Proximity space

A set $ P $ with a binary relation $ \delta $ on the set of all its subsets which satisfies the following axioms:

1) $ A \delta B $ is equivalent to $ B \delta A $( symmetry);

2) $ A \delta ( B \cup C) $ is equivalent to $ A \delta B $ or $ A \delta C $( additivity);

3) $ A \delta A $ is equivalent to $ A \neq \emptyset $( reflexivity). The relation $ \delta $ defines a proximity structure or simply a proximity on $ P $; if $ A {\overline \delta \; } B $, where $ \overline \delta \; $ means "non-d" , $ A $ and $ B $ are called remote sets. Proximity spaces were introduced in 1936 (published in 1951 [1]). The properties of proximity spaces are a generalization of the uniform properties of a metric space — in analogy to the generalization of the latter's continuity properties to a topological space. An attempt to introduce a structure somewhat similar to that of the proximity structure dates back [3] to the times when the concept of a topological space was not yet rigorously defined, and a derived set was considered rather than the closure; the inter-set relation introduced at that time corresponded to a common (possibly to an "ideal" ) point of contact.

The more substantial concepts of a proximity satisfy not only the axioms 1)–3), but also additional axioms analogous to the separation axioms; these include, for example, a Hausdorff proximity, which satisfies the axiom: $ \{ x \} \delta \{ y \} $ is equivalent to $ x = y $( instead of 3) it is sufficient to accept its corollary $ \emptyset \overline \delta \; \emptyset $); and a normal proximity, which is distinguished by the axiom: If $ A {\overline \delta \; } B $, then there exist disjoint sets $ U $ and $ V $ such that $ A {\overline \delta \; } ( P \setminus U) $ and $ B {\overline \delta \; } ( P \setminus V) $.

The appearance of the axioms of a proximity space, which are a natural symmetrization of the axioms of a topological space, couched in the terms of closure (i.e. the proximity of a set and a point) became possible after it had been established that the property of mappings $ f: M _ {1} \rightarrow M _ {2} $ between metric spaces, consisting in the fact that any sets located in $ M _ {1} $ at zero distance have images in $ M _ {2} $ that are also infinitely near, is exactly equivalent to uniform continuity. (The similar topological property $ f[ K] \subset [ fK] $, where $ [ K] $ is the closure of $ K $, is sometimes taken as the definition of continuity.) Thus, any metric, dist, on a set $ P $ generates a proximity on it ( $ A \delta B $ is equivalent to $ \mathop{\rm dist} ( A, B) = 0 $), and $ \delta $- continuity in the sense of the latter is equivalent to uniform continuity [2]; the proximity spaces for which such a metric is possible are said to be metrizable. A proximity structure generates a topological structure (topology): The closure of a set $ K $ is defined as follows: $ x \in [ K] $ if and only if $ \{ x \} \delta K $. The $ \delta $- continuity of a mapping implies its continuity in this topology. Proximity spaces which generate the same topology are not necessarily $ \delta $- isomorphic; thus, Euclidean space and Lobachevskii space are not $ \delta $- isomorphic, even though they are homeomorphic [2]. The topology of a Hausdorff proximity is Hausdorff; on the contrary, a normal proximity is merely completely regular (cf. Completely-regular space): closed disjoint sets are not necessarily distant. Moreover, all completely-regular topologies are generated by a normal proximity; in the case of a compact space this proximity is unique. Since any distant sets in a proximity space may be separated by a $ \delta $- continuous function [2], they are functionally separable in the sense of its topology; proximity spaces for which the opposite is true are known as Stone–Čech proximity spaces.

The presence of a topology in proximity spaces implies certain generalizations of the proximity structure, which usually consist in replacing the normality axiom by a weaker one. Examples are the Lodato proximity: If $ A {\overline \delta \; } B $, then $ A {\overline \delta \; } [ B] $; the Fedorchuk proximity ( $ \theta $- proximity): $ A {\overline \delta \; } B $ is equivalent to the existence of a closed set $ C \supset A $ for which the interior of the closure, $ \mathop{\rm Int} [ C] $, coincides with $ A $ and, in addition, $ A {\overline \delta \; } ( X \setminus [ C]) $; etc.

That the concept of a proximity space is natural is also manifested by the fact that any proximity space has a unique compactification; thus there is a one-to-one correspondence between homeomorphic proximity spaces and the compactifications of the topological space they generate. A $ \delta $- continuous mapping is a mapping $ f: P \rightarrow Q $ such that for any $ A, B \subset P $ it follows from $ A \delta B $ that $ fA \delta fB $, and only such a mapping can be extended to a continuous mapping $ \overline{P}\; \rightarrow \overline{Q}\; $ between the compactifications. These theorems, which were first formulated by Yu.M. Smirnov [4], had been actually demonstrated as early as 1948 by P. Samuel. He studied the compactification of uniform spaces and found [10] that it is only some rather than all uniform spaces — the so-called pre-compact spaces, i.e. spaces with a compact completion, that are uniformly continuously imbeddable in a compactum, but that each uniform space has a unique compactification (an $ S $- reflection) $ r: P \rightarrow SP $( the inverse mapping is continuous, but not necessarily uniformly continuous), and the extension can be made to include all neighbourhoods of a certain type, for example, all neighbourhoods of the type $ P \times P \setminus A \times B $. Thus, uniform structures are subdivided into equivalence classes — two uniformities are equivalent if they have the same $ S $- reflection. A proximity in a uniform space is introduced by the condition: $ A \delta B $ if for any neighbourhood $ \Omega $ the relation $ ( A \times B) \cap \Omega \neq \emptyset $ is valid; just like the above equivalence, the construction of imbedding is a $ \delta $- isomorphism, that is, a one-to-one $ \delta $- continuous mapping.

The one-to-one correspondence between proximities and compactifications is responsible for the fact that, during a long period of time following the establishment of this correspondence, the principal objects of study were the particular properties of these spaces which can be directly formulated in terms of compactifications. These are, for example, the dimensions $ \delta d $( but not $ \Delta d $) [6], the proximity weight, proximal connectedness, etc. G. Cantor was the first to note the simple property of proximal connectedness, viz. $ A \times ( P \setminus A) \neq \emptyset $ implies $ A \delta P \setminus A $. Cantor defined a continuum (his continuum was introduced only at a later date) as a proximally-connected complete subspace in $ E ^ {n} $. Even though, in principle, all properties of proximity spaces $ P $ are comprised in the properties of the injection $ P \rightarrow \overline{P}\; $, it must be borne in mind that, firstly, not all of them by far are comprised in the properties of $ \overline{P}\; $ itself and, secondly, the particular properties of the injection $ P \rightarrow \overline{P}\; $ which correspond to such properties of $ P $ as, say, metrizability, completeness or regularity, are not known. Proximity spaces are valuable since they serve to study compactifications, but the converse is not true. Properties of proximity spaces which cannot be directly described in topological terms are said to be uniform. The first uniform property of proximity spaces which was systematically studied was completeness: attempts to introduce Cauchy filters or fundamental sequences in terms of compactifications were unsuccessful.

A covering (cf. Covering (of a set)) $ \omega $ of a proximity space $ P $ is said to be a uniform covering if it is the start of a star-refining sequence of coverings $ \omega \gg \omega _ {1} \gg \omega _ {2} \gg \dots $( i.e. $ \mathop{\rm St} ( x, \omega _ {n+ 1 } ) \subset U \in \omega _ {n} $), none of which disconnects proximal sets, i.e. always $ L {\overline \delta \; } P \setminus \mathop{\rm St} ( L, \omega _ {n} ) $.

The set of uniform coverings of a proximity space is identical with the union of all uniform structures compatible with this space [4]. One may also define uniform coverings as inverse images under $ \delta $- continuous mappings into metric spaces of coverings with a positive Lebesgue number.

Completeness defined using Cauchy filters — filters $ \Phi $ such that $ \Phi \cap \omega \neq \emptyset $ for any uniform covering $ \omega $— corresponds to intuitive concepts and is identical with metric completeness for metric spaces. For a proximity space to be complete, it is sufficient that any uniform space compatible with it be complete. It is not known (1977) whether or not this is also a necessary condition; in any case, examples to the contrary can only be provided by incorrect (see below) proximity spaces. Completions

of proximity spaces have been constructed as the least (no longer unique) complete extensions; at the same time completions are the largest ones admitting extensions of all uniform coverings to a uniform covering and also of all $ \delta $- continuous mappings into complete proximity spaces (in other words, the subcategory of complete proximity spaces, and also the subcategory of compact spaces, is a reflective category (cf. Reflexive category). Spaces with compact completions (i.e. pre-compact spaces) are characterized by the fact that a finite uniform subcovering may be chosen out of each one of their uniform coverings.

The product of proximity spaces $ P $ and $ Q $ was originally introduced by inducing on the set-theoretic product the proximity from the topological product $ \overline{P}\; \times \overline{Q}\; $ of their compactifications. Such a product, though identical with the product in the sense of the category of proximity spaces, is geometrically inconvenient and mainly serves in the construction of outlandish examples. E.g., this product (which is usually denoted by $ P \cdot Q $) of two infinite discrete spaces is not discrete and is not even metrizable; the product of two straight lines is non-metrizable and an-isotropic: a rotation by an acute angle of the "plane" will not be a $ \delta $- isomorphism, etc.

The proximity product of two (and, similarly, of any number of) proximity spaces $ P $ and $ Q $ is the product with the coarsest proximity, [7], in which all Cartesian products of uniform coverings of the factors, i.e. all coverings of the type $ \omega ( \times ) \psi = \{ {U \times V } : {U \in \omega, V \in \psi } \} $ are uniform. The requirement that both projections $ P \times Q \rightarrow P $ and $ P \times Q \rightarrow Q $ be $ \delta $- continuous is equivalent to the corresponding condition for finite uniform coverings. For uniform spaces both definitions are equivalent, unlike for proximity spaces, since the subcategory of metric spaces is not closed with respect to Cartesian products in the category of proximity spaces, though it is closed in the categories of topological and uniform spaces. The proximity product may be understood as a natural extension of the product functor from the subcategory of metrizable proximity spaces to all proximity spaces, i.e. the proximity of the product $ Q _ {1} \times Q _ {2} $ is the coarsest proximity in which the $ \delta $- continuity of an arbitrary mapping $ f _ {1} \times f _ {2} : Q _ {1} \times Q _ {2} \rightarrow M _ {1} \times M _ {2} $, where $ M _ {i} $ are metric spaces, follows from the $ \delta $- continuity of both mappings $ f _ {i} : Q _ {i} \rightarrow M _ {i} $, if $ M _ {1} \times M _ {2} $ is understood to mean the ordinary product of metric spaces [7].

An odd, but inevitable consequence of the fact that the subcategory of metric spaces is not closed is: A "vector-function" $ P \rightarrow A \times B $ in which both "coordinate" functions are $ \delta $- continuous is not necessarily $ \delta $- continuous (irrespective of whether $ A $ and $ B $ are understood to mean arbitrary proximity spaces or metrizable proximity spaces only), if the product of metric spaces is understood in its ordinary sense. Proximity spaces for which this never happens are said to be correct. Correct proximity spaces may also be defined as spaces in which the projection of the diagonal of $ P \times P $ is a $ \delta $- isomorphism. In correct proximity spaces and only in such spaces is the intersection $ \omega ( \cap ) \psi = \{ U \cap V, U \in \omega, V \in \psi \} $ of two uniform coverings again a uniform covering, and the collection of such intersections is a uniformity [7].

Any proximity space $ P $ has a coarsest correct space $ P! $ which majorizes $ P $— the so-called correction. The correction $ P! $ is at the same time the finest proximity space to which any mapping of type $ M \rightarrow P $, where $ M $ is metrizable, can be extended (i.e. the mapping $ M \rightarrow P! $ is $ \delta $- continuous if the mapping $ M \rightarrow P $ is $ \delta $- continuous). The theorem remains valid if $ M $ is replaced by an arbitrary correct proximity space $ Q $; thus, the sets of $ \delta $- continuous mappings $ Q \rightarrow P $ and $ Q \rightarrow P! $ are in a natural one-to-one correspondence, i.e. the subcategory of correct proximity spaces is coreflective, and the functor "!" is a coreflector. Metrizable proximity spaces are correct (and the projection $ P! \rightarrow P $ is a homeomorphism), and the subcategory of metric spaces is closed with respect to the Cartesian product in the category of correct spaces. Not only metric spaces, but also pre-compact spaces are correct; moreover, the theorem: If for all $ Q $ it follows from $ Q! = P $ that $ Q = P $, is equivalent to saying that $ P $ is pre-compact.

The corrections of the products "" and "×" are identical: $ ( P \cdot Q) ! = ( P \times Q)! = ( P! \times Q!)! $. Thus, the product $ P \cdot Q $ is almost always incorrect, since $ P \cdot Q = P \times Q $ is valid if and only if one of the factors is pre-compact; it is not known (1977) if the product of correct proximity spaces is correct.

The dimension theory of proximity spaces shows certain special features. First of all, proximity spaces are considered to have two dimensions "along the coverings" $ \delta d $ and $ \Delta d $( the definition is analogous to that of the topological dimension dim, but with the use of finite or, respectively, arbitrary uniform coverings), and only one inductive dimension $ \delta { \mathop{\rm Ind} } $, disjoint sets being replaced by remote ones [11]. However, the proximity analogue of a partition is non-trivial: A set $ H $" liberating setliberates" remote sets $ A $ and $ B $ if $ H {\overline \delta \; } A \cup B $ and if it follows from $ H {\overline \delta \; } U \supset A \cup B $ that $ U = V \cup W $ and $ A \subset V {\overline \delta \; } W \supset B $. Not a single case has as yet (1977) been encountered in which any two of these three dimensions are not identical. The dimension $ \delta d $ is finitely additive, and $ \delta dP = { \mathop{\rm dim} } P $; if $ P $ is dense in $ Q $, then $ \delta dP = \delta dQ $. The dimension $ \delta \mathop{\rm Ind} $ is not less than the dimension $ \delta d $ and cannot decrease under transition to a dense subspace, but it is not known (1977) if it may increase in doing so; it remains unchanged under completion. For metrizable spaces $ \delta { \mathop{\rm Ind} } M \leq \Delta dM $, and in an arbitrary space either $ \Delta dP = \delta dP $ or $ \Delta dP = \infty $ is true. A number of examples of uniform spaces with non-identical dimensions have been constructed, but none of these constructions is applicable to proximity spaces. In the proximity product $ N \times N $, where $ N $ is discrete and countable, the dimensions $ \Delta d $ and $ \delta d $ are equal if and only if the proximity space is correct [8]. At the same time, if $ N \times N $ is incorrect, it follows that $ \Delta d $ is no longer monotone (because $ \Delta d ( N \times N) = 0 $).

References

[1]	V.A. Efremovich, "Infinitesimal spaces" Dokl. Akad. Nauk SSSR , 76 (1951) pp. 341–343 (In Russian)
[2]	V.A. Efremovich, "Geometry of proximities I" Mat. Sb. , 31 (73) : 1 (1952) pp. 189–200 (In Russian)
[3]	F. Riesz, "Stetigkeitsbegriff und abstrakte Mengen" , Internat. Congress Mathematicians IV (Rome) , 2 , R. Accad. Lincei , Rome (1909) pp. 18–24
[4]	Yu.M. Smirnov, "On proximity spaces" Mat. Sb. , 31 (73) : 3 (1952) pp. 543–574 (In Russian)
[5a]	Yu.M. Smirnov, "On the completeness of proximity spaces I" Trudy Moskov. Mat. Obshch. , 3 (1954) pp. 271–306 (In Russian)
[5b]	Yu.M. Smirnov, "On the completeness of proximity spaces II" Trudy Moskov. Mat. Obshch. , 4 (1955) pp. 421–438 (In Russian)
[6]	Yu.M. Smirnov, "On dimensions of proximity spaces" Mat. Sb. , 38 (80) : 3 (1956) pp. 283–302 (In Russian)
[7]	V.Z. Polyakov, "Regularity and the product of proximity spaces" Mat. Sb. , 67 (109) : 3 (1965) pp. 428–439 (In Russian)
[8]	V.Z. Polyakov, "On the regularity of the proximity product of regular spaces" Mat. Sb. , 68 (110) : 2 (1965) pp. 242–250 (In Russian)
[9]	V.Z. Polyakov, "On extension of Cartesian products" Math. USSR-Sb. , 5 : 4 (1968) pp. 559–569 Mat. Sb. , 76 (118) : 4 (1968) pp. 593–604
[10]	P. Samuel, "Ultrafilters and compactifications of uniform spaces" Trans. Amer. Math. Soc. , 64 (1948) pp. 100–132
[11]	J.R. Isbell, "On finite-dimensional uniform spaces" Pacific J. Math. , 9 (1959) pp. 107–121
[a1]	Somashekhar A. Naimpally, Brian D. Warrack, "Proximity spaces", Cambridge Tracts in Mathematics 59, Cambridge University Press (1970) ISBN 0-521-07935-7 Zbl 0206.24601; reprinted (2008) ISBN 978-0-521-09183-1 Zbl 1184.54027