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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.

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 [

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

Given a topological space

Given a set

There is the smallest bornology on

A bornology

Clearly each bornology

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 [

Let

Uniform continuity can be expressed in the following equivalent form:

Let

Let

Let

We shall use the notation

Proximal and strong proximal continuity on a set can be equivalently expressed in the following way:

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 [

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 [

Given a proximity space

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

Let

Take

Let

We only need to prove that (1) implies (2). Fix

Since the bornology

In Beer [

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.

Let

Let

Strong uniform convergence on

Let

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

Let

Let

Let

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

In the case the bornology

Property

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

In this example we see some pathological phenomenon associated to properties

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

Let

If

If

In this example we show that implication

On the other hand, on

In light of Example

With some (weak) assumptions on the functions

Let

If

If

If

(2) Suppose

(3) Let

The following Corollary is immediate.

Let

property

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.

Let

for every

Let

Observe that in the above condition the set

From Lemma

Let

This is an immediate consequence of Lemma

We now introduce two new properties on a pair

Let

The next example shows that a constant net does not need to fulfill the property

Let

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 [

Let

In the sequel we shall naturally say, for sets

Let

Property

Let

It is enough to prove that

Let

Obviously we only need to prove that (2) implying (1), since (1) implies (2) is always true, with no assumptions on

As a conclusion, observe that on

Let

From the previous considerations the following corollary is immediate.

Let

property

property

property

property

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

Let

Fix

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

Let

Fix

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

A basic upper neighborhood of

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

Given a topological space

Let

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

Here is our first result.

Let

The next result instead deals with the same issue, but when the function

Let

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

Let

Let

The following corollary generalizes Theorem 2.2. in Di Maio et al. [

Let

The next results instead deal with convergences on a bornology.

Let

Suppose

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

Let

Let

To show that (1) implies (2), we take

Specializing the above results to the bornology

Let

We remind of that, as shown in Naimpally [