Abstract and Applied AnalysisVolume 2011 (2011), Article ID 857278, 9 pagesdoi:10.1155/2011/857278

Research Article

Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

Agata Caserta

Department of Mathematics, SUN, 81100 Caserta, Italy

Received 20 June 2011; Accepted 23 August 2011

Academic Editor: Ljubisa Kocinac

Copyright © 2011 Agata Caserta. 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.

We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology.

1. Introduction

Since the end of the nineteenth century several outstanding papers appeared to formulate a set of conditions, which are both necessary and sufficient, to be added to pointwise convergence of a sequence of continuous functions, to preserve continuity of the limit. Indeed, all classical kinds of convergences of sequences of functions between metric spaces (Dini, Arzelà, Alexandroff) are based on the pointwise convergence assumption that has been always considered a preliminary one. Recently, in [1, 2], Caserta et al. proposed a new model to investigate convergences in function spaces: the statistical one. Actually, they obtained results parallel to the classical ones, concerning the continuity of the limit, in spite of the fact that statistical convergence has a minor control of the whole set of functions. In [2] they proved that continuity of the limit of a sequence of functions is equivalent to several modes of statistical convergence which are similar, but weaker than the classical ones. A parallel to the classical results is expected since, after all, in [3] the authors found the statistical convergence to be the same as a very special regular triangular matrix summability method for bounded (and some unbounded) sequences. Thus many new results concerning statistical convergence follow from the corresponding known results for matrix summability.

In 1969 Bouleau [4, 5] defined the sticking topology as the weakest topology finer than pointwise convergence to preserve continuity. In [6], Beer presented two new topologies on , finer than the topology of pointwise convergence, which are indeed the decomposition of the sticking topology in upper and lower halves. For its upper part the closure of is the set of all upper semicontinuous functions, and the pointwise convergence of a sequence in to an upper semicontinuous limit automatically ensures convergence in this finer topology. Beer also pointed out that the pointwise convergence within the set of all upper semicontinuous functions does not ensure this convergence, even if the limit is continuous.

We realize that the pointwise convergence is a too strong assumption to analyze the upper and lower semicontinuous limit of functions, and as a result continuity, in fact the whole pointwise convergence conceals the properties that functions and limits must satisfy. We offer a decomposition of the pointwise convergence in upper and lower halves that will enable us to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity of limits and functions. We introduce two new properties for a sequence of functions with respect to a prospective limit function and give a characterization of upper (resp., lower) semicontinuity of the limit in terms of this new properties of the whole sequence.

The decomposition of the pointwise convergence is necessary to expose the salient features of these new important properties and display the asymmetric role of the upper and lower decomposition of the topology of pointwise convergence with respect to the upper and lower decomposition of the sticking topology. This phenomenon is completely concealed if we consider the whole pointwise convergence within the set of all continuous functions.

Thanks to a careful analysis of the behavior of this decomposition and its interplay with upper and lower semicontinuity of functions and limits, we can extend the results in [2]. We prove that whole statistical pointwise convergence in ensures convergence in the two symmetric topologies of the decomposition of the sticking topology, and therefore the statistical pointwise convergence of a sequence of continuous functions to a continuous function forces the convergence in a wider class of finer topologies.

2. Preliminaries

Let be metric space. We denote the power set of by , the nonempty (resp., nonempty finite) subsets of by (resp., ). Let be topological space, denotes the set of all real valued continuous functions on . Recall that a function is upper (resp., lower) semicontinuous at if for every there is such that for all then (resp., ). Also is upper semicontinuous if it is upper semicontinuous at every . By (resp., ) we denote the set of all real valued lower (resp., upper) semicontinuous functions defined on ; evidently .

A quas-iuniformity for a set is a collection reflexive relations on that forms a filter and such that for every there is such that . The pair is called quasi-uniform space. A subfamily of is a base for the quasi-uniformity if it is cofinal in with respect to the inclusion; that is, for every , there is such that .

If is a quasi-uniformity for , then where , is also a quasi-uniformity for . The collections and are called conjugate quasi-uniformities [7].

The quasi-uniform space will always be considered to be a topological space with the topology obtained by using as the family of all neighborhoods of a point all sets of the form , where runs over . Such a topology is called topology of the quasi-uniformity. Note that each topology on is induced by a quasi-uniformity (see [8, 9]), the most familiar of which is the Pervin quasi-uniformity [8].

A quasi-uniform space is said to be a uniform space, and the family will be called a uniformity for , if and only if the quasi-uniformity satisfies the symmetric relation: if , then . For a quasi-uniformity the smallest uniformity containing has a base all sets of the form where runs over a prescribed base for . Moreover, a topology is induced by a uniformity if and only if the space is completely regular [7, 10].

Let us familiarize the reader with the notion of statistical convergence, that first appeared in 1935 under the name of almost convergence in the celebrated monograph of Zygmund [11]. The definition of statistical convergence for sequences of real numbers was given by Fast in [12] and is based on the notion of asymptotic density of a subset of natural numbers. Let and . Put . Then one defines as the lower and upper asymptotic density of , respectively. If , then is the asymptotic (or natural) density of .

All the three densities, if they exist, are in . We recall also that for . A set is said to be statistically dense if . Let us mention that the union and intersection of two statistically dense sets in are also statistically dense. For additional properties of the asymptotic density, in a more general setting, the reader might consult [13].

A sequence in a topological space is said to converge statistically (or shortly, -converge) to , if for every neighborhood of , . This will be denoted by , where is a topology on .

It was shown [14] (see [15, 16] for ) that for first countable spaces this definition is equivalent to the statement: there exists a subset of with such that the sequence converges to . Recently in [17], Çakalli and Khan pointed out that the first countability is not a necessary condition.

3. Decomposition of Pointwise Convergence and Weakly Exhaustiveness in

Given and , a base for the standard uniformity for the topology of pointwise convergence on consists of all entourages of the form In what follows, we offer a decomposition of the pointwise convergence in upper and lower part to better visualize their hidden and opposite roles with respect to the lower and upper semicontinuity of limits and functions.

Definition 3.1. Let be metric space, , and . Consider the quasi-uniformity on having as a base all sets of the form The induced upper pointwise topology on is denoted by .

The conjugate quasi-uniformity has as a base all sets We denote the induced lower pointwise topology on by . Clearly the topology of pointwise convergence can be written as .

In the next definition we also give a decomposition of the statistical weakly exhaustive property for a sequence of functions. Statistical weakly exhaustiveness is a variation of the classical definition of weakly exhaustiveness [18] and has been deeply investigated in [1, 2].

Definition 3.2. A sequence in is said to be cofinally upper (resp., lower) weakly exhaustive at , simply -upper (resp., lower) weakly exhaustive at , if for every there is such that for all there exists a cofinal subset of such that for all we have (resp., ). The sequence is -upper weakly exhaustive if it is -upper weakly exhaustive at every .In case is statistically dense in we say that is statistically upper (resp., lower) weakly exhaustive at , shortly -upper (resp., lower) weakly exhaustive at .

We point out that upper (resp., lower) weakly exhaustiveness are weaker notion than what is called equisemicontinuity, in that the cofinal set of indexes depends on the point .

We now introduce two new properties for a sequence of real valued functions with respect to a prospective limit function at a point that play a significant role in our investigation (see Theorem 4.9).

Definition 3.3. Let , in , and . The sequence is said to be cofinally almost below (resp., above) around , shortly -almost below (resp., above) around ; if for every there is such that for all there exists a cofinal subset of such that for all we have (resp., ).When is statistically dense in we say that is statistically almost below (resp., above) around , shortly -almost below (resp., above) around .

In the next propositions we show that each of the notions -weakly exhaustive at and -almost below around are forced by one of the -convergence and -convergence and the other property.

Proposition 3.4. Let be a sequence of functions in that is -convergent to . If the sequence is -almost below around , then (i) is -upper weakly exhaustive at , and (ii) is upper semicontinuous at .

Proof. Let be fixed. Since is -convergent to at , there exists such that for all we have that ; hence for every , . Since is -almost below around , there is a such that for every there exists cofinal in , with for every , and for all , . Thus for all and we have . Let be fixed. Since is -almost below around , there is a such that for every there exists cofinal in and for all , . Since is -convergent to at , there exists such that for all we have that . By cofinality of the index set, let be such that . It follows that .

Proposition 3.5. Let be a sequence of functions in , that is, -convergent to . If is -upper weakly exhaustive at , then it is -almost below around .

Proof. Let be fixed. By assumption there is a such that for every there is cofinal in and for all , . Since is -convergent to at , there exists such that for all , . It follows that for all with , .

Proposition 3.6. Let be a sequence of functions in . If is -convergent to and is upper semicontinuous at , then is -almost below around .

Proof. Let be fixed, there is a such that for every , . Since is -convergent to at , there exists such that for all , . Thus for every , the subset is cofinal in and .

Therefore, if we assume the whole pointwise convergence the two properties, -upper weakly exhaustive at and -almost below around , associated to the upper and lower pointwise convergence, coincide.

Corollary 3.7. Let be a sequence of functions in that is pointwise convergent to and . The following are equivalent: (i) is upper semicontinuous at , (ii) is -upper weakly exhaustive at , (iii) is -almost below around .

All the previous propositions hold if we consider statistical convergences of the sequences of functions. Therefore, next corollary, similar to that in [1], improves the result weakening the condition on the sequence.

Corollary 3.8. Let be a sequence of functions in , that is, - convergent to . The sequence is -weakly exhaustive at if and only if is continuous at .

4. Decomposition of Sticking Topology in

First, let us recall some definitions and results given in [6].

Definition 4.1. Let be metric space, , and . Consider the quasi-uniformity on having as a base all sets of the form We denote the induced topology on as . Also we denote by the topology induced on by the conjugate quasi-uniformity. If we take the uniformity generated by our standard uniformity for and its conjugate we get a uniformity having as a base all sets of the form The topology induced by this uniformity is called the sticking topology by Bouleau in [4, 5] and the topology of strong pointwise convergence by Beer and Levi in [19]. This topology has the intrinsic property to preserve continuity: is -closed in , and the -convergence reduces to pointwise convergence on itself. In [20] the authors gave a complete characterization of continuity for the pointwise limit of continuous functions. In [1, 2], it has been proved that similar results about continuity of the limit function are true for statistical pointwise convergence of sequences of functions between metric spaces.

In [1], a statistical version of the classical Alexandroff convergence, introduced in 1948 in [21] (see [20]), and the well-known quasi-uniform convergence, introduced by Arzelà [22] in 1883 and extended by Bartle [23] in 1955, were defined (see also [20]).

Definition 4.2. A sequence in is said to be statistically Alexandroff convergent to , denoted by being -, provided and for every and every statistically dense set there exist an infinite set and an open cover such that for every we have .

Definition 4.3. A sequence in is said to be statistically Arzelà convergent to , denoted by being - convergent to , if is - convergent to , and for every and every statistically dense set there exists a finite set such that for every it holds for at least one .

In [2] (Theorem 2.3) the authors proved the following characterization for the continuity of a statistical pointwise limit of continuous functions.

Theorem 4.4 (see [2]). Let be a sequence of functions in , that is, - convergent to . The following are equivalent: (i) is continuous, (ii) is - convergent to on compacta, (iii) is - convergent to , (iv) is - convergent to .

We continue our analysis with respect to the statistical version of these convergences, and we prove that the whole statistical pointwise convergence in ensures convergence in the two symmetric topologies of the decomposition of the sticking topology.

In [6] (Proposition 3.2), the author proved that if is a sequence of functions in , that is -convergent to , then the sequence is -convergent to . And he shows that the -convergence in does not ensure -convergence, even if the limit is continuous and convergences are statistical. In the next proposition we clarify what exactly must be added to the -convergence of a sequence of lower semicontinuous functions with an upper semicontinuous limit to ensure the -convergence even from the statistical point of view.

Proposition 4.5. Let be a sequence of functions in . If is --convergent to and is -almost below in , then is - convergent to .

Proof. Fix a finite subset of , and . We show that there is statistically dense such that for every it follows that . Since is --convergent to , there is statistically dense, such that for every and for all we have . Let , since is lower semicontinuous at each ; there exists such that for every , . By upper semicontinuity of at each , there exists such that for every , . Also is -almost below at each ; therefore for every , there is such that for every , there is statistically dense, and for all , . Set . For every we have . Let . For all and each it follows that and ; thus is statistically -convergent to .

We underline that there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking topology. In fact with the same assumptions on the functions and the limit, the necessary condition to get the - convergence is guaranteed by the - convergence.

Proposition 4.6. Let be a sequence of functions in and . If is - convergent to , then (i) is continuous, (ii) is - convergent to , (iii) is -almost below .

Proof. To prove , it is sufficient to prove lower semicontinuity of . Fix that and . By assumption there is statistically dense, and for all , . Thus there is such that for all , and . Let . Since is lower semicontinuous, we may assume that for every we have also . It follows that for , To prove , fix and . Let , since statistically - converges to ; there is statistically dense, such that for all and , we have and . By upper semicontinuity of at each there exists such that for all , . Let . For all and it follows that .

From Propositions 4.5 and 4.6 we have the following corollaries.

Corollary 4.7. Let be a sequence of functions in and . If is - convergent to , then is - convergent to .

Corollary 4.8. Let be a sequence of functions in and . If is --convergent to , then is - convergent to .

If we restrict our attention to continuous functions, we can relate results of the previous section due to the pointwise convergence in , namely, Corollary 3.7, to the sticking topology in .

Theorem 4.9. Let be in . If is - convergent to , then the following are equivalent: (i) is continuous, (ii) is -almost below and -almost above ,(iii) is -weakly exhaustive, (iv) is - convergent to .

Combining Corollaries 4.7 and 4.8, we show that assuming the statistical pointwise convergence in , the convergence in one of the two symmetric topologies and automatically ensures convergence in the whole sticking topology, since they coincide.

Corollary 4.10. Let and . If is - convergent to , then the following are equivalent: (i) is - convergent to , (ii) is - convergent to .

Acknowledgments

The paper is supported by GNSAGA. The author is very grateful to Professor Giuseppe Di Maio for the valuable comments and suggestions that have become improvements. The author also thanks the referees for the careful reading and suggestions.

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