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

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 ℝ^𝑋, 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 𝐶(𝑋) is the set of all upper semicontinuous functions, and the pointwise convergence of a sequence in 𝐶(𝑋) 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 𝐶(𝑋) 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 (𝑋,𝑑) be metric space. We denote the power set of 𝑋 by 𝒫(𝑋), the nonempty (resp., nonempty finite) subsets of 𝑋 by 𝒫₀(𝑋) (resp., ℱ₀(𝑋)). Let 𝑋 be topological space, 𝐶(𝑋) denotes the set of all real valued continuous functions on 𝑋. Recall that a function 𝑓∈ℝ^𝑋 is upper (resp., lower) semicontinuous at 𝑥∈𝑋 if for every 𝜖>0 there is 𝛿>0 such that for all 𝑦∈𝑆(𝑥,𝛿) then 𝑓(𝑦)<𝑓(𝑥)+𝜖 (resp., 𝑓(𝑦)>𝑓(𝑥)−𝜖). Also 𝑓 is upper semicontinuous if it is upper semicontinuous at every 𝑥∈𝑋. By ℒ(𝑋) (resp., 𝒰(𝑋)) we denote the set of all real valued lower (resp., upper) semicontinuous functions defined on 𝑋; evidently 𝒰(𝑋)∩ℒ(𝑋)=𝐶(𝑋).

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

If 𝒰 is a quasi-uniformity for 𝑋, then 𝒰^′={𝑈⁻¹∶𝑈∈𝒰} where 𝑈⁻¹={(𝑥,𝑦)∶(𝑦,𝑥)∈𝑈}, is also a quasi-uniformity for 𝑋. The collections 𝒰 and 𝒰^′ are called conjugate quasi-uniformities [7].

The quasi-uniform space (𝑋,𝒰) will always be considered to be a topological space with the topology obtained by using as the family of all neighborhoods of a point 𝑥₀∈𝑋 all sets of the form 𝑈(𝑥₀)={𝑥∈𝑋∶(𝑥₀,𝑥)∈𝑈}, where 𝑈 runs over 𝒰. Such a topology is called topology of the quasi-uniformity. Note that each topology on 𝑋 is induced by a quasi-uniformity (see [8, 9]), the most familiar of which is the Pervin quasi-uniformity [8].

A quasi-uniform space (𝑋,𝒰) is said to be a uniform space, and the family 𝒰 will be called a uniformity for 𝑋, if and only if the quasi-uniformity satisfies the symmetric relation: if 𝑈∈𝒰, then 𝑈⁻¹∈𝒰. For a quasi-uniformity 𝒰 the smallest uniformity containing 𝒰 has a base all sets of the form 𝑈∩𝑈⁻¹ where 𝑈 runs over a prescribed base for 𝒰. 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 𝐴⊂ℕ and 𝑛∈ℕ. Put 𝐴(𝑛)∶={𝑘∈𝐴∶𝑘≤𝑛}. Then one defines 𝜕(𝐴)∶=liminf_𝑛→∞|𝐴(𝑛)|𝑛,𝜕(𝐴)∶=limsup_𝑛→∞|𝐴(𝑛)|𝑛, (2.1) as the lower and upper asymptotic density of 𝐴, respectively. If 𝜕(𝐴)=𝜕(𝐴), then 𝜕(𝐴)=lim_𝑛→∞|𝐴(𝑛)|𝑛 (2.2) is the asymptotic (or natural) density of 𝐴.

All the three densities, if they exist, are in [0,1]. We recall also that 𝜕(ℕ⧵𝐴)=1−𝜕(𝐴) for 𝐴⊂ℕ. A set 𝐴⊂𝑋 is said to be statistically dense if 𝜕(𝐴)=1. Let us mention that the union and intersection of two statistically dense sets in ℕ are also statistically dense. For additional properties of the asymptotic density, in a more general setting, the reader might consult [13].

A sequence (𝑥_𝑛)_𝑛∈ℕ in a topological space 𝑋 is said to converge statistically (or shortly, 𝗌𝗍-converge) to 𝑥∈𝑋, if for every neighborhood 𝑈 of 𝑥, 𝜕({𝑛∈ℕ∶𝑥_𝑛∉𝑈})=0. This will be denoted by (𝑥_𝑛)−−−→_𝑛∈ℕ^{𝗌𝗍-𝜏}𝑥, where 𝜏 is a topology on 𝑋.

It was shown [14] (see [15, 16] for 𝑋=ℝ) that for first countable spaces this definition is equivalent to the statement: there exists a subset 𝐴 of ℕ with 𝜕(𝐴)=1 such that the sequence (𝑥_𝑛)_𝑛∈𝐴 converges to 𝑥. 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 ℝ^𝑋

Given 𝐹∈ℱ₀(𝑋) and 𝜖>0, a base for the standard uniformity for the topology of pointwise convergence 𝜏_𝑝 on ℝ^𝑋 consists of all entourages of the form [𝐹;𝜖]_𝑝∶=𝑓,𝑔∶∀𝑥∈𝐹𝑓(𝑥)−𝑔(𝑥)<𝜖. (3.1) 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 (𝑋,𝑑) be metric space, 𝐹∈ℱ₀(𝑋), and 𝜖>0. Consider the quasi-uniformity on ℝ^𝑋 having as a base all sets of the form [𝐹;𝜖]^𝑢_𝑝∶=𝑓,𝑔∶∀𝑥∈𝐹𝑓(𝑥)<𝑔(𝑥)+𝜖. (3.2) The induced upper pointwise topology on ℝ^𝑋 is denoted by 𝜏^𝑢_𝑝.

The conjugate quasi-uniformity has as a base all sets [𝐹;𝜖]^𝑙_𝑝∶=𝑓,𝑔∶∀𝑥∈𝐹𝑓(𝑥)>𝑔(𝑥)−𝜖. (3.3) We denote the induced lower pointwise topology on ℝ^𝑋 by 𝜏^𝑙_𝑝. Clearly the topology of pointwise convergence 𝜏_𝑝 can be written as 𝜏_𝑝=𝜏^𝑢_𝑝∨𝜏^𝑙_𝑝.

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 (𝑓_𝑛)_𝑛∈𝜔 in ℝ^𝑋 is said to be cofinally upper (resp., lower) weakly exhaustive at 𝑥₀∈𝑋, simply 𝖼𝖿-upper (resp., lower) weakly exhaustive at 𝑥₀, if for every 𝜖>0 there is 𝛿>0 such that for all 𝑦∈𝑆(𝑥₀,𝛿) there exists a cofinal subset {𝑛_𝑦𝑘}_𝑘 of ℕ such that for all 𝑛∈{𝑛_𝑦𝑘}_𝑘 we have 𝑓_𝑛(𝑦)<𝑓_𝑛(𝑥₀)+𝜖 (resp., 𝑓_𝑛(𝑦)>𝑓_𝑛(𝑥₀)−𝜖). The sequence (𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-upper weakly exhaustive if it is 𝖼𝖿-upper weakly exhaustive at every 𝑥₀∈𝑋.
In case {𝑛_𝑦𝑘}_𝑘 is statistically dense in ℕ we say that (𝑓_𝑛)_𝑛∈𝜔 is statistically upper (resp., lower) weakly exhaustive at 𝑥₀∈𝑋, shortly 𝗌𝗍-upper (resp., lower) weakly exhaustive at 𝑥₀.

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 𝑦.

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

Definition 3.3. Let (𝑓_𝑛)_𝑛∈𝜔, 𝑓 in 𝑌^𝑋, and 𝑥₀∈𝑋. The sequence (𝑓_𝑛)_𝑛∈𝜔 is said to be cofinally almost below (resp., above) 𝑓 around 𝑥₀∈𝑋, shortly 𝖼𝖿-almost below (resp., above) 𝑓 around 𝑥₀; if for every 𝜖>0 there is 𝛿>0 such that for all 𝑦∈𝑆(𝑥₀,𝛿) there exists a cofinal subset {𝑛_𝑦𝑘}_𝑘 of ℕ such that for all 𝑛∈{𝑛_𝑦𝑘}_𝑘 we have 𝑓_𝑛(𝑦)<𝑓(𝑥₀)+𝜖 (resp., 𝑓_𝑛(𝑦)>𝑓(𝑥₀)−𝜖).
When {𝑛_𝑦𝑘}_𝑘 is statistically dense in ℕ we say that (𝑓_𝑛)_𝑛∈𝜔 is statistically almost below (resp., above) 𝑓 around 𝑥₀∈𝑋, shortly 𝗌𝗍-almost below (resp., above) 𝑓 around 𝑥₀.

In the next propositions we show that each of the notions 𝖼𝖿-weakly exhaustive at 𝑥₀ and 𝖼𝖿-almost below 𝑓 around 𝑥₀ are forced by one of the 𝜏^𝑢_𝑝-convergence and 𝜏^𝑢_𝑙-convergence and the other property.

Proposition 3.4. Let (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in ℝ^𝑋 that is 𝜏^𝑢_𝑝-convergent to 𝑓∈ℝ^𝑋. If the sequence (𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-almost below 𝑓 around 𝑥₀, then (i)(𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-upper weakly exhaustive at 𝑥₀, and (ii)𝑓 is upper semicontinuous at 𝑥₀.

Proof. (𝑖) Let 𝜖>0 be fixed. Since (𝑓_𝑛)_𝑛∈𝜔 is 𝜏^𝑢_𝑝-convergent to 𝑓 at 𝑥₀, there exists 𝑛₁ such that for all 𝑛≥𝑛₁ we have that 𝑓_𝑛∈[{𝑥₀},𝜖/4]^𝑢_𝑝(𝑓); hence for every 𝑛≥𝑛₁, 𝑓(𝑥₀)≤𝑓_𝑛(𝑥₀)+𝜖/4. Since (𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-almost below 𝑓 around 𝑥₀, there is a 𝛿₀>0 such that for every 𝑦∈𝑆(𝑥₀,𝛿₀) there exists {𝑛_𝑦𝑘}_𝑘 cofinal in ℕ, with 𝑛_𝑦𝑘≥𝑛₁ for every 𝑘∈𝜔, and for all 𝑛∈{𝑛_𝑦𝑘}, 𝑓_𝑛(𝑦)≤𝑓(𝑥₀)+𝜖/4. Thus for all 𝑛∈{𝑛_𝑦𝑘} and 𝑦∈𝑆(𝑥₀,𝛿₀) we have 𝑓_𝑛(𝑦)≤𝑓(𝑥₀)+𝜖/4≤𝑓_𝑛(𝑥₀)+𝜖/2.
(𝑖𝑖) Let 𝜖>0 be fixed. Since (𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-almost below 𝑓 around 𝑥₀, there is a 𝛿₀>0 such that for every 𝑦∈𝑆(𝑥₀,𝛿₀) there exists {𝑛_𝑦𝑘}_𝑘 cofinal in ℕ and for all 𝑛∈{𝑛_𝑦𝑘}, 𝑓_𝑛(𝑦)≤𝑓(𝑥₀)+𝜖/2. Since (𝑓_𝑛)_𝑛∈𝜔 is 𝜏^𝑢_𝑝-convergent to 𝑓 at 𝑦, there exists 𝑛_𝑦 such that for all 𝑛≥𝑛_𝑦 we have that 𝑓(𝑦)≤𝑓_𝑛(𝑦)+𝜖/2. By cofinality of the index set, let 𝑛∈{𝑛_𝑦𝑘} be such that 𝑛≥𝑛_𝑦. It follows that 𝑓(𝑦)≤𝑓_𝑛(𝑦)+𝜖/2<𝑓(𝑥₀)+𝜖.

Proposition 3.5. Let (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in ℝ^𝑋, that is, 𝜏^𝑙_𝑝-convergent to 𝑓∈ℝ^𝑋. If (𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-upper weakly exhaustive at 𝑥₀, then it is 𝖼𝖿-almost below 𝑓 around 𝑥₀.

Proof. Let 𝜖>0 be fixed. By assumption there is a 𝛿>0 such that for every 𝑦∈𝑆(𝑥₀,𝛿) there is {𝑛_𝑦𝑘}_𝑘 cofinal in ℕ and for all 𝑛∈{𝑛_𝑦𝑘}, 𝑓_𝑛(𝑦)≤𝑓_𝑛(𝑥₀)+𝜖/4. Since (𝑓_𝑛)_𝑛∈𝜔 is 𝜏^𝑙_𝑝-convergent to 𝑓 at 𝑥₀, there exists 𝑛₀ such that for all 𝑛≥𝑛₀, 𝑓_𝑛(𝑥₀)≤𝑓(𝑥₀)+𝜖/4. It follows that for all 𝑛∈{𝑛_𝑦𝑘}_𝑘 with 𝑛≥𝑛₀, 𝑓_𝑛(𝑦)≤𝑓_𝑛(𝑥₀)+𝜖/4≤𝑓(𝑥₀)+𝜖/2.

Proposition 3.6. Let (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in ℝ^𝑋. If (𝑓_𝑛)_𝑛∈𝜔 is 𝜏^𝑙_𝑝-convergent to 𝑓∈ℝ^𝑋 and 𝑓 is upper semicontinuous at 𝑥₀, then (𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-almost below 𝑓 around 𝑥₀.

Proof. Let 𝜖>0 be fixed, there is a 𝛿₀>0 such that for every 𝑦∈𝑆(𝑥₀,𝛿₀), 𝑓(𝑦)<𝑓(𝑥₀)+𝜖/2. Since (𝑓_𝑛)_𝑛∈𝜔 is 𝜏^𝑙_𝑝-convergent to 𝑓 at 𝑦, there exists 𝑛_𝑦 such that for all 𝑛≥𝑛_𝑦, 𝑓_𝑛(𝑦)≤𝑓(𝑦)+𝜖/2. Thus for every 𝑦∈𝑆(𝑥₀,𝛿₀), the subset {𝑛_𝑦𝑘}_𝑘={𝑛∶𝑛≥𝑛_𝑦} is cofinal in ℕ and 𝑓_𝑛(𝑦)<𝑓(𝑦)+𝜖/2<𝑓(𝑥₀)+𝜖.

Therefore, if we assume the whole pointwise convergence the two properties, 𝖼𝖿-upper weakly exhaustive at 𝑥₀ and 𝖼𝖿-almost below 𝑓 around 𝑥₀, associated to the upper and lower pointwise convergence, coincide.

Corollary 3.7. Let (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in ℝ^𝑋 that is pointwise convergent to 𝑓∈ℝ^𝑋 and 𝑥₀∈𝑋. The following are equivalent: (i)𝑓 is upper semicontinuous at 𝑥₀, (ii)(𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-upper weakly exhaustive at 𝑥₀, (iii)(𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-almost below 𝑓 around 𝑥₀.

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 (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in ℝ^𝑋, that is, 𝗌𝗍-𝜏_𝑝 convergent to 𝑓∈ℝ^𝑋. The sequence (𝑓_𝑛)_𝑛∈𝜔 is 𝖼𝖿-weakly exhaustive at 𝑥₀ if and only if 𝑓 is continuous at 𝑥₀.

4. Decomposition of Sticking Topology in 𝐶(𝑋)

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

Definition 4.1. Let (𝑋,𝑑) be metric space, 𝐹∈ℱ₀(𝑋), and 𝜖>0. Consider the quasi-uniformity on ℝ^𝑋 having as a base all sets of the form [𝐹;𝜖]^𝑢𝑓(𝑥)(𝑥(𝑥)(𝑥).∶=𝑓,𝑔∶∀𝑥∈𝐹−𝑔)<𝜖,∃aneighborhood𝑉of𝐹suchthat∀𝑥∈𝑉,𝑔<𝑓+𝜖 (4.1) We denote the induced topology on ℝ^𝑋 as 𝜏^𝑢. Also we denote by 𝜏^𝑙 the topology induced on ℝ^𝑋 by the conjugate quasi-uniformity. If we take the uniformity generated by our standard uniformity for 𝜏^𝑢 and its conjugate 𝜏^𝑙 we get a uniformity having as a base all sets of the form [𝐹;𝜖]^□∶=𝑓,𝑔∶∃aneighborhood𝑉of𝐹suchthat∀𝑥∈𝑉,𝑓(𝑥)−𝑔(𝑥)<𝜖𝐹∈ℱ₀(𝑋),𝜖>0. (4.2) The topology 𝜏_□ 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: 𝐶(𝑋) is 𝜏_□-closed in ℝ^𝑋, and the 𝜏_□-convergence reduces to pointwise convergence on 𝐶(𝑋) 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 (𝑓_𝑛)_𝑛∈𝜔 in 𝐶(𝑋,𝑌) is said to be statistically Alexandroff convergent to 𝑓∈𝑌^𝑋, denoted by (𝑓_𝑛)_𝑛∈𝜔 being 𝗌𝗍-𝜏_Al, provided (𝑓_𝑛)−−−−→_𝑛∈𝜔^{𝗌𝗍-𝜏𝑝}𝑓 and for every 𝜖>0 and every statistically dense set 𝐴⊂ℕ there exist an infinite set 𝑀_𝐴={𝑛₁<𝑛₂<⋯𝑛_𝑘<⋯}⊂𝐴 and an open cover 𝒰={𝑈_𝑛∶𝑛∈𝐴} such that for every 𝑥∈𝑈_𝑘 we have 𝜌(𝑓_𝑛𝑘(𝑥),𝑓(𝑥))<𝜖.

Definition 4.3. A sequence (𝑓_𝑛)_𝑛∈𝜔 in 𝐶(𝑋,𝑌) is said to be statistically Arzelà convergent to 𝑓∈𝑌^𝑋, denoted by (𝑓_𝑛)_𝑛∈𝜔 being 𝗌𝗍-𝜏_Arz convergent to 𝑓, if (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏_𝑝 convergent to 𝑓, and for every 𝜖>0 and every statistically dense set 𝐴⊂ℕ there exists a finite set {𝑛₁,𝑛₂,…,𝑛_𝑘}⊂𝐴 such that for every 𝑥∈𝑋 it holds 𝜌(𝑓_𝑛𝑖(𝑥),𝑓(𝑥))<𝜖 for at least one 𝑖≤𝑘.

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 (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in 𝐶(𝑋), that is, 𝗌𝗍-𝜏_𝑝 convergent to 𝑓∈ℝ^𝑋. The following are equivalent: (i)𝑓 is continuous, (ii)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏_Arz convergent to 𝑓 on compacta, (iii)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏_□ convergent to 𝑓, (iv)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏_Al convergent to 𝑓.

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

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

Proposition 4.5. Let (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in ℒ(𝑋). If (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑢_𝑝-convergent to 𝑔∈𝒰(𝑋) and (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-almost below 𝑔 in 𝑋, then (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑢 convergent to 𝑔.

Proof. Fix 𝐹={𝑥₁,…,𝑥_𝑛} a finite subset of 𝑋, and 𝜖>0. We show that there is 𝐴⊂ℕ statistically dense such that for every 𝑛∈𝐴 it follows that 𝑓_𝑛∈[𝐹;𝜖]^𝑢(𝑔). Since (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑢_𝑝-convergent to 𝑔, there is 𝑀⊂ℕ statistically dense, such that for every 𝑥∈𝐹 and for all 𝑛∈𝑀 we have 𝑔(𝑥)≤𝑓_𝑛(𝑥)+𝜖/3. Let 𝑛₁∈𝑀, since 𝑓_𝑛1 is lower semicontinuous at each 𝑥_𝑖∈𝐹; there exists 𝑈_𝑥𝑖 such that for every 𝑦∈𝑈_𝑥𝑖, 𝑓_𝑛1(𝑦)>𝑓_𝑛1(𝑥_𝑖)−𝜖/3. By upper semicontinuity of 𝑔 at each 𝑥_𝑖∈𝐹, there exists 𝑊_𝑥𝑖 such that for every 𝑦∈𝑊_𝑥𝑖, 𝑔(𝑦)<𝑔(𝑥_𝑖)+𝜖/3. Also (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-almost below 𝑔 at each 𝑥_𝑖∈𝐹; therefore for every 𝑖≤𝑛, there is 𝛿_𝑖>0 such that for every 𝑦∈𝑆(𝑥_𝑖,𝛿_𝑖), there is 𝐵_𝑦⊂ℕ statistically dense, and for all 𝑛∈𝐵_𝑦, 𝑓_𝑛(𝑦)<𝑔(𝑥_𝑖)+𝜖/3. Set 𝑉=⋃^𝑛_𝑖=1(𝑊_𝑥𝑖∩𝑉_𝑥𝑖∩𝑆(𝑥_𝑖,𝛿_𝑖)). For every 𝑣∈𝑉 we have 𝑔(𝑣)<𝑔(𝑥_𝑖)+𝜖/3<𝑓_𝑛1(𝑥_𝑖)+2𝜖/3<𝑓_𝑛1(𝑣)+𝜖. Let 𝐴=(⋂^𝑛_𝑖=1𝐵_𝑥𝑖)∩𝑀. For all 𝑛∈𝐴 and each 𝑥_𝑖∈𝐹 it follows that 𝑓_𝑛(𝑥_𝑖)≤𝑔(𝑥_𝑖)+𝜖/3 and 𝑔(𝑥_𝑖)<𝑓_𝑛(𝑥_𝑖)+𝜖; thus (𝑓_𝑛)_𝑛∈𝜔 is statistically 𝜏^𝑢-convergent to 𝑔.

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 𝗌𝗍-𝜏^𝑢 convergence is guaranteed by the 𝗌𝗍-𝜏^𝑙 convergence.

Proposition 4.6. Let (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in ℒ(𝑋) and 𝑔∈𝒰(𝑋). If (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑙 convergent to 𝑔, then (i)𝑔 is continuous, (ii)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑢_𝑝 convergent to 𝑔, (iii)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-almost below 𝑔.

Proof. To prove (𝑖), it is sufficient to prove lower semicontinuity of 𝑔. Fix that 𝑥₀∈𝑋 and 𝜖>0. By assumption there is 𝐴⊂ℕ statistically dense, and for all 𝑛≥𝐴, 𝑓_𝑛∈[{𝑥₀},𝜖/3]^𝑙(𝑔). Thus there is 𝛿>0 such that for all 𝑦∈𝑆(𝑥₀,𝛿), 𝑔(𝑦)>𝑓_𝑛(𝑦)−𝜖/3 and |𝑔(𝑥₀)−𝑓_𝑛(𝑥₀)|<𝜖/3. Let 𝑛₀∈𝐴. Since 𝑓_𝑛0 is lower semicontinuous, we may assume that for every 𝑦∈𝑆(𝑥₀,𝛿) we have also 𝑓_𝑛0(𝑦)>𝑓_𝑛0(𝑥₀)−𝜖/3. It follows that for 𝑦∈𝑆(𝑥₀,𝛿), 𝑔𝑥₀<𝑓_𝑛0𝑥₀+𝜖3<𝑓_𝑛0𝑦+2𝜖3<𝑔𝑦+𝜖. (4.3) To prove (𝑖𝑖𝑖), fix 𝑥₀∈𝑋 and 𝜖>0. Let 𝐹={𝑥₀}, since (𝑓_𝑛)_𝑛∈𝜔 statistically 𝜏^𝑙- converges to 𝑔; there is 𝐴⊂ℕ statistically dense, 𝛿₁>0 such that for all 𝑦∈𝑆(𝑥₀,𝛿₁) and 𝑛∈𝐴, we have 𝑔(𝑦)>𝑓_𝑛(𝑦)−𝜖/2 and |𝑔(𝑥₀)−𝑓_𝑛(𝑥₀)|<𝜖/2. By upper semicontinuity of 𝑔 at each 𝑥₀ there exists 𝛿₀>0 such that for all 𝑦∈𝑆(𝑥₀,𝛿₀), 𝑔(𝑦)<𝑔(𝑥₀)+𝜖/2. Let 𝛿=min{𝛿₀,𝛿₁}. For all 𝑛∈𝐴 and 𝑦∈𝑆(𝑥₀,𝛿) it follows that 𝑓_𝑛(𝑦)≤𝑔(𝑦)+𝜖/2<𝑔(𝑥₀)+𝜖.

From Propositions 4.5 and 4.6 we have the following corollaries.

Corollary 4.7. Let (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in 𝐶(𝑋) and 𝑔∈ℒ(𝑋). If (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑢 convergent to 𝑔, then (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑙 convergent to 𝑔.

Corollary 4.8. Let (𝑓_𝑛)_𝑛∈𝜔 be a sequence of functions in 𝐶(𝑋) and 𝑔∈𝒰(𝑋). If (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑙-convergent to 𝑔, then (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑢 convergent to 𝑔.

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

Theorem 4.9. Let (𝑓_𝑛)_𝑛∈𝜔 be in 𝐶(𝑋). If (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏_𝑝 convergent to 𝑔, then the following are equivalent: (i)𝑔 is continuous, (ii)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-almost below and 𝗌𝗍-almost above 𝑔,(iii)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-weakly exhaustive, (iv)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏_□ convergent to 𝑔.

Combining Corollaries 4.7 and 4.8, we show that assuming the statistical pointwise convergence in 𝐶(𝑋), the convergence in one of the two symmetric topologies 𝜏^𝑢 and 𝜏^𝑙 automatically ensures convergence in the whole sticking topology, since they coincide.

Corollary 4.10. Let (𝑓_𝑛)_𝑛∈𝜔 and 𝑔∈𝐶(𝑋). If (𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏_𝑝 convergent to 𝑔, then the following are equivalent: (i)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑢 convergent to 𝑔, (ii)(𝑓_𝑛)_𝑛∈𝜔 is 𝗌𝗍-𝜏^𝑙 convergent to 𝑔.

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.