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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 412796, 10 pages

http://dx.doi.org/10.1155/2013/412796

## Strong Proximal Continuity and Convergence

^{1}Department of Mathematics, Seconda Università degli Studi di Napoli, 81100 Caserta, Italy^{2}Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy^{3}96 Dewson Street, Toronto, ON, Canada M3J 1P3

Received 16 October 2012; Accepted 9 January 2013

Academic Editor: Yuriy Rogovchenko

Copyright © 2013 Agata Caserta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In several situations the notion of uniform continuity can be strengthened to strong uniform continuity to produce interesting properties, especially in constrained problems. The same happens in the setting of proximity spaces. While a parallel theory for uniform and strong uniform convergence was recently developed, and a notion of proximal convergence is present in the literature, the notion of strong proximal convergence was never considered. In this paper, we propose several possible convergence notions, and we provide complete comparisons among these concepts and the notion of strong uniform convergence in uniform spaces. It is also shown that in particularly meaningful classes of functions these notions are equivalent and can be considered as natural definitions of strong proximal convergence. Finally we consider a function acting between two proximity spaces and we connect its continuity/strong continuity to convergence in the respective hyperspaces of a natural functor associated to the function itself.

#### 1. Introduction

In topology and analysis the concepts of uniform continuity and uniform convergence on compacta play a central role. It is well known that a continuous function restricted to a compact set is uniformly continuous on that set. In the setting of metric spaces, Beer and Levi [1] observed that a continuous function is actually uniformly continuous on a sufficiently small enlargement of every compact set. This led them to introduce the concept of strong uniform continuity on a set that is not necessarily compact. The concepts of uniform continuity and strong uniform continuity agree on the whole space, but they may differ on a subset. This phenomenon is interesting since it is rare; generally one comes across concepts which differ globally and agree locally. It is also known that uniform convergence preserves uniform continuity but on the other hand this may fail for strong uniform continuity. Thus in Beer and Levi [1] and Caserta et al. [2] a definition of strong uniform convergence is considered, enjoying the property of preserving strong uniform continuity. This concept is related to the sticking convergence defined by Bouleau [3] (see also [4]). In Beer [5], the author extends the concept of strong uniform continuity to the more general setting of Hausdorff uniform spaces and develops the rudiments of the theory. Since in any uniform space a natural proximity is associated to the uniformity, it is then useful to consider notions of continuity and convergence directly connected to the proximity structure. Thus a notion of proximal continuity has been proposed, with the property of preserving the nearness of two sets. This is a stronger notion than simple continuity, which preserves nearness of a point and a set. The equality of the two properties provides a powerful tool in studying the problem of extension of continuous functions from dense subspaces; see Gagrat and Naimpally [6]. In analogy with the uniform case, a natural concept of convergence is defined in this setting. It is called proximal convergence, or also Leader convergence, since it was defined by Leader [7]. In his investigation Leader in particular showed that uniform convergence implies proximal convergence and that the converse in general does not hold. However, when considering functions acting between the two proximity spaces and , it is known that two convergence modes are equal in the following three cases: (a) is totally bounded, (b) the converging net is a sequence or the directed set of the net is linearly ordered, and (c) is compact (see [7–10]). Furthermore, proximal convergence preserves continuity as well as proximal continuity. Then Beer and Levi [1] introduced some new concepts of local-type proximal continuity and local-type proximal convergence, in the setting of metric spaces, and provided some characterizations of strong uniform continuity and strong uniform convergence in terms of the analogous proximal properties. These characterizations rely on convergence of various types on bornologies. Some extensions are also provided in Beer [5].

In this paper, after revising the various concepts of continuity in uniform and proximal spaces, we introduce several forms of strong proximal convergences, and we investigate their connections in the setting of Tychonoff spaces with compatible uniformities and proximities. We also compare them with uniform convergences, and we study them on bornologies. Then we connect uniform and proximal continuity and convergences of functions with the behavior of a natural functor in the hyperspace associated to a given function , that is, the functor . In the hyperspace we consider the proximal topology and the Hausdorff-Bourbaki uniformity. Our results generalize similar ones in Beer and Levi [1] and Di Maio et al. [11–13].

#### 2. Preliminaries

Given a topological space , we will denote by () the set of all subsets (closed subsets) of , and by () the set of all nonempty (nonempty closed) subsets. A central role in the paper will be played by the notion of bornology. Thus, let us start with the following definition.

*Definition 1. *Given a set , a bornology on is a family of subsets of which is a covering, closed under finite union, and hereditary. By a base for a bornology one means a subfamily of that is cofinal with respect to inclusion.

There is the smallest bornology on , denoted by , containing (only) the *finite *subsets of . Of course is the greatest bornology on . Another important bornology is the bornology of the relatively compact subsets of . In the sequel we shall constantly assume that bornologies have a closed base. We introduce now some relevant properties of bornologies, introduced and studied in [5, 14, 15].

*Definition 2. *A bornology on a Hausdorff uniform space is said to be stable under small enlargements if it contains an enlargement of each of its members ; is said to be shielded from closed sets if for every there is such that and each neighborhood of contains an enlargement of .

Clearly each bornology that is stable under small enlargements is shielded from closed sets, but the converse does not hold: is always shielded but stable under small enlargements if and only if all points are isolated, and is stable under small enlargement if and only if is locally compact.

In the paper we focus only on Tychonoff topological spaces with associated Efremovič symmetric uniformities. For a complete reference on proximity spaces see Naimpally and Warrack [16]. It is well known that to any uniformity on a uniform space a natural proximity is associated: if there is no a symmetric such that . In the other way around the situation is different: given a proximity space , there are in general several, not equivalent uniformities on , having as a natural proximity. Thus, we shall consider uniform spaces when we deal with uniform structures, and the associated proximity spaces , when we speak about proximal properties. We shall use indifferently the notation and to indicate that is false. We now introduce some continuity notions we shall use throughout the paper.

*Definition 3. *Let be uniform spaces, and a bornology on . One says that a function is uniformly continuous if for every there exists such that for every :
Given a nonempty subset of , is uniformly continuous on if its restriction is uniformly continuous. Finally, we say that is uniformly continuous on if is uniformly continuous on , for every .

*Remark 4. *Uniform continuity can be expressed in the following equivalent form:
This last remark suggests the following enforcement of uniform continuity.

*Definition 5. *Let be uniform spaces, and a bornology on . let be a nonempty subset of . One says that the function is strongly uniformly continuous on if for every there exists such that
We say that is strongly uniformly continuous on if it is strongly uniformly continuous on , for every .

*Definition 6. *Let and be proximity spaces, and a bornology on . We say that the function is *proximally continuous on * if
Given a nonempty subset of , is *proximally continuous on * if its restriction is proximally continuous. Finally, we say that is proximally continuous on if is proximally continuous on , for every *. *

*Definition 7. *Let and be proximity spaces; let be a nonempty subset of and a bornology on . One says that the function is strongly proximally continuous on if
Finally, one says that is strongly proximally continuous on if is strongly proximally continuous on , for every .

We shall use the notation to denote the family of the functions from to which are strongly proximally continuous on the bornology .

*Remark 8. *Proximal and strong proximal continuity on a set can be equivalently expressed in the following way: is proximally continuous on if
is strongly proximally continuous on if

We now recall some connections among the introduced continuity notions. Strong uniform continuity on a set implies uniform continuity on that set. The converse does not hold in general; for a complete characterization of equivalence among the two continuities in uniform spaces, see Beer [5]; strong proximal continuity implies proximal continuity, and in general the two notions do not coincide; furthermore (strong) uniform continuity implies (strong) proximal continuity, for the natural proximity associated to the uniformity. On the contrary a proximally continuous function needs not to be uniformly continuous for the uniformities compatible with the proximity. As an example, take a proximity space and uniformities and on such that is strictly finer than . The identity map is not uniformly continuous, but the associated map from the proximal space into itself is so. However, if , are metric spaces and is proximally continuous with respect to the induced proximities, then it is also uniformly continuous with respect to their underlying uniformities (see [17, Corollary ]). More generally, it is clear that on the whole bornology the two notions of strong continuity do coincide and are equivalent to continuity of .

For the coincidence of the two notions of uniform continuity and strong uniform continuity on a bornology, let us recall the result proved in Beer [5, Theorem 3.5] that a continuous function acting between two uniform spaces, if it is uniformly continuous on a bornology shielded from closed sets, is automatically strongly uniformly continuous on the bornology. To provide a similar result in the proximity setting, we rephrase the notion of bornology stable under small enlargements, given in Beer [5, Definition 3.2] in a uniform setting, in order to have the analogous property in proximal spaces.

*Definition 9. *Given a proximity space and a bornology on it, one says that is *stable under small enlargements* if for every in there is in such that .

The next Lemma is useful to prove when proximal continuity and strong proximal continuity coincide on a bornology.

Lemma 10. *Let and be proximity spaces. Let such that . Suppose that for each such that it holds . Then is strongly proximally continuous on .*

*Proof. *Take such that . Let . We claim that . Otherwise, setting , it is , but then is such that , against the assumption. Thus and then by assumption . Since , it follows that .

Proposition 11. *Let and be two proximity spaces. Let be a bornology on stable under small enlargements. Then the following are equivalent: *(1)is proximally continuous on ; (2) is strongly proximally continuous on .

*Proof. *We only need to prove that (1) implies (2). Fix and let be such that . From Lemma 10 it is enough to prove that if is such that , then . But this is an immediate consequence of the fact that is proximally continuous on .

Since the bornology is not stable under small enlargements, but it is shielded from closed sets, it is easily seen that in the above proposition the assumption on the bornology of being stable under small enlargements cannot be weakened, since any function is proximally continuous on , but only the continuous functions are strongly proximally continuous on .

In Beer [5, Theorem 3.5] a result similar to Proposition 11 in uniform setting is proved.

#### 3. Convergences

In this section we recall some notions of uniform and proximal convergence, and related properties. Some of the definitions are classical; some other are new. We also make several comparisons among them.

We first recall the well-known definition of uniform convergence between uniform spaces.

*Definition 12. *Let be uniform spaces, and let be a bornology on . Let be a directed set and , . One says that the net uniformly converges to on if
uniformly converges to on if it converges uniformly on every .

*Definition 13 (see Beer [5]). *Let be uniform spaces, and let be a bornology on . Let be a directed set and , . One says that the net strongly uniformly converges to on if
strongly uniformly converges to on if it converges strongly uniformly on every .

*Remark 14. *Strong uniform convergence on can be equivalently expressed in the following way:
Evidently for each Hausdorff uniform codomain the strong uniform convergence is finer than the uniform convergence on ; they collapse on if and only if the bornology is stable under small enlargements, Beer [5, Theorem 3.1]; moreover they coincide on if and only if the bornology is shielded from closed sets, Beer [5, Theorem 3.4].

*Definition 15 (see Leader [7]). *Let and be proximity spaces, and a bornology on . Let be a directed set and , . The net proximally converges to on if
The net proximally converges on if it proximally converges on for every .

We introduce now the following definitions, in order to investigate strong proximal convergence on bornology.

*Definition 16. *Let and be proximity spaces, and a bornology on . Let be a directed set and let , . The net outer proximally converges to on if
The net *outer proximally converges* on if it outer proximally converges to on , for every .

*Definition 17. *Let be a topological space, let be a proximity space, and let be a bornology on . Let be a directed set and let , . One says that satisfies property on if
One says that satisfies property on if it satisfies property on every .

*Definition 18. *Let and be proximity spaces, and let be a bornology on . Let be a directed set and let , . One says that satisfies property on if
One says that satisfies property on if it satisfies property on every .

We observe at first that there is an equivalent way, often used in the sequel, to express the above definitions. For instance, outer proximal convergence of to on can be equivalently expressed in the following way: let and be proximity spaces, and a bornology on . Let be a directed set and , . The net outer proximally converges to on if

*Remark 19. *In the case the bornology is the bornology of the finite subsets of , proximal convergence coincides with pointwise convergence. This is no longer true for outer proximal convergence, as Example 23 shows. However it is clearly true in the case the limit function is continuous, but in such a case a more general result holds (see Proposition 25).

*Remark 20. *Property usually does not define topological convergence, at least on the whole space . For instance, it can happen that does not fulfill property . Moreover, the next Example 21 shows that on the whole space the limit of a net does not need to be unique. However both conditions are interesting and natural properties to consider and become topological convergences on important subsets of . In particular, is a sticking convergence type, as defined in Bouleau [3] and Bouleau [4], while looks like a natural type of proximal convergence, since it directly appeals to proximity properties and not to topological properties of the two proximity spaces.

In the sequel, when dealing with metric spaces, we always intend that the proximity is the natural proximity associated with the metric.

*Example 21. *In this example we see some pathological phenomenon associated to properties and , from the point of view of convergence of nets. Let , , and the bornology of the bounded sets. Let
Then does not satisfy property . Let , . Let be the zero function, for every . Then observe that satisfies for any valued zero on a dense subset of .

Coming back to the relations between the above properties/convergences, we immediately have the following.

Proposition 22. *Let and be proximity spaces, and a bornology on . Let be a directed set and , . *(1)If satisfies on a set , then outer proximally converges on the set , whence the same holds on a bornology.(2)If satisfies on a set , then proximally converges on the set , whence the same holds on a bornology.

*Example 23. *In this example we show that implication in Proposition 22 is not an equivalence. Let and . Define
Then outer proximal converges to , but does not proximal converge on ; thus in particular does not hold.

On the other hand, on proximal and outer convergence are independent, as the following example shows.

*Example 24. *In light of Example 23, we only need to produce a sequence proximally converging, but not outer proximally converging on . Let let and the zero function. Let be an enumeration of the rational numbers and set
Taking any open set the set fulfills , while for any , and thus there is no outer proximal convergence. Observe that the limit function is continuous, while the approximating functions are not.

With some (weak) assumptions on the functions further relations hold.

Proposition 25. *Let and be proximity spaces, and a bornology on . Let be a directed set and , . *(1)If and outer proximally on , then proximally on , whence the same holds on ; (2)If and outer proximally on , then satisfies on , whence the same holds on ; (3)If and if proximally on , then outer proximally on , whence the same holds on .

*Proof. * Let and let be such that . There is such that . Let . Thus . By assumption there is such that there are such that and , and thus for all .

(2) Suppose . Then there is such that . We conclude by applying the definition of outer proximal convergence to .

(3) Let and let be such that there is with and . By assumption there is such that for all , . Thus there are for all open sets such that . Let . Hence and .

The following Corollary is immediate.

Corollary 26. *Let and be proximity spaces. Then in the space the following are equivalent: *(1)* for the proximal convergence; *(2)* for the outer proximal convergence; *(3)*property holds for . *

Thus in the class of continuous functions proximal convergence is a sticking type convergence described also by means of property .

The next Lemma is useful to characterize outer proximal convergence.

Lemma 27. *Let be two proximity spaces, a bornology on let and . Then the following are equivalent: *(1)* is strongly proximally continuous on ; *(2)*for every such that ,
*

*Proof. *Let be such that . Then by (2) there is such that and . Thus and so (2) implies (1). Conversely, let be such that . Then there is such that . Then by (1) , and there is such that ; it follows that .

Observe that in the above condition the set can be equivalently supposed to be open.

From Lemma 27 we immediately get the following corollary.

Corollary 28. *Let and be proximity spaces and a bornology on . Then in the class a net proximally converges if and only if property holds.*

*Proof. *This is an immediate consequence of Lemma 27.

We now introduce two new properties on a pair , where as usual , with a directed set. Similar to the case of property , they do not define a topological convergence on , but on important subclasses of they do.

*Definition 29. *Let and be proximity spaces. Let and be a bornology on . One says that satisfies property on if
One says that satisfies property on if it satisfies property on each .

The next example shows that a constant net does not need to fulfill the property . Thus the same considerations made for property in Remark 20 apply to as well. Naturally, we shall call property * equiproximal *convergence, when it is a topological convergence.

*Example 30. *Let be an infinite dimensional separable Hilbert space with orthonormal basis (e.g., with ). Let be the following function:
Setting , the unit ball in the Hilbert space, happens that , while for any *enlargement ** of **. Thus one more time the constant net ** does not *converge to .

The second property is meaningful instead only when a bornology is specified. For this reason we do not make comparisons of this property with the other ones with specific examples on sets.

To prove our results, we start by giving a new definition, in the context of proximity spaces: it is the natural adaptation in this setting of the definition given by Beer in [5] in the uniform case.

*Definition 31. *Let be a proximity space and a bornology on it. One says that is shielded from closed sets if for each there is such that each open set containing is such that .

In the sequel we shall naturally say, for sets and as above, that is a *shield *for .

*Definition 32. *Let be a topological space, let be a proximity space, and let be a bornology on . Let be a directed set and let , . One says that satisfies property on if

Property is a stronger notion than both property and proximal convergence. However in the next propositions we see that on bornologies with particular features the new properties coincide with previous convergences.

Theorem 33. *Let and be two proximity spaces; let be a directed set and , . Let be a bornology on and stable under small enlargements; let be strongly proximally continuous on . Then the following are equivalent: *(1)* satisfies property on ; *(2)* proximally converges to on ; *(3)* satisfies property on . ** On the other hand, if for every which is strongly proximally continuous on , and are equivalent on , then is stable under small enlargements. *

*Proof. *It is enough to prove that and are equivalent. Let and let us prove that if proximally on then satisfies property on . Thus, suppose is such that . Since is strongly proximally continuous on and is stable under small enlargements, from Lemma 27 we know that there is such that . Since on , then eventually , and this concludes the proof of the first part. Let us now prove the second statement. Suppose is not stable under small enlargements. We shall produce a net proximally converging to a strongly proximally continuous limit function on , but on the other hand does not satisfy property on . There is such that for every it is not the case that . Let , directed by inclusion. Observe that, for every such that , for every there is . Now define
We now prove that proximally converges on to the function valued zero everywhere, but on the other hand does not satisfy property on . The first statement is clear, since for all such that , it is on , whence on . About the second setting , it is , but for any such that , for every , , and this ends the proof.

Theorem 34. *Let and be two proximity spaces let be a directed set and , . Let be a bornology on shielded from closed sets; let be strongly proximally continuous on . Then the following are equivalent: *(1)* satisfies property ; *(2)* proximally converges to on . ** On the other hand, if is a normal space, if for every which is strongly proximally continuous on , (1) and (2) are equivalent on , then is shielded from closed sets. *

*Proof. *Obviously we only need to prove that (2) implying (1), since (1) implies (2) is always true, with no assumptions on and . Let be a net proximally converging to on , let be strongly proximally continuous on , let , and suppose . Then there is such that . Since is strongly proximally continuous on , it is . Take such that shields . It is easy to see then that shields . Moreover . Since proximally on , then eventually , and this concludes the proof of the first part. Suppose now is not shielded from closed sets. We shall produce a net proximally converging to a strongly proximally continuous limit function on , but on the other hand does not satisfy property on . Thus, there is such that for every there exists a closed set with and . Let , directed by inclusion. Now, for every find a continuous function which is valued on and on . Mimicking the proof of Theorem 34, it is easy to see that proximally converges to on (since has a closed base), but does not satisfy property on .

As a conclusion, observe that on convergences proximal, outer proximal and property coincide, and in the subset of of the strongly proximal continuous functions also is equivalent to proximal convergence. In the next example we see that can be in general different from other convergences in .

*Example 35. *Let be the following subset of :
Let . Consider the bornology of the sets of the form , with and . Finally, consider , . Then proximally, but does not satisfy property . If instead we take , , we see that does not proximally converge on , while holds on .

From the previous considerations the following corollary is immediate.

Corollary 36. *Let and be two proximity spaces let be a bornology on stable under small enlargements. Then in the space the following are equivalent: *(1)* for the proximal convergence; *(2)* for the outer proximal convergence; *(3)*property holds for (sticking convergence); *(4)*property holds for (equiproximal convergence); *(5)*property holds for ; *(6)*property holds for . *

We end the section showing some connections among strong uniform convergence and the proximal convergences introduced before.

Proposition 37. *Let be uniform spaces, with associated proximities ; let be a bornology on . Let , , a directed set, be a net strongly uniformly converging to on . Then satisfies property on .*

*Proof. *Fix and let be such that there is such that and ; hence there is such that . Let be such that . Since strongly uniformly converges to on , there exists such that for all there are such that for all , . Let such that . There is and there is be such that . Thus . Since , it follows that . Moreover, for all , , that is, .

The next proposition provides a similar, yet independent, result.

Proposition 38. *Let be uniform spaces, with associated proximities let be a bornology on . Let be a net strongly uniformly converging to on . Then outer proximal converges to on . *

*Proof. *Fix and let be such that there is with and ; thus there is such that . Let be such that . Since strongly uniformly converges to on , there exists such that for all there are such that for all , . Let . Then all are open in and for all , , that is, .

#### 4. Convergence on Hyperspaces

In this section we show that the notions of strong uniform continuity (convergence) and strong proximal continuity (convergence) restricted to a bornology are equivalent to continuity (convergences) on hyperspaces with suitable topologies.

Given a function acting from a topological space to another topological space, it is natural to consider the two following functors associated to and acting on hyperspaces:
defined as
We remember that by *hyperspace* the set (or a subset of it, like ) is intended, endowed with some topology making continuous the embedding of in . In particular, in this section we shall deal with a net of functions , and a function , and we want to connect convergence on a bornology, in various senses, of to , with convergence of the associated nets and . Our focus will be on the proximal topology in the hyperspace of a proximal space, and on the Hausdorff Bourbaki uniform topology when considering uniform spaces. As it is well known, these topologies are the join of two topologies, a so-called upper topology and a so-called lower topology.

A basic upper neighborhood of , where is a proximity space, is
for some with , and is from a given fixed family of subsets of . Equivalently, a basic neighborhood of is
where and and is from a given fixed family of subsets of . The natural lower topology associated with the former upper one is the lower Vietoris topology (depending *only *from the topology on , and indicated by ) whose join with the upper Hausdorff topology gives rise to the well-known proximal topology, indicated from now on by . We remember that a subbase for the lower Vietoris topology is given by , where is an open set in and indicates the family of all sets such that . Instead, if is a uniform space, in the hyperspace it is possible to consider the so-called Hausdorff-Bourbaki uniformity. As usual, it consists of a lower and an upper part. A typical neighborhood of a set in the upper uniform topology is the following family of sets:
where . Similarly, a typical neighborhood of a set in the lower uniform topology is the following family of sets:
where . It is known that, when considering the proximity naturally associated with the uniformity , the upper proximal Hausdorff topology agrees with the upper uniform topology (see [18], page 50).

Our first intent is to show that we can limit our analysis to one of the two functors. Our choice will be on .

*Definition 39. *Given a topological space , let be a topology on the set of the nonempty subsets of . One shall say that is a compatible topology on if for each and , is a neighborhood of if and only if it is a neighborhood of any set such that .

Proposition 40. *Let and be two proximity spaces, let be a function and let be compatible topologies on and , respectively. Let be a bornology on and denote by the set . Then the following are equivalent: *(1) is -continuous; (2) is -continuous.

It is clear that the hyperspace topologies introduced above are compatible topologies.

Here is our first result.

Proposition 41. *Let be proximity spaces, and let be a given function. The following are equivalent: *(1) is proximally continuous on ; (2) is continuous; (3) is continuous.

*Proof. * Without loss of generality, we can assume . Let us prove that, given , for every neighborhood of there is a neighborhood of such that . We can assume , for some such that . Thus there is such that . Then . Take . Then for all it is ; thus .

By contradiction, there are such that and . Let be the following neighborhood of : . Let be a neighborhood of : we can suppose , with . Since there exists . Thus but , so that for no neighborhood of in it happens that , and as a result is not continuous at .

It is enough to prove that proximally continuous implies is continuous. But actually continuity of on is enough to get it. For, let be an open set such that . Take such that . Since is continuous at , there is an open set in such that for all . Thus is a neighborhood of such that, for all , .

The next result instead deals with the same issue, but when the function is supposed to be strongly proximal continuous, rather than merely proximally continuous. We do not provide the proof, since it mimics that one of Proposition 41. Compare the result of Proposition 42 with Theorem 3.3 in Beer and Levi [1].

Proposition 42. *Let be proximity spaces, and let be a given function. The following are equivalent: *(1) is strongly proximally continuous at every ; (2) is continuous at every ; (3) is continuous at every .

We now provide analogous results, in uniform spaces. We shall prove only the second result; the proof of the first one is similar.

Proposition 43. *Let be uniform spaces, and let be a given function. The following are equivalent: *(1) is uniformly continuous on ; (2) is continuous; (3) is continuous.

Proposition 44. *Let be uniform spaces, and let be a given function. The following are equivalent: *(1) is strongly uniformly continuous on ; (2) is continuous at every ; (3) is continuous at every .

*Proof. * Suppose by contradiction that (1) is not true. Then there is such that for all there exist and points , with
Since is by assumption continuous with respect to the upper Hausdorff uniformities, given as above, there is such that for all subsets of . But this contradicts (1), for .

By contradiction, suppose there is such that there is such that there exists with and ; that is, there are and with . But this contradicts strong uniform continuity of at .

The following corollary generalizes Theorem 2.2. in Di Maio et al. [11], which establishes the same result in metric spaces.

Corollary 45. *Let be uniform spaces, and let be a given function. The following are equivalent: *(1)* is strongly uniformly continuous on ; *(2)* is uniformly continuous on ; *(3)* is continuous; *(4)* is continuous. *

The next results instead deal with convergences on a bornology.

Proposition 46. *Let and be two proximity spaces; let be a directed set and , . Let be a bornology on . Then the following are equivalent: *(1) proximally converges to on ; (2) pointwise converges to at every .

*Proof. *Suppose proximally converges to on , and let be a neighborhood of . We suppose , for some , such that . Then eventually, so that eventually. The opposite implication follows the same pattern.

In the sequel we shall use the following definition of convergence, that we shall call topological convergence.

*Definition 47. *Let and be topological spaces. A net , , a directed set, topologically converges to if for every , for every neighborhood of there is a neighborhood of and such that for every .

Proposition 48. *Let and be two proximity spaces; let be a directed set and , . Let be a bornology on . Then the following are equivalent: *(1) equiproximally converges to on ; (2) topologically converges to at the members of ; (3) topologically converges to at the members of .

*Proof. *To show that (1) implies (2), we take and a neighborhood of in . Without loss of generality there exists such that , with . Then there exist and such that for all . Thus is a neighborhood of such that , and this shows the claim. For the converse relation, let and let be such that . Consider the following neighborhood of :
From (2) there are and an open set such that for all . Since is open in and contains there is such that . There is such that . Thus and so . To conclude the proof, it is enough to show that equiproximal convergence implies convergence of for the lower Vietoris topology, on and . Thus, take an open set such that . Then there is such that . Thus , and there exists , that without loss of generality we can suppose to be open, such that . This implies that for all , , and this ends the proof.

Specializing the above results to the bornology and collecting some previous statements we can establish the following interesting corollary.

Corollary 49. *Let and be two proximity spaces; let be a directed set and , . Then the following are equivalent: *(1)* proximally converges to on ; *(2)* equiproximally converges to on ; *(3)* topologically converges to ; *(4)* topologically converges to ; *(5)* pointwise converges to at every .*

*Proof. *We remind of that, as shown in Naimpally [19], continuity with respect to upper Vietoris topology on graphs automatically implies pointwise convergence, and this implies continuity with respect to the lower Vietoris topology: exactly the same proof applies when considering upper proximal topology instead of upper Vietoris topology.

#### References

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