetry (weak)if AδBA \;\delta\; B then BδAB \;\delta\; Aif ABA \bowtie B then BAB \bowtie Aif ABA \ll B then B cA cB^{\mathsf{c}} \ll A^{\mathsf{c}}
Symmetry (strong)AδBA \;\delta\; B iff BδAB \;\delta\; AABA \bowtie B iff BAB \bowtie AABA \ll B iff B cA cB^{\mathsf{c}} \ll A^{\mathsf{c}}
Local DecomposabilityIf {x}B\{x\}\bowtie B, then for some CC and DD, {x}D\{x\}\bowtie D, CBC\bowtie B, and CD=XC\cup D = X.If {x}B\{x\}\ll B, then for some CC and DD, {x}C\{x\}\ll C, DBD\ll B, and C cD=XC^{\mathsf{c}}\cup D = X.
DecomposabilityIf ABA\bowtie B, then for some CC and DD, ADA\bowtie D, CBC\bowtie B, and CD=XC\cup D = X.If ABA\ll B, then for some CC and DD, ACA\ll C, DBD\ll B, and C cD=XC^{\mathsf{c}}\cup D = X.
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 AC iA \ll C_i for all ii, then A iC iA \ll \bigcap_i C_i

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 \ll is unreasonably strong (as is binary Additivity for δ\delta), 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 δ\delta because the other axioms for δ\delta 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.

Warnings

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 XX and YY are (quasi)-(pre)-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.

In constructive mathematics

As remarked above, the strong Symmetry axiom for \ll is too strong in constructive mathematics. In fact, if it is satisfied by a nontrivial space XX, then excluded middle follows: for any truth value PP such that ¬¬P\neg\neg P is true, let A=XA=X and B={xXP}B = \{ x\in X \mid P \}; then B c==A cB^{\mathsf{c}} = \emptyset \ll \emptyset = A^{\mathsf{c}}, so strong Symmetry gives ABA\ll B, hence ABA\subseteq B and so PP holds.

On the other hand, while the Symmetry axioms for \bowtie and δ\delta may seem stronger than the allowable weak symmetry axiom for \ll, this is arguably an illusion. For given a proximal neighborhood space with \ll (and weak Symmetry), if we define ABA\bowtie B to mean AB cA\ll B^{\mathsf{c}}, we obtain a proximal apartness space (with Symmetry). In fact, the Transitivity axiom for \bowtie implies that if ABA\bowtie B then A(B c) cA\bowtie (B^{\mathsf{c}})^{\mathsf{c}}, so that (with Symmetry) \bowtie 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 AδBA\;\delta \; B to mean ¬(AB)\neg(A\bowtie B), but with this definition we cannot even prove binary Additivity for δ\delta. In fact, binary Additivity for δ\delta seems too strong: it is apparently not satisfied even by a metric space.

We can, however, show the following constructively.

Theorem

(Quasi-)proximal apartness spaces satisfying Decomposability are equivalent to (quasi-)proximal neighborhood spaces satisfying Decomposability.

Proof

As above, in one direction we define ABA\bowtie B to mean AB cA\ll B^{\mathsf{c}}. In the other, we define ABA\ll B to mean that there exists a CC with ACA\bowtie C and BC=XB\cup C = X. Most of the axioms are easy. To prove right binary Additivity for \ll, given ABA\ll B and ACA\ll C, we have ADA\bowtie D and AEA\bowtie E with DB=XD\cup B = X and EC=XE\cup C = X, whence ADEA\bowtie D\cup E with (DE)(BC)=X(D\cup E) \cup (B\cap C) = X by distributing \cup over \cap.

To prove Decomposability (and hence Transitivity) for \ll, given ABA\ll B, we have ACA\bowtie C with BC=XB\cup C = X, whence by Decomposability for \bowtie we have D,ED,E with ADA\bowtie D, ECE\bowtie C, and DE=XD\cup E = X. By Decomposability again, we have D 1,D 2,E 1,E 2D_1,D_2,E_1,E_2 with AD 1A\bowtie D_1, D 2DD_2\bowtie D, and D 1D 2=XD_1\cup D_2 = X, while EE 1E\bowtie E_1, E 2CE_2\bowtie C, and E 1E 2=XE_1\cup E_2=X. Then AD 2A\ll D_2 (because of D 1D_1) and E 2BE_2 \ll B (because of CC), while DD 2 cD\subseteq D_2^{\mathsf{c}} (since D 2DD_2\bowtie D) and EE 2E\subseteq E_2 (because EE 1E\bowtie E_1 and so EE 1 cE 2E\subseteq E_1^{\mathsf{c}} \subseteq E_2) and so X=DE=D 2 cE 2X = D\cup E = D_2^{\mathsf{c}} \cup E_2.

Lastly, if \bowtie satisfies Symmetry and ABA\ll B, we have ACA\bowtie C with BC=XB\cup C = X, whence by Decomposability for \bowtie we have D,ED,E with ADA\bowtie D, ECE\bowtie C, and DE=XD\cup E = X. Then by Symmetry, we have B cCEB^{\mathsf{c}} \subseteq C\bowtie E while X=EDEA cX = E\cup D \subseteq E\cup A^{\mathsf{c}}, so B cA cB^{\mathsf{c}} \ll A^{\mathsf{c}}.

To show that these constructions are inverses, if we start with \ll then by a round-trip we obtain ABA\ll' B defined to mean that there exists CC with AC cA\ll C^{\mathsf{c}} and BC=XB\cup C = X. Now on one hand, BC=XB\cup C = X implies C cBC^{\mathsf{c}}\subseteq B, so AC cBA\ll C^{\mathsf{c}} \subseteq B. But on the other, if ABA\ll B then Decomposability gives CC and DD with ACA\ll C, DBD\ll B, and C cD=XC^{\mathsf{c}}\cup D = X, whence A(C c) cA \ll (C^{\mathsf{c}})^{\mathsf{c}} and X=DC cBC cX = D \cup C^{\mathsf{c}} \subseteq B \cup C^{\mathsf{c}}.

On the other side, if we start with \bowtie then by a round-trip we obtain ABA\bowtie' B defined to mean that there exists CC with ACA\bowtie C and CB c=XC \cup B^{\mathsf{c}}= X. Now on one hand, if CB c=XC \cup B^{\mathsf{c}}= X then BCB \subseteq C, so ACA\bowtie C implies ABA\bowtie B. But on the other, if ABA\bowtie B then Decomposability gives CC and DD with ACA\bowtie C and DBD\bowtie B and CD=XC\cup D = X, whence DB cD\subseteq B^{\mathsf{c}} and so X=CDCB cX = C\cup D \subseteq C \cup B^{\mathsf{c}}.

In other words, the two constructively reasonable ways to define a (quasi-)proximity space give the same result in the Decomposable case.

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 Transitivity, \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.

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.

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 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 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).

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 ABA \ll B if Aint(B)A \subseteq int(B). 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 xAx\bowtie A then x{y¬(yA)}x\bowtie \{ y \mid \neg(y\bowtie A) \}), 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 ABA \ll B if Aint(B)A \subseteq int(B) as above in the second case, and ABA\bowtie B if xAxBx\in A \Rightarrow x\bowtie B 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.

Unlike in the classical case, the topology underlying a proximity space (which, recall, includes Symmetry) apparently need not be completely regular or even regular. However, we can say:

Theorem

Let XX be a proximal apartness or neighborhood space (with Symmetry). Then:

  • If XX satisfies Local Decomposability, its underlying topological space is regular.
  • If XX satisfies Decomposability and countable choice holds, then its underlying topological space is completely regular.
Proof

Suppose first that XX is a Locally Decomposable proximal neighborhood space and UU is an open set containing xx, which means {x}U\{x\} \ll U. By Local Decomposability, we have sets CC and DD such that {x}C\{x\} \ll C and DUD\ll U and X=DC cX = D \cup C^{\mathsf{c}}. Transitivity then gives us a set BB with {x}BC\{x\} \ll B \ll C, and Symmetry then tells us that C cB cC^{\mathsf{c}} \ll B^{\mathsf{c}}. Let V=int(B c)V = int(B^{\mathsf{c}}) and W=(B)W = \int(B); then VW=V\cap W = \emptyset, WW is an open neighborhood of xx, and X=DC cUVX = D \cup C^{\mathsf{c}} \subseteq U \cup V. Thus, XX is regular.

Now suppose XX is a Decomposable proximal neighborhood space and we have countable choice. Then given any opens A,BA,B with ABA\ll B, essentially the same proof as above shows that AA is “well-inside” BB. 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 f:Xf:X\to \mathbb{R} separating AA and BB, 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 XX is a completely regular topological space, then the above definition that ABA \bowtie B iff they are separated by a continuous function f:X[0,1]f: X\to [0,1] is also constructively valid, and produces a proximity space satisfying Symmetry and Decomposability.

Uniform spaces

If YY is 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 defines a biperfect proximity structure on YY. In terms of the other relations, this means ABA\bowtie B iff V(A×B)=V \cap (A \times B) = \emptyset for some entourage VV, and ABA\ll B iff V[A]BV[A] \subseteq B, where V[A]={yxA,(x,y)V}V[A] = \{ y \mid \exists x\in A, (x,y)\in V \}.

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 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 δ,,\delta,\bowtie,\ll may be defined as above from a uniformity. All the axioms of all three kinds of proximity are then provable, except Binary Additivity for δ\delta, Strong Symmetry for \ll, 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 ABA\bowtie B if there is a uniformly 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. 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 \ll) yields a proximity space satisfying Decomposability, which has the same underlying topology if our original uniform space was already completely regular.

Syntopogenous spaces

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.

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.
  4. σ\sigma is nonempty.

Note that conditions (1) and (2), together with Isotony for δ\delta, imply that any cluster σ\sigma is upwards-closed. Together with nullary Additivity for δ\delta, they also imply the nullary version of (3): σ\emptyset \notin\sigma. However, a cluster is not generally a filter: it is not closed under binary intersections. This is most obvious when we consider how XX embeds in its compactification, taking each point xx to the “principal cluster” σ x={A{x}δA}\sigma_x = \{ A \mid \{x\}\;\delta\; A \}, i.e. the collection of sets whose closure contains xx, which is certainly not generally closed under intersections. (If it were a filter, of course, condition (3) would say that it is prime.)

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 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 \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 Transitivity 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?

Generalized uniform structures

proarrowmonadpro-monadsymmetric versions
binary relationpreorderquasiuniformitysymmetric relationequivalence relationuniformity
binary function to [0,)[0,\infty)quasipseudometricquasiprometricsymmetric binary functionpseudometricprometric
topogenyquasiproximitysyntopogenysymmetric topogenyproximitysymmetric syntopogeny

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 January 6, 2017 09:12:09 by Mike Shulman (76.167.222.204)