# Set Convergences via bornology

Yogesh Agarwal and Varun Jindal  Yogesh Agarwal & Varun Jindal: Department of
Mathematics, Malaviya National Institute of Technology Jaipur, Jaipur-302017,
Rajasthan, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper examines the equivalence between various set convergences, as
studied in [7, 13, 22], induced by an arbitrary bornology $\mathcal{S}$ on a
metric space $(X,d)$. Specifically, it focuses on the upper parts of the
following set convergences: convergence deduced through uniform convergence of
distance functionals on $\mathcal{S}$ ($\tau_{\mathcal{S},d}$-convergence);
convergence with respect to gap functionals determined by $\mathcal{S}$
($G_{\mathcal{S},d}$-convergence); and bornological convergence
($\mathcal{S}$-convergence). In particular, we give necessary and sufficient
conditions on the structure of the bornology $\mathcal{S}$ for the coincidence
of $\tau_{\mathcal{S},d}^{+}$-convergence with
$\mathsf{G}_{\mathcal{S},d}^{+}$-convergence, as well as
$\tau_{\mathcal{S},d}^{+}$-convergence with $\mathcal{S}^{+}$-convergence. A
characterization for the equivalence of $\tau_{\mathcal{S},d}^{+}$-convergence
and $\mathcal{S}^{+}$-convergence, in terms of certain convergence of nets,
has also been given earlier by Beer, Naimpally, and Rodriguez-Lopez in [13].
To facilitate our study, we first devise new characterizations for
$\tau_{\mathcal{S},d}^{+}$-convergence and $\mathcal{S}^{+}$-convergence,
which we call their miss-type characterizations.

###### Key words and phrases:

Bornology, Uniform convergence, Gap functional, Wijsman topology, Attouch-Wets
topology, Hausdorff metric topology, Bornological convergence

###### 2020 Mathematics Subject Classification:

Primary 54B20; Secondary 54A10, 54E35, 54A20

##  1\. Introduction

A topology on a collection of subsets of a topological space $(X,\tau)$ is
called a hyperspace topology. In the literature, a number of hyperspace
topologies have been studied on the collection $\mathcal{P}_{0}(X)$ of all
nonempty subsets of a topological space $(X,\tau)$, and on $CL(X)$, the family
of all nonempty closed subsets of $(X,\tau)$ [3, 24]. The theory of
hyperspaces is fundamental to various mathematical branches such as convex
analysis, functional analysis, optimization theory, and variational analysis
[3, 14, 23, 25].

For a metric space $(X,d)$, the two most studied hyperspace topologies on
$CL(X)$ are the Vietoris and the Hausdorff metric topologies [20, 26, 27].
However, these topologies are too strong while dealing with unbounded sets.
For example, the sequence of lines $L_{n}=\\{(x,y):y=\frac{x}{n}\\}$ appears
naturally converging to the horizontal axis yet does not converge with respect
to either of these two topologies. This prompted researchers to define several
weaker set convergences and hyperspace topologies. In the realm of metric
spaces, the interplay between set convergence and geometric set functionals
has proven to be an important approach for studying various set convergences
and corresponding topologies on subsets of a metric space.

The distance functional is the most useful of the set functionals. For a
nonempty subset $A$ of a metric space $(X,d)$, the distance functional
$d(\cdot,A)$ on $X$ corresponding to $A$ is defined as

$x\to d(x,A)=\inf\\{d(x,a):a\in A\\}\text{ for }x\in X.$

So through distance functional, we can identify a closed subset $A$ of $(X,d)$
with the continuous function $d(\cdot,A)\in C(X)$. Using this identification,
in 2008, Beer et al. [13] gave a unified approach to define hyperspace
topologies on $\mathcal{P}_{0}(X)$ (or $CL(X)$). In particular, for a family
$\mathcal{S}$ of nonempty subsets of a metric space $(X,d)$, they studied a
hyperspace topology corresponding to which a net $(A_{\lambda})$ in $CL(X)$
converges to $A\in CL(X)$ if and only if the associated net
$(d(\cdot,A_{\lambda}))$ of distance functionals converges uniformly on
members of $\mathcal{S}$ to $d(\cdot,A)$. This topology on $CL(X)$ is denoted
by $\tau_{\mathcal{S},d}$. When $\mathcal{S}$ is $\mathcal{P}_{0}(X)$ and
$\mathcal{B}_{d}(X)$ (the family of all $d$-bounded sets in $(X,d)$)
respectively, the topology $\tau_{\mathcal{S},d}$ reduces to the classical
Hausdorff metric topology $(\tau_{H_{d}})$ and Attouch-Wets topology
$(\tau_{AW_{d}})$ respectively. Furthermore, when
$\mathcal{S}=\mathcal{F}(X)$, the collection of all nonempty finite subsets of
$X$, we obtain the well-known Wijsman topology $(\tau_{W_{d}})$. For more on
these classical topologies, see ([1, 2, 17, 19, 28]).

Two important extensions of the distance functional are the gap and excess
functionals. These extensions play a central role in providing weak
formulations of many hyperspace topologies (see, [3, 8, 9, 12]). For a
nonempty subset $S$ of a metric space $(X,d)$, the gap functional determined
by $S$ is defined as

$A\to D_{d}(S,A)=\inf\\{d(x,a):x\in S\text{ and }a\in A\\}\text{ for
}A\in\mathcal{P}_{0}(X).$

Another unified approach to hyperspace topologies on subsets of a metric space
$(X,d)$ is through the weak topology generated by a family of gap functionals
determined by members of a family $\mathcal{S}\subseteq\mathcal{P}_{0}(X)$.
Beer, Constantini, and Levi considered this approach formally in [7]. The weak
topology determined by such gap functionals is known as the gap topology,
denoted by $\mathsf{G}_{\mathcal{S},d}$. In particular, when
$\mathcal{S}=\mathcal{F}(X)$, $\mathcal{B}_{d}(X)$, and $\mathcal{P}_{0}(X)$,
the gap topology reduces to Wijsman topology, bounded proximal topology, and
proximal topology, respectively.

For a metric space $(X,d)$ and a family $\mathcal{S}$ of nonempty subsets of
$X$, in 2004, Lechicki et al. [22] defined a new kind of set convergence known
as the bornological convergence (denoted by $\mathcal{S}$-convergence). The
bornological convergence generalizes two classical set convergences: the
Hausdorff metric convergence (when $\mathcal{S}=\mathcal{P}_{0}(X)$), and the
Attouch-Wets convergence (when $\mathcal{S}=\mathcal{B}_{d}(X)$). The
bornological convergence was further studied in [4, 5, 6, 9, 10, 11, 13].

It is to be noted that for $\mathcal{S}=\mathcal{P}_{0}(X)$ or
$\mathcal{B}_{d}(X)$, $\tau_{\mathcal{S},d}$-convergence and
$\mathcal{S}$-convergence coincide, however in either case, the
$\mathsf{G}_{\mathcal{S},d}$-convergence is weaker. As customary to hyperspace
convergences, if we decompose each of these convergences into two halves,
upper ($+$) and lower ($-$), then for $\mathcal{S}=\mathcal{P}_{0}(X)$ or
$\mathcal{B}_{d}(X)$, we have
$\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$ but
$\mathsf{G}_{\mathcal{S},d}^{-}$ may be strictly weaker than
$\mathcal{S}^{-}=\tau_{\mathcal{S},d}^{-}$.

In this paper, we are considering the above mentioned three types of set
convergences on $CL(X)$ for an arbitrary bornology $\mathcal{S}$ (see, [21])
on a metric space $(X,d)$. There has been tremendous interest in studying the
relation among these set convergences [9, 10, 13]. Beer, Naimpally, and
Rodriguez-Lopez characterized the equivalence of
$\tau_{\mathcal{S},d}^{+}$-convergence and $\mathcal{S}^{+}$-convergence in
[13] while in [9] Beer and Levi characterized the equivalence of
$G_{\mathcal{S},d}^{+}$-convergence and $\mathcal{S}^{+}$-convergence.
However, the equivalence between $\tau_{\mathcal{S},d}^{+}$ and
$G_{\mathcal{S},d}^{+}$ is not known. We fill this void in the present paper.
We also provide a new independent characterization for the equivalence of
$\tau_{\mathcal{S},d}^{+}$ and $\mathcal{S}^{+}$-convergences. To study these
equivalences, we first propose new characterizations for
$\tau_{\mathcal{S},d}^{+}$-convergence and $\mathcal{S}^{+}$-convergence.
While studying $\tau_{\mathcal{S},d}^{+}$-convergence vis-à-vis other
convergences, we discover in this paper that we require a variational type of
enlargement of $S\in\mathcal{S}$ in which the situation demands enlargements
of different sizes at each point of $S$. We formulate this in terms of
enlargement of $S\in\mathcal{S}$ by positive functions. This idea is central
to our investigations.

The paper is organized as follows. In section 2, we define various hyperspace
convergences and Section 3 delves into basic convergence results. Section 4 is
devoted to provide a new representation for the
$\tau_{\mathcal{S},d}^{+}$-convergence. Some classical results are obtained as
corollaries to our main theorem (Theorem 4.1) of this section. The main result
(Theorem 5.5) of Section 5 unfolds the necessary and sufficient condition for
the coincidence $\tau_{\mathcal{S},d}^{+}=G_{\mathcal{S},d}^{+}$. This
necessary and sufficient condition turns out to be a covering condition (see,
Definition 5.3) on enlargements of members of $\mathcal{S}$ by positive
functions. In the final section, we present a new perspective to look at
$\mathcal{S}^{+}$-convergence. Then the coincidence of
$\tau_{\mathcal{S},d}^{+}$-convergence and $\mathcal{S}^{+}$-convergence is
characterized. Several examples and counter-examples are given throughout the
paper to support the findings.

##  2\. Preliminaries

In this section, we give definitions of various set convergences and other key
concepts. We start with the definition of a bornology which is central to our
analysis. For a metric space $(X,d)$, a nonempty family
$\mathcal{S}\subseteq\mathcal{P}_{0}(X)$ which is hereditary, closed under
finite union, and that forms a cover of $X$ is called a bornology on $X$. Some
important bornologies on a metric space $(X,d)$ are $\mathcal{F}(X),$
$\mathcal{B}_{d}(X),$ and $\mathcal{P}_{0}(X)$. Two other useful bornologies
on a metric space $(X,d)$ are $\mathcal{K}(X)$, the set of all nonempty
relatively compact subsets of $X$, and $\mathcal{T}\mathcal{B}_{d}(X)$ the set
of all nonempty $d$-totally bounded subsets of $X$.

The open (closed) ball centered at $x\in X$ and radius $\epsilon>0$ in $(X,d)$
is denoted by $B_{d}(x,\epsilon)$ ($\overline{B_{d}}(x,\epsilon)$). For
$A\subseteq X$ and $\epsilon>0$, the $\epsilon$-enlargement of $A$, denoted by
$B_{d}(A,\epsilon)$, is defined as $B_{d}(A,\epsilon)=\\{x\in
X:d(x,A)<\epsilon\\}$. Note that for $\epsilon>0,r>0$, we have
$B_{d}(B_{d}(A,\epsilon),r)\subseteq B_{d}(A,\epsilon+r)$.

######  Definition 2.1.

A metric space $(X,d)$ is called an almost convex metric space if for
$A\in\mathcal{P}_{0}(X)$ and $\epsilon>0,r>0$, we have
$B_{d}(B_{d}(A,\epsilon),r)=B_{d}(A,\epsilon+r)$.

Clearly, every normed linear space is an almost convex metric space.

An interesting representation of the Attouch-Wets convergence in terms of
enlargements of sets is as follows: a net $(A_{\lambda})$ in
$\mathcal{P}_{0}(X)$ $\tau_{AW_{d}}$-converges to $A$ provided $\forall$
$\epsilon>0$ and $B\in\mathcal{B}_{d}(X)$, $A_{\lambda}\cap B\subseteq
B_{d}(A,\epsilon)$ and $A\cap B\subseteq B_{d}(A_{\lambda},\epsilon)$
eventually.

Lechicki et al. [22] generalized this representation of Attouch-Wets
convergence by replacing $\mathcal{B}_{d}(X)$ with any nonempty family
$\mathcal{S}$ of subsets of a metric space $(X,d)$. The resulting set
convergence is known as bornological convergence.

######  Definition 2.2.

Suppose $\mathcal{S}$ is a nonempty family of subsets of a metric space
$(X,d)$. A net $(A_{\lambda})$ is said to be lower bornological convergent to
$A$ in $\mathcal{P}_{0}(X)$ denoted by $\mathcal{S}^{-}$-convergence, if for
each $\epsilon>0$ and $S\in\mathcal{S}$, eventually $A\cap S\subseteq
B_{d}(A_{\lambda},\epsilon)$.

We say $(A_{\lambda})$ is upper bornological convergent to $A$ in
$\mathcal{P}_{0}(X)$ denoted by $\mathcal{S}^{+}$-convergence, if for each
$\epsilon>0$ and $S\in\mathcal{S}$, eventually $A_{\lambda}\cap S\subseteq
B_{d}(A,\epsilon)$.

A net $(A_{\lambda})$ $\mathcal{S}$-converges to $A$ provided it is both
$\mathcal{S}^{-}$-convergent and $\mathcal{S}^{+}$-convergent to $A$. The
$\mathcal{S}$-convergence is known as the bornological convergence.

Clearly, for $\mathcal{S}=\mathcal{B}_{d}(X)$ the $\mathcal{S}$-convergence is
compatible with the Attouch-Wets topology $\tau_{AW_{d}}$. However, in
general, the bornological convergence corresponding to a bornology
$\mathcal{S}$ need not be topological. In the past, several authors have shown
interest in exploring the conditions under which this convergence becomes
topological [6, 10, 22].

In 2008, G. Beer et al. in [13] gave a unified approach to study hyperspace
topologies through uniform convergence of distance functionals on members of a
family $\mathcal{S}\subseteq\mathcal{P}_{0}(X)$.

######  Definition 2.3 ([15]).

Let $(X,d)$ be a metric space and $\mathcal{S}\subseteq\mathcal{P}_{0}(X)$.
Then for any $S\in\mathcal{S}$ and $\epsilon>0$ sets of the form

$[S,\epsilon]^{+}=\\{(A,C)\in\mathcal{P}_{0}(X)\times\mathcal{P}_{0}(X):d(x,A)-d(x,C)<\epsilon~{}~{}\forall
x\in S\\}$

forms a base for a quasi-uniformity on $\mathcal{P}_{0}(X)$. The corresponding
topology is denoted by $\tau_{\mathcal{S},d}^{+}$.

Similarly, $\tau_{\mathcal{S},d}^{-}$ is the topology on $\mathcal{P}_{0}(X)$
generated by the quasi-uniformity given by sets of the form

$[S,\epsilon]^{-}=\\{(A,C)\in\mathcal{P}_{0}(X)\times\mathcal{P}_{0}(X):d(x,C)-d(x,A)<\epsilon~{}~{}\forall
x\in S\\}.$

The topology $\tau_{\mathcal{S},d}$ is the supremum of
$\tau_{\mathcal{S},d}^{-}$ and $\tau_{\mathcal{S},d}^{+}$. This topology is a
uniformizable topology on $\mathcal{P}_{0}(X)$ and the family
$\\{U_{\mathcal{S},\epsilon}:S\in\mathcal{S},\epsilon>0\\}$ forms a base for a
compatible uniformity for $\tau_{\mathcal{S},d}$, where

$U_{\mathcal{S},\epsilon}=\\{(A,B)\in\mathcal{P}_{0}(X)\times\mathcal{P}_{0}(X):|d(x,A)-d(x,B)|<\epsilon\
\ \forall x\in S\\}.$

So a net $(A_{\lambda})$ in $\mathcal{P}_{0}(X)$ is
$\tau_{\mathcal{S},d}$-convergent to a nonempty set $A$ if and only if for any
$\epsilon>0$ and $S\in\mathcal{S}$, eventually,
$|d(x,A)-d(x,A_{\lambda})|<\epsilon$ for all $x\in S$, that is, if the net
$(d(\cdot,A_{\lambda}))$ converges to $d(\cdot,A)$ uniformly on members of
$\mathcal{S}$.

In [7], the authors studied the weak topology generated by a family of gap
functionals with fixed left argument from an arbitrary family $\mathcal{S}$ of
subsets of a metric space $(X,d)$. This topology is known as the gap topology.

######  Definition 2.4.

For an arbitrary family $\mathcal{S}$ of nonempty subsets of a metric space
$(X,d)$, the gap topology is defined as the weakest topology on
$\mathcal{P}_{0}(X)$ for which all functionals of the form $C\rightarrow
D_{d}(S,C)\hskip 7.11317pt(S\in\mathcal{S})$ are continuous. It is denoted by
$\mathsf{G}_{\mathcal{S},d}$. The two halves of the gap topology
$\mathsf{G}_{\mathcal{S},d}$ are:

  1. (i)

the upper gap topology is the weakest topology on $\mathcal{P}_{0}(X)$ for
which each member of the family of gap functionals
$\\{D_{d}(S,\cdot):S\in\mathcal{S}\\}$ is lower semi-continuous. It is denoted
by $\mathsf{G}_{\mathcal{S},d}^{+}$.

  2. (ii)

the lower gap topology is the weakest topology on $\mathcal{P}_{0}(X)$ for
which each member of the family of gap functionals
$\\{D_{d}(S,\cdot):S\in\mathcal{S}\\}$ is upper semi-continuous. It is denoted
by $\mathsf{G}_{\mathcal{S},d}^{-}$.

So a subbase for a compatible uniformity for the topology
$\mathsf{G}_{\mathcal{S},d}$ consists of all sets of the kind:

$\\{(A,C)\in\mathcal{P}_{0}(X)\times\mathcal{P}_{0}(X):|D_{d}(S,A)-D_{d}(S,C)|<\epsilon\\}\
\ (S\in\mathcal{S},\epsilon>0).$

######  Remark 2.5.

It is to be noted that whenever the family $\mathcal{S}$ contains all
singletons the lower gap topology reduces to the lower Vietoris topology
([7]).

Let $\mathcal{S}\subseteq\mathcal{P}_{0}(X)$. Then for
$S_{1},\ldots,S_{n}\in\mathcal{S}$ and $\epsilon_{1},\ldots,\epsilon_{n}>0$,
define,

$\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{n};\epsilon_{1},\ldots,\epsilon_{n})=\\{C\in\mathcal{P}_{0}(X):D_{d}(C,S_{i})>\epsilon_{i}~{}\text{for}~{}i=1,\ldots,n\\}.$

The collection
$\\{\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{n};\epsilon_{1},\ldots,\epsilon_{n}):n\in\mathbb{N},S_{i}\in\mathcal{S},\epsilon_{i}>0,1\leq
i\leq n\\}$ together with $\mathcal{P}_{0}(X)$ forms a base for the upper gap
topology on $\mathcal{P}_{0}(X)$ ([7]).

In [7], the authors gave the following important characterization for the
$\mathsf{G}_{\mathcal{S},d}^{+}$-convergence.

######  Theorem 2.6.

Let $(X,d)$ be a metric space, and let
$\mathcal{S}\subseteq\mathcal{P}_{0}(X)$. Suppose $(A_{\lambda})$ is a net in
$\mathcal{P}_{0}(X)$ and $A\in\mathcal{P}_{0}(X)$. Then the following
assertions are equivalent:

  1. (i)

$(A_{\lambda})$ $\mathsf{G}_{\mathcal{S},d}^{+}$-converges to $A$;

  2. (ii)

for $S\in\mathcal{S}$, and $0<\alpha<\epsilon$, whenever $A\cap
B_{d}(S,\epsilon)=\emptyset$, then $A_{\lambda}\cap B_{d}(S,\alpha)=\emptyset$
eventually.

In this paper, we give analogous characterizations for the
$\tau_{\mathcal{S},d}^{+}$-convergence (see, Theorem 4.1) and
$\mathcal{S}^{+}$-convergence (see, Theorem 6.1). Moreover, we study all these
convergences on $CL(X)$ and when $\mathcal{S}$ is a bornology of subsets of
$X$.

##  3\. Basic Relations

In this section, we give relations between the hyperspace convergences defined
in the previous section (see, [9]). We also provide some examples showing the
non-equivalence of these convergences in general.

######  Proposition 3.1.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Then $\mathsf{G}_{\mathcal{S},d}^{+}\subseteq\tau_{\mathcal{S},d}^{+}$ on
$CL(X)$ ($\mathcal{P}_{0}(X)$).

###### Proof.

Take $A\in CL(X)$, $S_{1},\ldots,S_{n}\in\mathcal{S}$, and
$\epsilon_{1},\ldots,\epsilon_{n}>0$ such that
$A\in\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{n};\epsilon_{1},\ldots,\epsilon_{n})$.
Let $r_{i}=D_{d}(S_{i},A)-\epsilon_{i}>0$ for $1\leq i\leq n$. Take
$r=\min\\{\frac{r_{i}}{2}:i=1,\ldots,n\\}$ and
$S=\cup_{i=1}^{n}S_{i}\in\mathcal{S}$. We claim that
$[S,r]^{+}(A)\subseteq\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{n};\epsilon_{1},\ldots,\epsilon_{n})$.
To see this, consider $C\in[S,r]^{+}(A)$. Then $d(x,A)-d(x,C)<r\ \ \forall
x\in S_{i}$ and for each $1\leq i\leq n$. So for each $i=1,\ldots,n$,
$D_{d}(S_{i},A)<r+d(x,C)\ \ \forall x\in S_{i}$, which implies,
$D_{d}(S_{i},A)-r\leq D_{d}(S_{i},C)$. Thus,
$C\in\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{n};\epsilon_{1},\ldots,\epsilon_{n})$.
∎

######  Proposition 3.2.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Then $\mathcal{S}^{+}\leq\mathsf{G}_{\mathcal{S},d}^{+}$ on $CL(X)$, that is,
every $\mathsf{G}_{\mathcal{S},d}^{+}$-convergent net is
$\mathcal{S}^{+}$-convergent.

###### Proof.

Let $(A_{\lambda})$ be a net that $\mathsf{G}_{\mathcal{S},d}^{+}$-converges
to $A$ in $CL(X)$. Suppose by contradiction, there exist $S_{1}\in\mathcal{S}$
and $\epsilon>0$ such that $A_{\lambda}\cap S_{1}\nsubseteq B_{d}(A,\epsilon)$
frequently. Then there is a cofinal set $\Lambda_{1}\subseteq\Lambda$ such
that $S_{2}\cap B_{d}(A,\epsilon)=\emptyset$, where $S_{2}=\\{a_{\lambda}\in
A_{\lambda}\cap S_{1}:\lambda\in\Lambda_{1}\\}$. Note that
$S_{2}\in\mathcal{S}$ and $A\in\mathscr{A}_{d}^{+}(S_{2};\frac{\epsilon}{2})$.
But $A_{\lambda}\cap S_{2}\neq\emptyset$ for all $\lambda\in\Lambda_{1}$,
which contradicts our assumption. ∎

Using Proposition 3.1 and Proposition 3.2, we have the following relation
between various set convergences.

######  Corollary 3.3.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Then $\tau_{\mathcal{S},d}^{+}$-convergence
$\Rightarrow\mathsf{G}_{\mathcal{S},d}^{+}$-convergence
$\Rightarrow\mathcal{S}^{+}$-convergence on $CL(X)$ $(\mathcal{P}_{0}(X))$.

We now give some examples showing that the reverse implications in Corollary
3.3 may not hold in general. We first present an example showing that
$\mathcal{S}^{+}\nRightarrow\mathsf{G}_{\mathcal{S},d}^{+}\nRightarrow\tau_{\mathcal{S},d}^{+}$
such an example seems not to be available in the literature.

######  Example 3.4.

Let $X=\\{(x,y):x\geq 0,y\geq 0\\}\subseteq\mathbb{R}^{2}$ and $d=d_{e}$, the
Euclidean metric. Let $\mathcal{S}=\\{B\subseteq S\cup
F:F\in\mathcal{F}(X)\\}$, where $S=\\{(0,n):n\in\mathbb{N}\\}\cup\\{(0,0)\\}$.
Then $\mathcal{S}$ is a bornology on $X$. Note that neither
$\overline{B_{d}}(S,\frac{1}{2})=\\{x\in
X:d(x,S)\leq\frac{1}{2.}\\}\notin\mathcal{S}$ and nor
$\overline{B_{d}}(S,\frac{1}{2})=X$. So by Theorem $3$ of [9],
$\mathcal{S}^{+}$-convergence $\nRightarrow$
$\mathsf{G}_{\mathcal{S},d}^{+}$-convergence on $CL(X)$.

Next we claim that $\tau_{\mathcal{S},d}^{+}$ is strictly finer than
$\mathsf{G}_{\mathcal{S},d}^{+}$ on $CL(X)$. Take $A=\\{(x,0):x\geq 5\\}$.
Then $A\in CL(X)$. Consider $[S,\frac{1}{2}]^{+}(A)$, a neighborhood of $A$ in
$\tau_{\mathcal{S},d}^{+}$. It is enough to show that no
$\mathsf{G}_{\mathcal{S},d}^{+}$-neighborhood of $A$ of the form
$\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{n};\epsilon_{1},\ldots,\epsilon_{n})$ is
contained in $[S,\frac{1}{2}]^{+}(A)$, where $S_{i}\in\mathcal{S}$ and
$\epsilon_{i}>0$ for all $1\leq i\leq n$. Choose $n_{0}\in\mathbb{N}$ such
that $n_{0}>\max\\{x:(x,y)\in S_{i},i=1,\ldots,n\\}+2r$, where
$r=\max\\{\epsilon_{i}:i=1,\ldots,n\\}$. Let $C=\\{(n_{0},y):y\geq 1\\}$.
Since for any $(n_{0},y)\in C$ and any $(x^{\prime},y^{\prime})\in S_{i}$ for
$1\leq i\leq n$,
$d((n_{0},y),(x^{\prime},y^{\prime}))\geq|n_{0}-x^{\prime}|\geq 2r$, we have
$D_{d}(S_{i},C)\geq 2r\ \ \forall i=1,\ldots,n$. So
$C\in\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{n};\epsilon_{1},\ldots,\epsilon_{n})$.
However, $C\notin[S,\frac{1}{2}]^{+}(A)$ as

$d((0,2n_{0}),A)-d((0,2n_{0}),(n_{0},2n_{0}))=\sqrt{4n_{0}^{2}+25}-n_{0}>n_{0}.$

∎

######  Example 3.5.

Let $(X,d)=(\mathbb{R},d_{u})$, where $d_{u}$ is the usual metric on
$\mathbb{R}$ and $\mathcal{S}=\mathcal{F}(\mathbb{R})$. Then
$\mathsf{G}_{\mathcal{S},d}^{+}=\tau_{W_{d}}^{+}$. Suppose
$A_{n}=\\{\frac{1}{n}\\}$ for $n\in\mathbb{N}$ and $A=\\{1\\}$. Then $(A_{n})$
is $\mathcal{F}(\mathbb{R})^{+}$-convergent to $A$. However,
$d(0,\\{1\\})-d(0,A_{n})=1-\frac{1}{n}>\frac{1}{2}$ for large $n$. Thus,
$(A_{n})$ is not $\tau_{W_{d}}^{+}$-convergent to $A$.∎

##  4\. Miss-type characterization for $\tau_{\mathcal{S},d}^{+}$-convergence

The Theorem 2.6 characterized $\mathsf{G}_{\mathcal{S},d}^{+}$-convergence in
terms of enlargements of members of $\mathcal{S}$ by positive constants. In
this section, we present a parallel criterion for the
$\tau_{\mathcal{S},d}^{+}$-convergence whenever $\mathcal{S}$ is a bornology.
This new characterization, in authors opinion, makes it easier to deal with
$\tau_{\mathcal{S},d}^{+}$-convergence. Surprisingly, in the case of
$\tau_{\mathcal{S},d}^{+}$-convergence, enlargements of members of
$\mathcal{S}$ by positive constants need not be sufficient (see, Example 3.4).

We start with some preliminary notations. Define,

$\mathbb{R}_{+}^{X}=\\{f:X\to\mathbb{R}:f(x)>0\text{ for all }x\in X\\}.$

For simplicity, we write $\mathcal{Z}^{+}=\mathbb{R}_{+}^{X}$. Given any
nonempty subset $A$ of $X$ and $f\in\mathcal{Z}^{+}$, the $f$-enlargement of
$A$ is denoted by $B_{d}(A,f)$, and is defined as

$B_{d}(A,f)=\cup_{x\in A}B_{d}(x,f(x)).$

######  Theorem 4.1.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Suppose $(A_{\lambda})$ is a net in $CL(X)$ and $A\in CL(X)$. Then the
following statements are equivalent:

  1. (i)

for $S\in\mathcal{S}$ and $f,g\in\mathcal{Z}^{+}$ with $\inf\\{g(x)-f(x):x\in
S\\}>0$, whenever $A$ misses $g$-enlargement of $S$, then $(A_{\lambda})$
misses the $f$-enlargement of $S$ eventually;

  2. (ii)

$(A_{\lambda})$ is $\tau_{\mathcal{S},d}^{+}$-convergent to $A$.

###### Proof.

$(i)\Rightarrow(ii)$. Let $S\in\mathcal{S}$ and $\epsilon>0$. If $S\subseteq
B_{d}(A,\epsilon)$, then we are done. Otherwise, consider $S^{\prime}=\\{x\in
S:x\notin B_{d}(A,\epsilon)\\}$. Define $f,g\in\mathcal{Z}^{+}$ such that
$g(x)=d(x,A)$ and $f(x)=d(x,A)-\frac{\epsilon}{2}$ for all $x\in S^{\prime}$.
Then $\inf_{x\in S^{\prime}}(g(x)-f(x))>0$ and $A\cap
B_{d}(S^{\prime},g)=\emptyset$. By the hypothesis, we get a $\lambda_{0}$ such
that $A_{\lambda}\cap B_{d}(S^{\prime},f)=\emptyset$ for all
$\lambda\geq\lambda_{0}$. This would imply that
$d(x,A)-d(x,A_{\lambda})<\epsilon$ for all $x\in S$ and
$\lambda\geq\lambda_{0}$. Hence the net $(A_{\lambda})$ is
$\tau_{\mathcal{S},d}^{+}$-convergent to $A$.

$(ii)\Rightarrow(i)$. Suppose $(A_{\lambda})$ is a net that
$\tau_{\mathcal{S},d}^{+}$-converges to $A$ in $CL(X)$. If $A=X$, then there
is nothing to prove. Otherwise, let $S\in\mathcal{S}$ and
$f,g\in\mathcal{Z}^{+}$ with $\inf\\{g(x)-f(x):x\in S\\}=r>0$ and $A\cap
B_{d}(S,g)=\emptyset$. Then by the hypothesis, there is a $\lambda_{0}$ such
that for all $x\in S$ and $\lambda\geq\lambda_{0}$, we have
$d(x,A)-d(x,A_{\lambda})<r$ . So whenever $\lambda\geq\lambda_{0}$ and $x\in
S$, we have $d(x,A_{\lambda})>f(x)$. Thus, $A_{\lambda}\cap
B_{d}(S,f)=\emptyset$ for all $\lambda\geq\lambda_{0}$.

∎

Theorem 4.1 can be viewed as miss-type characterization of
$\tau_{\mathcal{S},d}^{+}$-convergence.

It is worth mentioning that in Theorem 2.6 the family $\mathcal{S}$ is not
assumed to be a bornology. However, Theorem 4.1 need not be true for an
arbitrary family $\mathcal{S}\subseteq\mathcal{P}_{0}(X)$ (see the next
example). So it may be interesting to study the set convergence given by
Theorem 4.1 $(i)$, specially in the context of normed linear spaces where one
usually encounters families of subsets which do not form a bornology.

######  Example 4.2.

Let $(X,d)=(\mathbb{R}^{2},d_{e})$. Define $A_{n}=\\{(x,0):x\leq
0\\}\cup\\{(x,y):y=1-\frac{x}{n},x>0\\}$ for each $n\in\mathbb{N}$, and
$A=\\{(x,0):x\leq 0\\}\cup\\{(x,y):x>0,y=1\\}$. Then $(A_{n})$ is a sequence
in $CL(X)$ and $A\in CL(X)$. Consider
$\mathcal{S}=\\{\\{(x,0):x\in\mathbb{R}\\}\\}\cup\mathcal{F}(\mathbb{R}^{2})$.
Then it is not hard to verify that the sequence $(A_{n})$ and $A$ satisfy the
convergence given in Theorem 4.1 $(i)$. However, the
$\tau_{\mathcal{S},d}^{+}$-convergence of sequence $(A_{n})$ to $A$ fails. As
for $S=\\{(x,0):x\in\mathbb{R}\\}$ and $\epsilon=\frac{1}{2}$, we have for any
$n_{0}\in\mathbb{N}$, $d((n_{0},0),A)-d((n_{0},0),A_{n_{0}})=1>\epsilon$. ∎

The following corollaries shows that for bornologies such as
$\mathcal{B}_{d}(X)$ and $\mathcal{F}(X)$, in Theorem 4.1 $(i)$, it is enough
to consider enlargements by positive constants rather than by members of
$\mathcal{Z}^{+}$, that is, for $\mathcal{S}=\mathcal{B}_{d}(X)$ or
$\mathcal{F}(X)$, we have
$\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}$ (thanks to Theorem
2.6). In the next section, we study the coincidence
$\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}$ for an arbitrary
bornology $\mathcal{S}$.

######  Proposition 4.3.

Let $(X,d)$ be a metric space and let $C$ be a bounded subset of $X$. If
$f\in\mathcal{Z}^{+}$, then $B_{d}(C,f)$ is either bounded or $X$.

###### Proof.

If $\sup\\{f(x):x\in C\\}=r<\infty$, then $B_{d}(C,f)\subseteq B_{d}(C,r)$.
Otherwise, we show that $B_{d}(C,f)=X$. If there is a $z\in X\setminus
B_{d}(C,f)$, then for any $x\in C$, $d(z,x)\geq f(x)$. Given $C$ is bounded,
$C\subseteq B_{d}(x_{0},r)$ for some $x_{0}\in X$ and $r>0$. Choose
$n\in\mathbb{N}$ such that $d(x_{0},z)<nr$. Since the set $\\{f(x):x\in C\\}$
is not bounded above, we can choose $x^{\prime}\in C$ such that
$f(x^{\prime})>(n+2)r$. Consequently, $d(x^{\prime},z)\leq
d(x^{\prime},x_{0})+d(x_{0},z)<(n+2)r<f(x^{\prime})$. We arrive at a
contradiction. ∎

######  Corollary 4.4.

Let $(X,d)$ be a metric space. Suppose $(A_{\lambda})$ is a net in $CL(X)$ and
$A\in CL(X)$. Then the following conditions are equivalent:

  1. (i)

for $S\in\mathcal{B}_{d}(X)$ and $0<\alpha<\epsilon$, whenever $A\cap
B_{d}(S,\epsilon)=\emptyset$, then $A_{\lambda}\cap B_{d}(S,\alpha)=\emptyset$
eventually;

  2. (ii)

for $S\in\mathcal{B}_{d}(X)$ and $f,g\in\mathcal{Z}^{+}$ with
$\inf\\{g(x)-f(x):x\in S\\}>0$, whenever $A\cap B_{d}(S,g)=\emptyset$, then
$A_{\lambda}\cap B_{d}(S,f)=\emptyset$ eventually;

  3. (iii)

$(A_{\lambda})$ is $\tau_{AW_{d}}^{+}$-convergent to $A$.

###### Proof.

$(i)\Rightarrow(ii)$. Let $S\in\mathcal{B}_{d}(X)$ and $f,g\in\mathcal{Z}^{+}$
with $\inf\\{g(x)-f(x):x\in S\\}=r>0$, satisfying, $A\cap
B_{d}(S,g)=\emptyset$. By Proposition 4.3, it follows that
$S^{\prime}=B_{d}(S,f)\in\mathcal{B}_{d}(X)$. Since
$B_{d}(S^{\prime},r)\subseteq B_{d}(S,g)$, we have $A\cap
B_{d}(S^{\prime},r)=\emptyset$. Thus, by the hypothesis for $0<\alpha<r$ there
is a $\lambda_{0}$ such that for all $\lambda\geq\lambda_{0}$, we have
$A_{\lambda}\cap B_{d}(S^{\prime},\alpha)=\emptyset$. Hence $(ii)$ is
established.

The implication $(ii)\Rightarrow(i)$ is immediate, and
$(ii)\Leftrightarrow(iii)$ follows from Theorem 4.1. ∎

######  Remark 4.5.

It is well-known that the $\mathsf{G}_{\mathcal{B}_{d}(X),d}^{+}$ coincides
with the upper bounded proximal topology (see, p.111 of [3]). So by Corollary
4.4 and Theorem 2.6, the upper bounded proximal topology and upper Attouch-
Wets topology coincides on $CL(X)$.

######  Corollary 4.6.

$($Lemma $2.1.2$, [3]$)$ Let $(X,d)$ be a metric space. Suppose
$(A_{\lambda})$ is a net in $CL(X)$ and $A\in CL(X)$. Then the following
assertions are equivalent:

  1. (i)

$(A_{\lambda})$ is $\tau_{W_{d}}^{+}$-convergent to $A\in CL(X)$;

  2. (ii)

for $0<\alpha<\epsilon$, whenever $A\cap B_{d}(x,\epsilon)=\emptyset$, then
$A_{\lambda}\cap B_{d}(x,\alpha)=\emptyset$ eventually.

##  5\. Coincidence of $\tau_{\mathcal{S},d}^{+}$ and
$\mathsf{G}_{\mathcal{S},d}^{+}$

We have seen earlier that $\tau_{\mathcal{S},d}^{+}$-convergence is stronger
than the $\mathsf{G}_{\mathcal{S},d}^{+}$-convergence in general (see,
Proposition 3.1). In this section, we explore the situations for the
equivalence of $\tau_{\mathcal{S},d}^{+}$-convergence and
$\mathsf{G}_{\mathcal{S},d}^{+}$-convergence on $CL(X)$. The following result
gives a sufficient condition for the coincidence of these convergences.

######  Theorem 5.1.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$. If
for each $S\in\mathcal{S}$ and $f\in\mathcal{Z}^{+}$ either
$B_{d}(S,f)\in\mathcal{S}$ or $\overline{B_{d}(S,f)}=X$, then
$\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}$ on $CL(X)$.

###### Proof.

Suppose $(A_{\lambda})$ is a net that
$\mathsf{G}_{\mathcal{S},d}^{+}$-converges to $A$ in $CL(X)$. Pick
$S\in\mathcal{S}$ and $f,g\in\mathcal{Z}^{+}$ with $\inf\\{g(x)-f(x):x\in
S\\}>0$ such that $A\cap B_{d}(S,g)=\emptyset$. Since
$\overline{B_{d}(S,f)}\subseteq B_{d}(S,g)$, we have
$B_{d}(S,f)\in\mathcal{S}$. Set $B_{d}(S,f)=S^{\prime}$, and
$r=\inf\\{g(x)-f(x):x\in S\\}$. Then $B_{d}(S^{\prime},\frac{r}{2})\subseteq
B_{d}(S,g)$. Consequently, $A\cap B_{d}(S^{\prime},\frac{r}{2})=\emptyset$. So
$A_{\lambda}\cap S^{\prime}=\emptyset$ eventually as $(A_{\lambda})$ is
$\mathsf{G}_{\mathcal{S},d}^{+}$-convergent to $A$. Therefore,
$A_{\lambda}\cap B_{d}(S,f)=\emptyset$ eventually. Thus, by Theorem 4.1,
$(A_{\lambda})$ is $\tau_{\mathcal{S},d}^{+}$-convergent to $A$. ∎

The converse of Theorem 5.1 need not be true in general. Consider
$(\mathbb{R},d_{u})$ and take $\mathcal{S}=\mathcal{F}(X)$. Then
$\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}=\tau_{W_{d}}^{+}$
while the condition given in Theorem 5.1 fails.

In [7], the authors introduced the notion of strictly $(\mathcal{S}-d)$
included to study the coincidence
$\mathsf{G}_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},\rho}^{+}$ for two
metrics $d,\rho$, and the coincidence
$\mathsf{G}_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{T},d}^{+}$ for two
families $\mathcal{S},\mathcal{T}$ of subsets of $X$. This notion is also
helpful in studying the coincidence
$\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}$.

######  Definition 5.2.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be an arbitrary family of
nonempty subsets of $X$. We say a subset $A$ of $X$ is strictly
$(\mathcal{S}-d)$ included in another nonempty subset $C$ of $X$ if there
exists a finite subset $\\{S_{1},\ldots,S_{n}\\}$ of $\mathcal{S}$, and for
every $i\in\\{1,\ldots,n\\}$ there are $\alpha_{i},\epsilon_{i}$ with
$0<\alpha_{i}<\epsilon_{i}$, such that

$A\subseteq\cup_{i=1}^{n}B_{d}(S_{i},\alpha_{i})\subseteq\cup_{i=1}^{n}B_{d}(S_{i},\epsilon_{i})\subseteq
C.$

The above definition is a generalization of the notion of strictly
$d$-included introduced by C. Costantini et al. in [16] while studying when
two equivalent metrics on $X$ determine the same Wijsman convergence on
$CL(X)$.

######  Proposition 5.3.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Suppose $A$ and $B$ are two nonempty subsets of $X$. Consider the following
statements:

  1. (i)

$A\in\mathcal{S}$ and $D_{d}(A,B)>0$ ;

  2. (ii)

$A$ is strictly $(\mathcal{S}-d)$ included in $X\setminus B$.

Then $(i)\Rightarrow(ii)$ holds.

###### Proof.

Let $D_{d}(A,B)=\epsilon$. Since $A\in\mathcal{S}$, $A\subseteq
B_{d}(A,\frac{\epsilon}{3})\subseteq B_{d}(A,\frac{\epsilon}{2})\subseteq
X\setminus B$. ∎

To prove our main result of this section, we first fix some notations and
prove a preliminary lemma. Let $\mathcal{S}$ be bornology on a metric space
$(X,d)$. Denote by $\mathcal{F}(\mathcal{S})$ the family of all nonempty
finite subsets of $\mathcal{S}$, and $\mathcal{F}(\mathbb{R}^{+})$ for the
family of all nonempty finite subsets of $\mathbb{R}^{+}=(0,\infty)$. For any
$F\in\mathcal{F}(\mathcal{S})$ choose $\alpha_{F}=\\{\alpha_{S}:S\in
F\\}\in\mathcal{F}(\mathbb{R}^{+})$. Let
$(F,\alpha_{F})=\\{(S,\alpha_{S}):S\in F\\}$. Consider the collection

$\mathcal{N}=\\{(F,\alpha_{F}):(F,\alpha_{F})\in\mathcal{F}(\mathcal{S})\times\mathcal{F}(\mathbb{R}^{+})\\}.$

Define a relation $\leqslant$ on $\mathcal{N}$ by
$(F,\alpha_{F})\leqslant(F^{\prime},\alpha_{F^{\prime}})$ if and only if
$\cup_{S\in F}B_{d}(S,\alpha_{S})\subseteq\cup_{S^{\prime}\in
F^{\prime}}B_{d}(S^{\prime},\alpha_{S^{\prime}})$.

Set $F^{\prime\prime}=F\cup F^{\prime}$ and
$\alpha_{F^{\prime\prime}}=\alpha_{F}\cup\alpha_{F^{\prime}}=\\{\alpha_{S^{\prime\prime}}:S^{\prime\prime}\in
F^{\prime\prime}\\}$. Then
$(F^{\prime\prime},\alpha_{F^{\prime\prime}})\in\mathcal{N}$. Moreover,
$(F,\alpha_{F})\leqslant(F^{\prime\prime},\alpha_{F^{\prime\prime}})$, and
$(F^{\prime},\alpha_{F^{\prime}})\leqslant(F^{\prime\prime},\alpha_{F^{\prime\prime}})$.
Observe that the relation $\leqslant$ is reflexive and transitive on
$\mathcal{N}$. Thus, $(\mathcal{N},\leqslant)$ is a directed set.

######  Lemma 5.4.

Let $C$ be a nonempty open subset of a metric space $(X,d)$ and let
$\mathcal{S}$ be a bornology on $X$. Then the collection
$\Omega\subseteq\mathcal{N}$ defined as,

$\Omega=\left\\{(F,\alpha_{F}):\exists~{}\epsilon_{F}\in\mathcal{F}(\mathbb{R}^{+})~{}\text{with}~{}\alpha_{S}<\epsilon_{S}\text{
for }S\in F\text{ and }\bigcup_{S\in F}B_{d}(S,\epsilon_{S})\subseteq
C\right\\},$

is a directed set under the relation $\leqslant$ defined above.

###### Proof.

Note that $\Omega$ is nonempty as $C$ is nonempty open and $\mathcal{S}$ is a
bornology. Since $\Omega\subseteq\mathcal{N}$, the relation $\leqslant$ is
reflexive as well as transitive on $\Omega$. It remains to show that for any
$(F,\alpha_{F}),(F^{\prime},\alpha_{F^{\prime}})\in\Omega$ there exists
$(F^{\prime\prime},\alpha_{F^{\prime\prime}})\in\Omega$ satisfying
$(F,\alpha_{F})\leqslant(F^{\prime\prime},\alpha_{F^{\prime\prime}})$, and
$(F^{\prime},\alpha_{F^{\prime}})\leqslant(F^{\prime\prime},\alpha_{F^{\prime\prime}})$.
Take $(F^{\prime\prime},\alpha_{F^{\prime\prime}})$ as above. By the above
argument, we just need to show that
$(F^{\prime\prime},\alpha_{F^{\prime\prime}})\in\Omega$.

Choose $\epsilon_{F}$, $\epsilon_{F^{\prime}}\in\mathcal{F}(\mathbb{R}^{+})$
satisfying the definition of $\Omega$ corresponding to $(F,\alpha_{F})$ and
$(F^{\prime},\alpha_{F^{\prime}})$ respectively. Let
$\epsilon_{F^{\prime\prime}}=\epsilon_{F}\cup\epsilon_{F^{\prime}}$. Then,
$\alpha_{S^{\prime\prime}}<\epsilon_{S^{\prime\prime}}$ for
$S^{\prime\prime}\in F^{\prime\prime}$, and $\cup_{S^{\prime\prime}\in
F^{\prime\prime}}B_{d}(S^{\prime\prime},\epsilon_{S^{\prime\prime}})\subseteq
C$. Therefore, $(\Omega,\leq)$ is a directed set.∎

######  Theorem 5.5.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Then the following statements are equivalent:

  1. (i)

$\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}$ on $CL(X)$;

  2. (ii)

for each $S\in\mathcal{S}$ and $f,g\in\mathcal{Z}^{+}$ with
$\inf\\{g(x)-f(x):x\in S\\}>0$, either $B_{d}(S,f)$ is strictly
$(\mathcal{S}-d)$ included in $B_{d}(S,g)$ or $B_{d}(S,g)=X$.

###### Proof.

$(i)\Rightarrow(ii)$. Suppose $(ii)$ fails. Choose an $S_{0}\in\mathcal{S}$
and $f,g\in\mathcal{Z}^{+}$ with $\inf\\{g(x)-f(x):x\in S_{0}\\}>0$ such that
neither $B_{d}(S_{0},g)=X$ nor $B_{d}(S_{0},f)$ is strictly $(\mathcal{S}-d)$
included in $B_{d}(S_{0},g)$. Consider

$\Omega=\left\\{(F,\alpha_{F})\in\mathcal{F}(\mathcal{S})\times\mathcal{F}(\mathbb{R}^{+}):\begin{split}\exists~{}\epsilon_{F}\in\mathcal{F}(\mathbb{R}^{+})~{}\text{with}~{}\alpha_{S}<\epsilon_{S}~{}\text{for}~{}S\in
F\\\ \text{and}\bigcup_{S\in F}B_{d}(S,\epsilon_{S})\subseteq
B_{d}(S_{0},g)\end{split}\right\\}.$

Clearly, $\Omega$ is nonempty as $(\\{x\\},\\{f(x)\\})\in\Omega$ for any $x\in
S_{0}$. By Lemma 5.4, $\Omega$ is a directed set under the relation
$\leqslant$. Since $B_{d}(S_{0},f)$ is not strictly $(\mathcal{S}-d)$ included
in $B_{d}(S_{0},g)$, for each $(F,\alpha_{F})\in\Omega$ we have
$x_{(F,\alpha_{F})}\in B_{d}(S_{0},f)\setminus(\cup_{S\in
F}B_{d}(S,\alpha_{S}))$. Put $A=X\setminus B_{d}(S_{0},g)$. Then $A\in CL(X)$.
Now for any $(F,\alpha_{F})\in\Omega$ define
$A_{(F,\alpha_{F})}=A\cup\\{x_{(F,\alpha_{F})}\\}$. Consequently,
$\left(A_{(F,\alpha_{F})}\right)$ is a net in $CL(X)$. We claim that the net
$(A_{(F,\alpha_{F})})$ $\mathsf{G}_{\mathcal{S},d}^{+}$-converges to $A$ while
its $\tau_{\mathcal{S},d}^{+}$-convergence fails. Take any finite subset
$F^{\prime}=\\{S^{\prime}_{1},\ldots,S^{\prime}_{m}\\}$ of $\mathcal{S}$ and
$\alpha_{F^{\prime}}=(\alpha^{\prime}_{1},\ldots,\alpha^{\prime}_{m})\in\mathcal{F}(\mathbb{R}^{+})$
such that
$A\in\mathscr{A}_{d}^{+}(S^{\prime}_{1},\ldots,S^{\prime}_{m};\alpha^{\prime}_{1},\ldots,\alpha^{\prime}_{m})$.
So $D_{d}(S^{\prime}_{i},A)>\alpha^{\prime}_{i}$ for each $i=1,\ldots,m$. Let
$r^{\prime}_{i}=\frac{D_{d}(S^{\prime}_{i},A)+\alpha^{\prime}_{i}}{2}$ for
$i=1,\ldots,m$. Then $D_{d}(S^{\prime}_{i},A)>r^{\prime}_{i}$ for
$i=1,\ldots,m$, which gives
$\cup_{i=1}^{m}B_{d}(S^{\prime}_{i},r^{\prime}_{i})\subseteq B_{d}(S_{0},g)$.
Thus, $(F^{\prime},\alpha_{F^{\prime}})\in\Omega$. So there exists
$\epsilon_{F^{\prime}}=(\epsilon^{\prime}_{1},\ldots,\epsilon^{\prime}_{m})\in\mathcal{F}(\mathbb{R}^{+})$
such that $\alpha^{\prime}_{i}<\epsilon^{\prime}_{i}$ for $1\leq i\leq m$ and
$(F^{\prime},\epsilon_{F^{\prime}})\in\Omega$. Then for any
$(F,\epsilon_{F})\in\Omega$ such that
$(F^{\prime},\epsilon_{F^{\prime}})\leqslant(F,\epsilon_{F})$ we have
$\\{x_{(F,\epsilon_{F})}\\}\in\mathscr{A}_{d}^{+}(S^{\prime}_{1},\ldots,S^{\prime}_{m};\alpha^{\prime}_{1},\ldots,\alpha^{\prime}_{m})$.
Consequently,
$A_{(F,\epsilon_{F})}\in\mathscr{A}_{d}^{+}(S^{\prime}_{1},\ldots,S^{\prime}_{m};\alpha^{\prime}_{1},\ldots,\alpha^{\prime}_{m})$
whenever $(F^{\prime},\epsilon_{F^{\prime}})\leqslant(F,\epsilon_{F})$. So the
net $(A_{(F,\alpha_{F})})_{(F,\alpha_{F})\in\Omega}$
$\mathsf{G}_{\mathcal{S},d}^{+}$-converges to $A$. Now $A\cap
B_{d}(S_{0},g)=\emptyset$ but for every $(F,\alpha_{F})\in\Omega$, we have
$A_{(F,\alpha_{F})}\cap B_{d}(S_{0},f)\neq\emptyset$. Thus, by Theorem 4.1,
the net $\left(A_{(F,\alpha_{F})}\right)_{(F,\alpha_{F})\in\Omega}$ does not
converge to $A$ with respect to $\tau_{\mathcal{S},d}^{+}$.

$(ii)\Rightarrow(i)$. Suppose $(A_{\lambda})$ is a net in $CL(X)$ that
$\mathsf{G}_{\mathcal{S},d}^{+}$-converges to $A\in CL(X)$. If $A=X$, then
$(A_{\lambda})$ is $\tau_{\mathcal{S},d}^{+}$-convergent to $A$. Otherwise,
consider $S\in\mathcal{S}$ and $f,g\in\mathcal{Z}^{+}$ with
$\inf\\{g(x)-f(x):x\in S\\}>0$ and $A\cap B_{d}(S,g)=\emptyset$. Since
$B_{d}(S,f)$ is strictly $(\mathcal{S}-d)$ included in $B_{d}(S,g)$, there
exist $S_{1},\ldots,S_{n}\in\mathcal{S}$, and $0<\alpha_{i}<\epsilon_{i}$ for
$i=1,\ldots,n$ such that
$B_{d}(S,f)\subseteq\cup_{i=1}^{n}B_{d}(S_{i},\alpha_{i})\subseteq\cup_{i=1}^{n}B_{d}(S_{i},\epsilon_{i})\subseteq
B_{d}(S,g)$. So $A\cap(\cup_{i=1}^{n}B_{d}(S_{i},\epsilon_{i}))=\emptyset$.
Then by Theorem 2.6,
$A_{\lambda}\cap(\cup_{i=1}^{n}B_{d}(S_{i},\alpha_{i}))=\emptyset$ eventually.
So $A_{\lambda}\cap B_{d}(S,f)=\emptyset$ eventually. Consequently, by Theorem
4.1, $(A_{\lambda})$ $\tau_{\mathcal{S},d}^{+}$-converges to $A$. ∎

Recall that for $\mathcal{S}=\mathcal{B}_{d}(X)$ and $\mathcal{P}_{0}(X)$, the
topology $\tau_{\mathcal{S},d}^{+}$ reduces to the upper Attouch-Wets
topology, and the upper Hausdorff metric topology, respectively. So we have
the following corollaries originally observed by Beer [3].

######  Corollary 5.6.

Let $(X,d)$ be a metric space. Then the upper Attouch-Wets topology on $CL(X)$
is the weakest topology such that each member of the family of set functionals
$\\{D_{d}(S,.\cdot):S\in\mathcal{B}_{d}(X)\\}$ is lower semi-continuous.

###### Proof.

Let $S\in B_{d}(X)$ and $f,g\in\mathcal{Z}^{+}$ with $\inf_{x\in
S}(g(x)-f(x))>0$. Then by Proposition 4.3, either
$B_{d}(S,g)\in\mathcal{B}_{d}(X)$ or $B_{d}(S,g)=X$. If $B_{d}(S,g)\neq X$,
then $B_{d}(S,f)\in\mathcal{B}_{d}(X)$. Further, by the choice of $f$ and $g$,
$B_{d}(S,f)$ is far from $X\setminus B_{d}(S,g)$. Consequently, by Proposition
5.3, $B_{d}(S,f)$ is strictly $(\mathcal{S}-d)$ included in $B_{d}(S,g)$.
Thus, the result follows from the Theorem 5.5. ∎

######  Corollary 5.7.

Let $(X,d)$ be a metric space. Then the upper Hausdorff metric topology on
$CL(X)$ is the weakest topology such that each member of the family of set
functionals $\\{D_{d}(S,\cdot):S\in\mathcal{P}_{0}(X)\\}$ is lower semi-
continuous.

######  Remark 5.8.

Note that each of the bornology $\mathcal{P}_{0}(X)$, $\mathcal{B}_{d}(X)$,
and $\mathcal{F}(X)$ is shielded from closed sets and in each case we have
$\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}$. However,
$\tau_{\mathcal{T}\mathcal{B}_{d}(X),d}^{+}=\mathsf{G}_{\mathcal{T}\mathcal{B}_{d}(X),d}^{+}$
while $\mathcal{T}\mathcal{B}_{d}(X)$ need not be shielded from closed sets in
general. For more on shields, see [6].

We now give an example of a sequence $(A_{n})$ of subsets of the Euclidean
space $(\mathbb{R}^{2},d_{e})$ that appears geometrically not converging to
$A\subseteq\mathbb{R}^{2}$ such that $(A_{n})$ does not converge to $A$ with
respect to $\tau_{\mathcal{S},d}^{+}$ but converges to $A$ with respect to
$\mathsf{G}_{\mathcal{S},d}^{+}$.

######  Example 5.9.

Let $(X,d)=(\mathbb{R}^{2},d_{e})$ and $\mathcal{B}=\\{S\subseteq S_{0}\cup
F:F\in\mathcal{F}(X)\\}$, where $S_{0}=\\{(x,y):x\geq 1,y=1\\}$. It is easy to
verify that $\mathcal{B}$ is a bornology on $X$. Define $A_{n}=\\{(x,y):x\geq
0,y=n\\}$ for each $n\in\mathbb{N}$ and $A=\\{(x,y):x=0,y\geq 0\\}$. Then
$(A_{n})$ is a sequence of closed sets. We claim that $(A_{n})$ converges to
$A$ in $\mathsf{G}_{\mathcal{B},d}^{+}$. Let
$S_{1},\ldots,S_{k}\in\mathcal{B}$ and $\epsilon_{1},\ldots,\epsilon_{k}>0$
such that
$A\in\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{k};\epsilon_{1},\ldots,\epsilon_{k})$.
Take $n_{0}=\max\\{y:(x,y)\in
S_{i},i=1,\ldots,k\\}+2\max\\{\epsilon_{i}:i=1,\ldots,k\\}$. Observe that, for
$n\geq n_{0}$,
$A_{n}\in\mathscr{A}_{d}^{+}(S_{1},\ldots,S_{k};\epsilon_{1},\ldots,\epsilon_{k})$.
Thus, we have established $\mathsf{G}_{\mathcal{B},d}^{+}$ convergence.

To see $(A_{n})$ does not converge to $A$ in $\tau_{\mathcal{B},d}^{+}$. Take
$S=\mathbb{N}\times\\{1\\}$ and let $f,g\in\mathcal{Z}^{+}$ defined by
$f((m,1))=\frac{m}{8}$ and $g((m,1))=\frac{m}{4}$ for each $m\in\mathbb{N}$.
Observe that $\inf\\{g((m,1))-f((m,1)):m\in\mathbb{N}\\}=\frac{1}{8}$. Also
$A\cap B_{d}(S,g)=\emptyset$.Since for any $n\in\mathbb{N}$, $(16n,n)\in
A_{n}\cap B_{d}((16n,1),\frac{16n}{8})$, $A_{n}\cap B_{d}(S,f)\neq\emptyset$
for every $n\in\mathbb{N}$. Thus, $(A_{n})$ does not converge to $A$ in
$\tau_{\mathcal{B},d}^{+}$. ∎

##  6\. Bornological Convergence

In [13], the authors characterized the coincidence of
$\mathcal{S}^{+}$-convergence and $\tau_{\mathcal{S},d}^{+}$-convergence for
an arbitrary bornology $\mathcal{S}$. They used the notion of
$\mathcal{S}$-convergent to infinity to characterize the aforesaid
coincidence. In this section, we first provide a new approach to determine
$\mathcal{S}^{+}$-convergence which is similar to the one given for the
$\tau_{\mathcal{S},d}^{+}$-convergence in Theorem 4.1. Then we give necessary
and sufficient conditions on the structure of the bornology $\mathcal{S}$ for
the coincidence of $\mathcal{S}^{+}$-convergence and
$\tau_{\mathcal{S},d}^{+}$-convergence on $CL(X)$.

######  Theorem 6.1 (Miss type characterization for bornological convergence).

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Suppose $(A_{\lambda})$ is a net in $\mathcal{P}_{0}(X)$ $(CL(X))$ and $A$ is
a nonempty (closed) subset of $X$. Then the following statements are
equivalent.

  1. (i)

$(A_{\lambda})$ $\mathcal{S}^{+}$-converges to $A$;

  2. (ii)

for $S\in\mathcal{S}$ and $\epsilon>0$, whenever $A$ misses
$\epsilon$-enlargement of $S$, then $(A_{\lambda})$ misses $S$ eventually.

###### Proof.

$(i)\Rightarrow(ii)$. Let $S\in\mathcal{S}$ and $\epsilon>0$ such that $A\cap
B_{d}(S,\epsilon)=\emptyset$. By the hypothesis, there is a
$\lambda_{0}\in\Lambda$ such that for all $\lambda\geq\lambda_{0}$, we have
$A_{\lambda}\cap S\subseteq B_{d}(A,\epsilon)$. Consequently, by the choice of
$S$ and $\epsilon$, $A_{\lambda}\cap S=\emptyset$ for all
$\lambda\geq\lambda_{0}$.

$(ii)\Rightarrow(i)$. Let $S\in\mathcal{S}$ and $\epsilon>0$. If $A\cap
B_{d}(S,\epsilon)=\emptyset$, then there is a $\lambda_{1}\in\Lambda$ such
that $A_{\lambda}\cap S=\emptyset$ for all $\lambda\geq\lambda_{1}$.
Consequently, $A_{\lambda}\cap S\subseteq B_{d}(A,\epsilon)$ for all
$\lambda\geq\lambda_{1}$. Otherwise, let $B=\\{x\in S:x\in
B_{d}(A,\epsilon)\\}$ and $S^{\prime}=S\setminus B$. When $B=S$, then
$A_{\lambda}\cap S\subseteq B_{d}(A,\epsilon)$ for all $\lambda$. If $B\neq
S$, then $S^{\prime}\neq\emptyset$. Moreover, $A\cap
B_{d}(S^{\prime},\epsilon)=\emptyset$. Thus, by the hypothesis there is a
$\lambda_{2}\in\Lambda$ such that $A_{\lambda}\cap S^{\prime}=\emptyset$ for
all $\lambda\geq\lambda_{2}$. Also for each $\lambda\in\Lambda$,
$A_{\lambda}\cap B\subseteq B_{d}(A,\epsilon)$. Consequently, $A_{\lambda}\cap
S\subseteq B_{d}(A,\epsilon)$ for all $\lambda\geq\lambda_{2}$. Hence
$(A_{\lambda})_{\lambda\in\Lambda}$ $\mathcal{S}^{+}$-converges to $A$. ∎

We would like to mention that the equivalence of Theorem 6.1 may not hold if
we do not assume $\mathcal{S}$ to be a bornology on $X$. This is shown by the
following example.

######  Example 6.2.

Suppose $X=\mathbb{R}^{2}$ and $d=d_{e}$, the Euclidean metric. Let
$\mathcal{S}=\\{B_{d}((0,0),2m):m\in\mathbb{N}\\}$. Then $\mathcal{S}$ is a
cover of $X$. Observe that $\downarrow\mathcal{S}=\\{B\subseteq
S:S\in\mathcal{S}\\}=\mathcal{B}_{d}(X)$. So by Corollary $3.3$ of [22], we
have $\mathcal{S}^{+}=\tau_{AW_{d}}^{+}$. Let
$A=\\{(x,y):y=\frac{1}{x},x>0\\}$, and
$A_{n}=\\{(x,y):y=\frac{1}{x}+\frac{1}{n},x<0\\}$ for each $n\in\mathbb{N}$.
Note that $A\cap B_{d}((0,0),2m)\neq\emptyset$ for any $m\geq 1$. Thus, the
sequence $(A_{n})$ and $A$ satisfy $(ii)$ of Theorem 6.1. On the other hand,
for any $0<\epsilon<1$ and $n\in\mathbb{N}$, $\emptyset\neq A_{n}\cap
B_{d}((0,0),2)\nsubseteq B_{d}(A,\epsilon)$. Therefore, the
$\tau_{AW_{d}}^{+}$-convergence of sequence $(A_{n})$ to $A$ fails.∎

Recall that for a bornology $\mathcal{S}$ on a metric space $(X,d)$, the upper
$\mathcal{S}$-proximal topology $\mu_{\mathcal{S}}^{++}$ on
$\mathcal{P}_{0}(X)$ is generated by all sets of the form:
$(S^{c})^{++}=\\{A\in\mathcal{P}_{0}(X):S\cap
B_{d}(A,\epsilon)=\emptyset~{}\text{for some}~{}\epsilon>0\\}$, where
$S\in\mathcal{S}$ ([22, 18]). The relation between
$\mathcal{S}^{+}$-convergence and $\mu_{\mathcal{S}}^{++}$-convergence is
given in Proposition $15$ of [13]. We reproduce its proof using Theorem 6.1.

######  Proposition 6.3.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Then the following statements are equivalent:

  1. (i)

for each $S\in\mathcal{S}$ there is an $\epsilon>0$ such that
$B_{d}(S,\epsilon)\in\mathcal{S}$, that is, $\mathcal{S}$ is stable under
small enlargements;

  2. (ii)

$\mathcal{S}^{+}$-convergence is compatible with
$\mu_{\mathcal{S}}^{++}$-convergence on $\mathcal{P}_{0}(X)$;

  3. (iii)

$\mathcal{S}^{+}$-convergence is topological on $\mathcal{P}_{0}(X)$.

###### Proof.

$(i)\Rightarrow(ii)$. Suppose $(A_{\lambda})$ is a net in $\mathcal{P}_{0}(X)$
that $\mathcal{S}^{+}$-converges to a nonempty set $A$. If $\overline{A}=X$,
then there is nothing to show. Otherwise, let $S\in\mathcal{S}$ be such that
$S\cap B_{d}(A,\epsilon)=\emptyset$, equivalently $A\cap
B_{d}(S,\epsilon)=\emptyset$ for some $\epsilon>0$. By assumption, we can find
$0<\delta<\frac{\epsilon}{2}$ such that $B_{d}(S,\delta)\in\mathcal{S}$. Then,
$A\cap B_{d}(B_{d}(S,\delta),{\frac{\epsilon}{2}})=\emptyset$. Applying
Proposition 6.1, it can be concluded that, $A_{\lambda}\cap
B_{d}(S,\delta)=\emptyset$ eventually. Thus, the net
$(A_{\lambda})_{\lambda\in\Lambda}$ $\mu_{\mathcal{S}}^{++}$-converges to $A$.

Conversely, suppose $(A_{\lambda})$ $\mu_{\mathcal{S}}^{++}$-converges to $A$.
Let $S\in\mathcal{S}$ and $\epsilon>0$ be such that $A\cap
B_{d}(S,\epsilon)=\emptyset$. Then $A_{\lambda}\in(S^{c})^{++}$ eventually.
Thus, $A_{\lambda}\cap S=\emptyset$ eventually. Hence by Theorem 6.1,
$(A_{\lambda})$ $\mathcal{S}^{+}$-converges to $A$.

The implications $(ii)\Rightarrow(iii)$ is immediate and for
$(iii)\Rightarrow(i)$ see Theorem $3.7$ of [22]. ∎

In [13], the authors gave an example (Example 20) showing the condition that a
family $\mathcal{S}$ being stable under enlargements by positive constants may
not be sufficient to enforce the equivalence of $\tau_{\mathcal{S},d}^{+}$ and
$\mathcal{S}^{+}$-convergences. So we may need a stronger stability condition
on $\mathcal{S}$ to achieve the required equivalence. We now give the main
result of this section that divulges this stronger stability condition on
$\mathcal{S}$.

######  Theorem 6.4.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$.
Then the following statements are equivalent:

  1. (i)

$\tau_{\mathcal{S},d}^{+}$-convergence $=\mathcal{S}^{+}$-convergence on
$CL(X)$;

  2. (ii)

for each $S\in\mathcal{S}$ and $f,g\in\mathcal{Z}^{+}$ with
$\inf\\{g(x)-f(x):x\in S\\}>0$, either $B_{d}(S,f)\in\mathcal{S}$ or
$B_{d}(S,g)=X$.

###### Proof.

$(i)\Rightarrow(ii)$. Suppose there exist $S_{0}\in\mathcal{S}$ and
$f,g\in\mathcal{Z}^{+}$ with $\inf_{x\in S_{0}}(g(x)-f(x))>0$ for which
$B_{d}(S_{0},g)\neq X$ as well as $B_{d}(S_{0},f)\notin\mathcal{S}$. Define

$\Omega=\\{S\in\mathcal{S}:\text{ there is an }\epsilon>0\text{ such that
}B_{d}(S,\epsilon)\subseteq B_{d}(S_{0},g)\\}.$

Note that $\Omega$ is nonempty as $S_{0}\in\Omega$. By assumption, for each
$S\in\Omega$ there is an $x_{S}\in B_{d}(S_{0},f)\setminus S$. Put
$A=X\setminus B_{d}(S_{0},g)$ and for $S\in\Omega$, let
$A_{S}=A\cup\\{x_{S}\\}$. Direct $\Omega$ by set inclusion. Then
$(A_{S})_{S\in\Omega}$ is a net in $CL(X)$. Since $A_{S}\cap
B_{d}(S_{0},f)\neq\emptyset$ for each $S\in\mathcal{S}$ and $A\cap
B_{d}(S_{0},g)=\emptyset$, by Theorem 4.1, the
$\tau_{\mathcal{S},d}^{+}$-convergence of $(A_{S})$ to $A$ fails. To see
$(A_{S})$ $\mathcal{S}^{+}$-converges to $A$, let $S_{1}\in\mathcal{S}$ and
$\epsilon_{1}>0$ such that $A\cap B_{d}(S_{1},\epsilon_{1})=\emptyset$. So
$S_{1}\in\Omega$. Then for any $S\in\Omega$ with $S\supseteq S_{1}$, we have
$A_{S}\cap S_{1}=\emptyset$. Therefore, by Theorem 6.1, the net $(A_{S})$ is
$\mathcal{S}^{+}$-convergent to $A\in CL(X)$. This is a contradiction.

$(ii)\Rightarrow(i)$. Suppose $(A_{\lambda})$ is a net
$\mathcal{S}^{+}$-converging to $A$ in $CL(X)$. If $A=X$, then there is
nothing to show. Otherwise, let $S\in\mathcal{S}$, and $f,g\in\mathcal{Z}^{+}$
with $\inf_{x\in S}(g(x)-f(x))=r>0$ such that $A\cap B_{d}(S,g)=\emptyset$. By
the hypothesis, we get $S^{\prime}=B_{d}(S,f)\in\mathcal{S}$. Since $A\cap
B_{d}(S^{\prime},r)=\emptyset$ and the net $(A_{S})$ is
$\mathcal{S}^{+}$-convergent to $A$, by Theorem 6.1, we obtain
$A_{\lambda}\cap S^{\prime}=\emptyset$ eventually. Thus, by Theorem 4.1, the
net $(A_{\lambda})$ is $\tau_{\mathcal{S},d}^{+}$-convergent to $A$. ∎

######  Corollary 6.5.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$. If
$B_{d}(S,f)\in\mathcal{S}$ for each $S\in\mathcal{S}$ and
$f\in\mathcal{Z}^{+}$, then $\tau_{\mathcal{S},d}^{+}$-convergence
$=\mathcal{S}^{+}$-convergence on $CL(X)$.

If $d$ is an unbounded metric on $X$ and
$\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$, then by Theorem 6.4, each ball in
$(X,d)$ is in $\mathcal{S}$. So whenever
$\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$ for an unbounded metric space
$(X,d)$, then $\mathcal{B}_{d}(X)\subseteq\mathcal{S}$. Moreover, in
Proposition $19$ of [13], it has been shown that whenever $(X,d)$ is unbounded
and $\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$, then $\mathcal{S}$ is stable
under enlargements. The proof of this proposition uses the notion of
$\mathcal{S}$-convergence to infinity. We next give a new proof of Proposition
$19$ of [13] in the vein of Theorem 6.4.

######  Corollary 6.6.

Let $(X,d)$ be an unbounded metric space and let $\mathcal{S}$ be a bornology
on $X$. If $\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$ on $CL(X)$, then
$\mathcal{S}$ is stable under enlargements.

###### Proof.

Let $S_{0}\in\mathcal{S}$ and $r>0$. If $B_{d}(S_{0},r+\epsilon)\neq X$ for
some $\epsilon>0$, then by Theorem 6.4, $B_{d}(S_{0},r)\in\mathcal{S}$.
Otherwise, $B_{d}(S_{0},r+\epsilon)=X$ for each $\epsilon>0$. Let $x_{0}\in
S_{0}$, and $S^{\prime}=S_{0}\setminus B_{d}(x_{0},3r)$. Since $(X,d)$ is
unbounded, $S^{\prime}\neq\emptyset$. Further, note that
$B_{d}(S^{\prime},\frac{5r}{2})\neq X$ as $d(x_{0},S^{\prime})\geq 3r$.
Consequently, by Theorem 6.4, we have $B_{d}(S^{\prime},2r)\in\mathcal{S}$.
Also $B_{d}(x_{0},5r)\in\mathcal{S}$. Then $B_{d}(S_{0},2r)\in\mathcal{S}$ as
$B_{d}(S_{0},2r)\subseteq B_{d}(S^{\prime},2r)\cup B_{d}(x_{0},5r)$. And, by
our assumption, $X=B_{d}(S_{0},2r)$. Hence $\mathcal{S}=\mathcal{P}_{0}(X)$. ∎

######  Proposition 6.7.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be bornology on $X$.
Suppose for any $S\in\mathcal{S}$ and $f\in\mathcal{Z}^{+}$ either
$B_{d}(S,f)\in\mathcal{S}$ or $\overline{B_{d}(S,f)}=X$. Then
$\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$ on $CL(X)$.

###### Proof.

Suppose $(A_{\lambda})$ is a net which $\mathcal{S}^{+}$-converges to $A$ in
$CL(X)$. If $A=X$, then there is nothing to show. Otherwise, let
$S\in\mathcal{S}$ and $f,g\in\mathcal{Z}^{+}$ with $\inf_{x\in
S}(g(x)-f(x))>0$ such that $A\cap B_{d}(S,g)=\emptyset$. Then by the
hypothesis, we get $B_{d}(S,f)\in\mathcal{S}$. Therefore, applying Theorem
6.1, we get $A_{\lambda}\cap B_{d}(S,f)=\emptyset$ eventually. Thus, the
coincidence $\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$ follows from Theorem
4.1. ∎

The following example shows that the condition in Proposition 6.7 need not be
necessary for the coincidence of the convergences $\tau_{\mathcal{S},d}^{+}$
and $\mathcal{S}^{+}$.

######  Example 6.8.

Let $X=[0,1]\cup E$ where $E=\cup_{p\in P}E_{p}$,
$E_{p}=\\{p^{n}:n\in\mathbb{N}\\}$, and $P=\\{p:p\text{ is prime}\\}$. Suppose
$d:X\times X\rightarrow\mathbb{R}$ is defined as

$d(x,y)=\begin{cases}2&\text{ if }x,y\in[0,1]\text{ or }x\in[0,1],y\in E,x\neq
y\\\ 1+\sum_{r=1}^{q}\frac{1}{2^{r}}&\text{ if }x\in E_{p},y\in E_{q},p\leq
q,x\neq y\\\ 0&\text{ if }x=y\end{cases}$

where $x,y\in X$. It is routine to verify that $d$ forms a metric on $X$.
Define the collection $\mathcal{A}=\\{E_{p}\cup F:p\in
P,F\in\mathcal{F}(X)\\}$. Then $\mathcal{A}$ forms a cover of $X$. By
Proposition $11.2$ of [5], $\mathcal{A}$ generates a bornology, say
$\mathcal{B}(\mathcal{A})$ on $X$.

First, we show that for $S\in\mathcal{B}(\mathcal{A})$ and
$f,g\in\mathcal{Z}^{+}$ with $\inf_{x\in S}(g(x)-f(x))>0$, either
$B_{d}(S,f)\in\mathcal{B}(\mathcal{A})$ or $B_{d}(S,g)=X$. Observe that any
$S\in\mathcal{B}(\mathcal{A})$ is a subset of $\cup_{i=1}^{n}E_{p_{i}}\cup F$
for $p_{i}\in P$, $1\leq i\leq n$ and $F\in\mathcal{F}([0,1])$. For
$f\in\mathcal{Z}^{+}$ there are two possibilities: $\sup_{x\in S}f(x)<2$ or
$\sup_{x\in S}f(x)\geq 2$. When $\sup_{x\in S}f(x)<2$, then we can find
$p_{0}\in P$ such that $p_{0}>\max_{1\leq i\leq n}p_{i}$ and $\sup_{x\in
S}f(x)<1+\sum_{r=1}^{p}\frac{1}{2^{r}}~{}\forall~{}p\geq p_{0}$. Therefore,
$B_{d}(S,f)\subseteq\cup_{p\leq p_{0}}E_{p}\cup F\in\mathcal{B}(\mathcal{A})$.
When $\sup_{x\in S}f(x)\geq 2$, then $g(x)>2$ for some $x\in S$. So
$B_{d}(S,g)=X$. Hence
$\tau_{\mathcal{B}(\mathcal{A}),d}^{+}=\mathcal{B}(\mathcal{A})^{+}$ follows
by Theorem 6.4.

Next, we show that the necessary condition in Proposition 6.7 fails. Consider
$E_{2}$ and $f\in\mathcal{Z}^{+}$ defined by $f(x=2^{n})=2-\frac{1}{n}$ for
$n\in\mathbb{N}$. We claim that $E\subseteq B_{d}(E_{2},f)$ and
$\overline{B_{d}(E_{2},f)}\neq X$. Let $y\in E_{q}$ for some $q\in P$. Then
there exists $n_{0}\in\mathbb{N}$ such that
$1+\sum_{r=1}^{q}\frac{1}{2^{r}}<2-\frac{1}{n_{0}}$. Consequently, $y\in
B_{d}(2^{n_{0}},2-\frac{1}{n_{0}})\subseteq B_{d}(E_{2},f)$. So $E\subseteq
B_{d}(E_{2},f)$. Therefore $B_{d}(E_{2},f)\notin\mathcal{B}(\mathcal{A})$.
Finally, note that for any $z\in[0,1]$, we have $B_{d}(z,1)\cap
B_{d}(E_{2},f)=\emptyset$. Hence $\overline{B_{d}(E_{2},f)}\neq X$.∎

######  Theorem 6.9.

Let $(X,d)$ be an almost convex metric space and let $\mathcal{S}$ be a
bornology on $X$. Then the following statements are equivalent:

  1. (i)

for $S\in\mathcal{S}$ and $f\in\mathcal{Z}^{+}$ either
$B_{d}(S,f)\in\mathcal{S}$ or $\overline{B_{d}(S,f)}=X$;

  2. (ii)

$\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$ on $CL(X)$.

###### Proof.

It is enough to prove $(ii)\Rightarrow(i)$. Suppose $(i)$ fails. So there
exist $S\in\mathcal{S}$ and $f\in\mathcal{Z}^{+}$ such that neither
$B_{d}(S,f)\in\mathcal{S}$ nor $\overline{B_{d}(S,f)}=X$. Let $p\in
X\setminus\overline{B_{d}(S,f)}$. Put $2r=d(p,\overline{B_{d}(S,f)})$.
Consider $g\in\mathcal{Z}^{+}$ with $g(x)=f(x)+r~{}\forall~{}x\in X$. Since
$(X,d)$ is an almost convex metric space,
$B_{d}(B_{d}(x,f(x)),r)=B_{d}(x,f(x)+r)$ for $x\in S$. Therefore,
$B_{d}(B_{d}(S,f),r)=B_{d}(S,g)$. Consequently, $p\notin B_{d}(S,g)$. Thus
$B_{d}(S,g)\neq X$. Which contradicts Theorem 6.4. ∎

If $\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$, then we also have
$\mathcal{S}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}$. However, in general
($\mathcal{S}^{+}=\mathsf{G}_{\mathcal{S},d}^{+})\nRightarrow(\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+})$.
This can be seen by the following example.

######  Example 6.10.

Let $X=\mathbb{R}^{2}$ and $d=d_{e}$. Let $\mathcal{S}$ be the bornology
defined as
$\mathcal{S}=\\{S\subseteq\mathbb{R}\times[-n,n]:n\in\mathbb{N}\\}$. It
follows from Example $20$ of [13] that
$\tau_{\mathcal{S},d}^{+}\neq\mathcal{S}^{+}$ on $CL(X)$. Since for any
$S\in\mathcal{S}$ and $r>0$, $\overline{B_{d}}(S,r)=\\{x\in X:d(x,S)\leq
r\\}\in\mathcal{S}$, by Theorem $3$ of [9],
$\mathsf{G}_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$ on $CL(X)$.

Now as applications of Theorem 6.4, we deduce the coincidence
$\tau_{\mathcal{S},d}^{+}=\mathcal{S}^{+}$ for some special bornologies. In
particular, we consider the relation of $\mathcal{S}^{+}$-convergence with
$\tau_{\mathcal{S},d}^{+}$-convergence for
$\mathcal{S}=\mathcal{F}(X),~{}\mathcal{K}(X)$,
$\mathcal{T}\mathcal{B}_{d}(X)$, $\mathcal{B}_{d}(X)$, and
$\mathcal{P}_{0}(X)$. Note that in each of these special cases, we get
different $\mathcal{S}^{+}$-convergences. However, when
$\mathcal{S}=\mathcal{F}(X)$, $\mathcal{K}(X)$, and
$\mathcal{T}\mathcal{B}_{d}(X)$, we have
$\tau_{\mathcal{S},d}^{+}=\tau_{W_{d}}^{+}$. So in these three cases, we
actually deduce the equivalence $\tau_{W_{d}}^{+}=\mathcal{S}^{+}$ using
Theorem 6.4. We would like to mention here that in Theorem 1 of [9], Beer and
Levi have proved a much more general result where they have compared the
$\tau_{W_{d}}^{+}$-convergence to $\mathcal{S}^{+}$-convergence for an
arbitrary bornology $\mathcal{S}$.

######  Corollary 6.11.

$($[3]$)$ Let $(X,d)$ be a metric space. Then the following statements hold.

  1. (i)

$\tau_{AW_{d}}^{+}$-convergence $=\mathcal{B}_{d}^{+}(X)$-convergence on
$CL(X)$;

  2. (ii)

$\tau_{H_{d}}^{+}$-convergence = $\mathcal{P}_{0}^{+}(X)$-convergence on
$CL(X)$.

###### Proof.

$(i)$. Let $B\in\mathcal{B}_{d}(X)$, and $f\in\mathcal{Z}^{+}$. Then by
Proposition 4.3, either $B_{d}(B,f)\in\mathcal{B}_{d}(X)$ or $B_{d}(B,f)=X$.
Therefore the result follows from Theorem 6.4.

$(ii)$. It is immediate from Theorem 6.4. ∎

In order to examine other cases, we first prove a following general result
which can be inferred from Theorem $8$ of [13]. For the readers convenience
and sake of completeness we record its proof below.

######  Lemma 6.12.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$. If
$\mathcal{S}\subseteq\mathcal{T}\mathcal{B}_{d}(X)$, then
$\tau_{\mathcal{S},d}^{+}=\tau_{W_{d}}^{+}$ on $CL(X)$.

###### Proof.

Suppose $(A_{\lambda})$ is a net in $CL(X)$ that $\tau_{W_{d}}^{+}$-converges
to $A\in CL(X)$. We show that the net $(A_{\lambda})$ is
$\tau_{\mathcal{S},d}^{+}$-convergent to $A$. Take $S\in\mathcal{S}$ and
$\epsilon>0$. Choose $F\in\mathcal{F}(X)$ such that $S\subseteq
B_{d}(F,\frac{\epsilon}{4})$. By assumption, there is a $\lambda_{0}$ such
that for all $\lambda\geq\lambda_{0}$ and $y\in F$, we have
$d(y,A)-d(y,A_{\lambda})<\frac{\epsilon}{4}$. Then for any $x\in S$ and
$\lambda\geq\lambda_{0}$ we have $d(x,A)<d(x,A_{\lambda})+\epsilon$. ∎

######  Proposition 6.13.

Let $(X,d)$ be a metric space and let $\mathcal{S}$ be a bornology on $X$ such
that $\mathcal{S}\subseteq\mathcal{T}\mathcal{B}_{d}(X)$. Then the following
statements are equivalent:

  1. (i)

each proper closed ball is in $\mathcal{S}$;

  2. (ii)

for $S\in\mathcal{S}$ and $f,g\in\mathcal{Z}^{+}$ with $\inf_{x\in
S}(g(x)-f(x))>0$ whenever $B_{d}(S,g)\neq X$, we have
$B_{d}(S,f)\in\mathcal{S}$.

###### Proof.

$(i)\Rightarrow(ii)$. Let $S\in\mathcal{S}$ and $f,g\in\mathcal{Z}^{+}$ with
$\inf_{x\in S}(g(x)-f(x))>0$ such that $B_{d}(S,g)\neq X$. By Lemma 6.12, we
have
$\tau_{\mathcal{S},d}^{+}=\tau_{W_{d}}^{+}\subseteq\mathsf{G}_{\mathcal{S},d}^{+}$.
So $\tau_{\mathcal{S},d}^{+}=\mathsf{G}_{\mathcal{S},d}^{+}$. Then by Theorem
5.5, we have $B_{d}(S,f)$ is strictly $(\mathcal{S}-d)$ included in
$B_{d}(S,g)$. Let $S_{1},\ldots,S_{n}\in\mathcal{S}$ and
$0<\alpha_{i}<\epsilon_{i}$ for $1\leq i\leq n$ such that

$B_{d}(S,f)\subseteq\cup_{i=1}^{n}B_{d}(S_{i},\alpha_{i})\subseteq\cup_{i=1}^{n}B_{d}(S_{i},\epsilon_{i})\subseteq
B_{d}(S,g).$

Choose $0<r_{i}<\epsilon_{i}-\alpha_{i}$ for $1\leq i\leq n$. Since each
$S_{i}$ is totally bounded, there exist $x_{i_{1}},..,x_{i_{k}}\in S_{i}$ such
that $S_{i}\subseteq\cup_{j=1}^{k}B_{d}(x_{i_{j}},r_{i})$. Consequently,

$B_{d}(S_{i},\alpha_{i})\subseteq\cup_{j=1}^{k}B_{d}(x_{i_{j}},\alpha_{i}+r_{i})\subseteq
B_{d}(S_{i},\epsilon_{i}).$

By the hypothesis,
$\cup_{j=1}^{k}B_{d}(x_{i_{j}},\alpha_{i}+r_{i})\in\mathcal{S}$. Thus,
$B_{d}(S_{i},\alpha_{i})\in\mathcal{S}$ for each $i=1,\ldots,n$. Hence
$B_{d}(S,f)\in\mathcal{S}$.

$(ii)\Rightarrow(i)$. Let $x\in X$ and $\epsilon>0$ be such that
$\overline{B_{d}}(x,\epsilon)\neq X$. Then there exists a $y\in X$ such that
$d(x,y)>\epsilon$. Put $d(x,y)-\epsilon=3r$. So $B_{d}(x,\epsilon+3r)\neq X$.
By the hypothesis, we have $B_{d}(x,\epsilon+2r)\in\mathcal{S}$. Hence
$\overline{B_{d}}(x,\epsilon)\in\mathcal{S}$. ∎

######  Corollary 6.14.

$($[3]$)$ Let $(X,d)$ be a metric space. Then $\tau_{W_{d}}^{+}$-convergence =
$\mathcal{K}(X)^{+}$-convergence if and only if $(X,d)$ have nice closed balls
(that is, each proper closed ball is compact).

###### Proof.

Since $\mathcal{K}(X)\subseteq\mathcal{T}\mathcal{B}_{d}(X)$, the result
follows from Lemma 6.12, Proposition 6.13 and Theorem 6.4. ∎

By Corollary $3.3$ of [22], we have $s(X)^{+}=\mathcal{F}(X)^{+}$, where
$s(X)$ = collection of all singletons in $X$.

######  Corollary 6.15.

$($Proposition $22$, [13]$)$ Let $(X,d)$ be a metric space. The following
statements are equivalent:

  1. (i)

$\mathcal{F}(X)^{+}$-convergence ensures $\tau_{W_{d}}^{+}$-convergence on
$CL(X)$;

  2. (ii)

either each closed ball in $(X,d)$ is finite or $(X,d)$ is bounded.

Our next corollary also follows from Lemma 6.12, Proposition 6.13 and Theorem
6.4. However, we give a direct proof without using Proposition 6.13. The proof
of $(ii)\Rightarrow(i)$ in the following corollary can be imitated to give an
alternative proof of $(1)\Rightarrow(2)$ in Theorem 1 of [9].

######  Corollary 6.16.

Let $(X,d)$ be a metric space. Then the following statements are equivalent:

  1. (i)

$\mathcal{T}\mathcal{B}_{d}(X)^{+}$-convergence ensures
$\tau_{W_{d}}^{+}$-convergence on $CL(X)$;

  2. (ii)

each proper closed ball in $(X,d)$ is totally bounded.

###### Proof.

$(i)\Rightarrow(ii)$. Let $x\in X$ and $\epsilon>0$ be such that
$\overline{B_{d}}(x,\epsilon)\neq X$. Then there exists a $y\in X$ such that
$d(x,y)>\epsilon$. Put $d(x,y)-\epsilon=3r$. So $B_{d}(x,\epsilon+3r)\neq X$.
Therefore by Theorem 6.4, we have,
$B_{d}(x,\epsilon+2r)\in\mathcal{T}\mathcal{B}_{d}(X)$. Hence
$\overline{B_{d}}(x,\epsilon)\in\mathcal{T}\mathcal{B}_{d}(X)$.

$(ii)\Rightarrow(i)$. Let $(A_{\lambda})$ be a net that
$\mathcal{T}\mathcal{B}_{d}(X)^{+}$-converges to $A$ in $CL(X)$. Suppose $x\in
X$ and $0<\alpha<\epsilon$ such that $A\cap B_{d}(x,\epsilon)=\emptyset$. Set
$\overline{B_{d}}(x,\alpha)=S$. Then by the hypothesis,
$S\in\mathcal{T}\mathcal{B}_{d}(X)$. Clearly, $A\cap
B_{d}(S,\frac{\epsilon-\alpha}{2})=\emptyset$. So by
$\mathcal{T}\mathcal{B}_{d}(X)^{+}$-convergence of net $(A_{\lambda})$ to $A$
and using Theorem 6.1, there exists $\lambda_{0}$ such that $A_{\lambda}\cap
S=\emptyset~{}\forall\lambda\geq\lambda_{0}$, that is, $A_{\lambda}\cap
B_{d}(x,\alpha)=\emptyset~{}\forall\lambda\geq\lambda_{0}$. Thus, the net
$(A_{\lambda})$ is $\tau_{W_{d}}^{+}$-convergent to $A$ in $CL(X)$. ∎

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