footnote

#  Variations of star selection principles on Hyperspaces111The author was
supported for this research by Postdoctoral Fellowship Program at UNAM.

JAVIER CASAS-DE LA ROSA

###### Abstract

In this paper we define some combinatorial principles to characterize spaces
$X$ whose hyperspace satisfies some variation of some classical star selection
principle. Specifically, the variations characterized are the selective and
absolute versions of the star selection principles for the Menger and
Rothberger cases; also, the hyperspaces considered in these characterizations
are $CL(X)$, $\mathbb{K}(X)$, $\mathbb{F}(X)$ and $\mathbb{CS}(X)$ in both
cases, endowed with either the Fell topology or the Vietoris topology.

_Key words._ Hyperspaces, Fell topology, Vietoris topology, star selection
principles, (absolutely) strongly star-Menger, (selectively) strongly star-
Menger, (absolutely) strongly star-Rothberger, (selectively) strongly star-
Rothberger.

_Mathematics Subject Classification_ : Primary 54B20, 54D20; Secondary 54A05,
54A25.

##  1 Introduction and preliminaries

Many branches from Selection Principles Theory have arisen after a systematic
research made in [17] by Scheepers. Nowadays, this theory has connections as
well as applications to several areas of mathematics as General Topology,
Function spaces, Hyperspaces, etc. As an example of it, in [9], [10],[15] and
[16], the authors studied the fundamental problem on Hyperspaces which
consists in establishing dualities between some topological properties or, as
it is in this case, between some classical selection principles under
different hyperspaces topologies. In other words, given two selection
principles $\mathcal{P}$ and $\mathcal{Q}$, one must determine if it holds
that a topological space $X$ satisfies the principle $\mathcal{P}$ if and only
if its hyperspace satisfies the principle $\mathcal{Q}$. Hence, this duality
problem can be viewed as a method to characterize some classical selection
principles of $X$ with selection principles on different hyperspaces of $X$,
using several well-known topologies.

On the other hand, several versions of the original selection principles have
been defined from its beginning. One of the most important versions that its
investigation has rapidly increased is the star versions of the classical
selection principles. These star versions were defined by Kočinac in [13] and
they gave rise to the star selection principles theory (see [14] and [12]).

The study of the duality problem involving some star selection principles on
hyperspaces initiated very recently in [4] and continued in [3], [6], [7] and
[8] have dealt with classical star selection principles only. In particular,
in these works several combinatorial principles have been defined to establish
characterizations for the (strongly) star-Menger property and the (strongly)
star-Rothberger property on several hyperspaces under different topologies. In
this paper we continue this line of investigation for some variations of some
classical star selection principles. In Section 2, the selective and absolute
versions of Menger-type and Rothberger-type star selection principle on
several hyperspaces with the Vietoris topology are characterized. In Section
3, analogous characterizations are given for several hyperspaces with the Fell
topology.

###  1.1 Hyperspaces

All spaces are assumed to be Hausdorff noncompact and, even, nonparacompact.
Given a topological space $X$ , the hyperspace of $X$, denoted by $CL(X)$, is
the set of all nonempty closed subsets of $X$. By $\mathbb{K}(X)$
($\mathbb{F}(X)$), we denote the family of all nonempty compact (all nonempty
finite) subsets of $X$. Also, by $\mathbb{CS}(X)$ we denote the family of all
convergent sequences of $X$.

We denote by $\omega$ the first infinite cardinal and for a set $A$,
$[A]^{<\omega}$ denotes the set of all finite subsets of $A$. For a subset
$U\subseteq X$ and a family $\mathcal{U}$ of subsets of $X$, we write:

$U^{-}$ | = |  $\\{A\in CL(X):A\cap U\neq\emptyset\\}$;
---|---|---
$U^{+}$ | = |  $\\{A\in CL(X):A\subseteq U\\}$;
$U^{c}$ | = |  $X\backslash U$;
$\mathcal{U}^{c}$ | = |  $\\{U^{c}:U\in\mathcal{U}\\}$;
$\mathcal{U}^{-}$ | = |  $\\{U^{-}:U\in\mathcal{U}\\}$;
$\mathcal{U}^{+}$ | = |  $\\{U^{+}:U\in\mathcal{U}\\}$.

In the literature there are many topologies that can be defined on $CL(X)$ or
on a subcollection of it. In this paper we will consider two well-known
topologies, the Vietoris topology, denoted by $\mathbf{V}$, and the Fell
topology, denoted by $\mathbf{F}$.

A basic open subset of the Fell topology is of the form:

$(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+},$

where $V_{1},\ldots,V_{n}$ are open subsets of $X$ and $K$ is a compact subset
of $X$.

A basic open subset of the Vietoris topology is a set of the form:

$\langle U_{1},\ldots,U_{n}\rangle=\\{A\in
CL(X):A\subseteq\bigcup_{i=1}^{n}U_{i},A\cap U_{i}\neq\emptyset\text{ for each
}i\leq n\\}$

where $U_{1},\ldots,U_{n}$ are open subsets of $X$, $n\in\omega$.

###  1.2 Variations of the classical star selection principles

For a set $A\subseteq X$ and a collection $\mathcal{U}$ of subsets of $X$, the
star of $A$ with respect to $\mathcal{U}$, denoted by $St(A,\mathcal{U})$, is
the set $\bigcup\\{U\in\mathcal{U}:U\cap A\neq\emptyset\\}$; for $A=\\{x\\}$
with $x\in X$, we write $St(x,\mathcal{U})$ instead of
$St(\\{x\\},\mathcal{U})$.

In recent years, different versions of the classical star selection principles
have been defined and studied in several articles (see for instance [1], [5]
and [2]). Some of these new star selection principles involve dense subsets of
the space and they are called as the absolute and selective versions of
classical star selection principles (see [14] for more information about the
absolute versions and see [2] for an overview about the selective versions).
The selective versions are stronger than the absolute versions and the
absolute versions are stronger than the classical star selection
principles222In [5], the authors introduced the absolute versions of classical
star selection principles in a general form with a different notation than the
one used in [2], where it was defined the absolute and selective versions of
star selection principles, also given in a general form.. The following
properties are the absolute versions of the strongly star selection principles
for the Menger and Rothberger cases.

######  Definition 1.1 ([5]).

We say that a space $X$ is:

  1. 1.

absolutely strongly star-Menger ($aSSM$) if for each sequence
$\\{\mathcal{U}_{n}:n\in\omega\\}$ of open covers of $X$ and each dense subset
$D$ of $X$, there is a sequence $\\{F_{n}:n\in\omega\\}$ of finite sets such
that $F_{n}\subseteq D$, $n\in\omega$, and
$\\{St(F_{n},\mathcal{U}_{n}):n\in\omega\\}$ is an open cover of $X$.

  2. 2.

absolutely strongly star-Rothberger ($aSSR$) if for each sequence
$\\{\mathcal{U}_{n}:n\in\omega\\}$ of open covers of $X$ and each dense subset
$D$ of $X$, there is a sequence $\\{x_{n}:n\in\omega\\}$ of points of $X$ such
that $x_{n}\in D$, $n\in\omega$, and
$\\{St(x_{n},\mathcal{U}_{n}):n\in\omega\\}$ is an open cover of $X$.

Now, we recall the selective versions of the strongly star selection
principles for the Menger and Rothberger cases333The Hurewicz case and some
other interesting properties are also given in [2].

######  Definition 1.2 ([2]).

We say that a space $X$ is:

  1. 1.

selectively strongly star-Menger ($selSSM$) if for each sequence
$\\{\mathcal{U}_{n}:n\in\omega\\}$ of open covers of $X$ and each sequence
$\\{D_{n}:n\in\omega\\}$ of dense sets of $X$, there exists a sequence
$\\{F_{n}:n\in\omega\\}$ of finite sets such that $F_{n}\subseteq D_{n}$,
$n\in\omega$, and $\\{St(F_{n},\mathcal{U}_{n}):n\in\omega\\}$ is an open
cover of $X$.

  2. 2.

selectively strongly star-Rothberger ($selSSR$) if for each sequence
$\\{\mathcal{U}_{n}:n\in\omega\\}$ of open covers of $X$ and each sequence
$\\{D_{n}:n\in\omega\\}$ of dense sets of $X$, there exists a sequence
$\\{x_{n}:n\in\omega\\}$ of points of $X$ such that $x_{n}\in D_{n}$,
$n\in\omega$, and $\\{St(x_{n},\mathcal{U}_{n}):n\in\omega\\}$ is an open
cover of $X$.

##  2 Absolute and selective versions on hyperspaces with the Vietoris
topology

In this section we present some technical principles that are useful to
characterize some variations of classical star selection principles ($selSSM$,
$selSSR$, $aSSM$ and $aSSR$) on several hiperspaces with the Vietoris
topology. To that end, we recall some notation and useful definitions to
establish these characterizations.

Using the notation of [16], $\zeta$ denotes the family:

$\zeta=\\{(V_{1},\ldots,V_{n}):V_{1},\ldots,V_{n}\text{ are }\text{open
subsets of }X,\ \ n\in\mathbb{N}\\}.$

In [16], Li defined the notion of $\pi_{V}$-networks in a space $X$ as
follows:

A family $\zeta$ is called a _$\pi_{V}$ -network of $X$_ if for each open
subset $U$ of $X$, with $U\neq X$, there exist $(V_{1},\ldots,V_{n})\in\zeta$
and a finite set $F$ with $F\cap V_{i}\neq\emptyset$ $(1\leq i\leq n)$ such
that $\bigcap_{i=1}^{n}V_{i}^{c}\subseteq U$ and $F\cap U=\emptyset$. The
collection of $\pi_{V}$-networks of a space $X$ is denoted by $\Pi_{V}$.

Henceforth, $\Delta$ will denote a subset of $CL(X)$ which is closed under
finite unions and contains all singletons. Using a family $\Delta$, in [8], it
was defined a modification of $\pi_{V}$-network which is called as
$\pi_{V}(\Delta)$-network of $X$.

######  Definition 2.1 ([8]).

A family $\zeta$ is called a $\pi_{V}(\Delta)$-network of $X$, if for each
$U\in\Delta^{c}$, there exist a $(V_{1},\ldots,V_{n})\in\zeta$ and
$F\in[X]^{<\omega}$ with $F\cap V_{i}\neq\emptyset$ $(1\leq i\leq n)$ such
that $\bigcap_{i=1}^{n}V_{i}^{c}\subset U$ and $F\cap U=\emptyset$. The
collection of all $\pi_{V}(\Delta)$-networks of $X$ is denoted by
$\Pi_{V}(\Delta)$.

As pointed out in [8], if we consider $\Delta$ to be $CL(X)$, then the
collections $\Pi_{V}(\Delta)$ and $\Pi_{V}$ coincide. But this fact not need
be true in general. In [8], the authors gave an example of a
$\pi_{V}(\Delta)$-network that is not a $\pi_{V}$-network (on a metrizable
space $X$ and a specific family $\Delta$). Here we present another example in
a non-metrizable space444This space was also used in [4] to show that
$\pi_{F}(\Delta)$-network and $\pi_{F}$-network are different notions; see
Section 3 for definitions..

######  Example 2.2.

There exists a family $\zeta$ on a certain space $X$ and there exists a family
$\Delta\subseteq CL(X)$ such that $\zeta$ is a $\pi_{V}(\Delta)$-network of
$X$ but is not a $\pi_{V}$-network of $X$.

###### Proof.

Let $\mathcal{A}$ be an uncountable almost disjoint family on $\omega$ with
$\omega=\bigcup\mathcal{A}$. We consider the Mrówka-Isbell space (see [11]),
$X=\Psi(\mathcal{A})$ and $\Delta=\mathbb{K}(X)$. Let

$\zeta=\\{(V_{1},\ldots,V_{n}):V_{i}=\\{A_{i}\\}\cup(A_{i}\setminus D_{i})$
$(1\leq i\leq n)$ where $A_{1},\ldots,A_{n}\in\mathcal{A}$,
$D_{1}\ldots,D_{n}\in[\omega]^{<\omega}$, $n\in\mathbb{N}\\}$

Note that $\zeta$ is properly defined, that is, for each
$i\in\\{1,\dots,n\\}$, $\\{A_{i}\\}\cup(A_{i}\setminus D_{i})$ is an open set
of $X$.

_Claim 1:_ $\zeta$ is a $\pi_{V}(\mathbb{K}(X))$-network of $X$.
Let $U\in[\mathbb{K}(X)]^{c}$. Suppose that $U=X\setminus K_{0}$ for some
$K_{0}\in\mathbb{K}(X)$. We consider the following two cases:
_Case I:_ $K_{0}$ is infinite.
Note that in this case, $K_{0}\cap\mathcal{A}$ is a nonempty finite set.
Moreover, the set $(K_{0}\cap\omega)\setminus(\bigcup\\{B:B\in
K_{0}\cap\mathcal{A}\\})$ is finite. For each
$m\in(K_{0}\cap\omega)\setminus(\bigcup\\{B:B\in K_{0}\cap\mathcal{A}\\})$,
let $B_{m}\in\mathcal{A}$ be such that $m\in B_{m}$. The collection
$\mathcal{B}=(K_{0}\cap\mathcal{A})\cup\\{B_{m}:m\in(K_{0}\cap\omega)\setminus(\bigcup\\{B:B\in
K_{0}\cap\mathcal{A}\\})\\}$ is finite. We enumerate this collection as
$\mathcal{B}=\\{A_{1},\ldots,A_{n}\\}$. We define, for each
$i\in\\{1,\ldots,n\\}$, $V_{i}=\\{A_{i}\\}\cup A_{i}$. Then, each $V_{i}$ is
an open set of $X$ and therefore, $(V_{1},\dots,V_{n})\in\zeta$. Note that
$K_{0}\subseteq\bigcup_{i=1}^{n}V_{i}$. Thus,
$\bigcap_{i=1}^{n}V_{i}^{c}\subseteq U$. Finally, we let
$F=(K_{0}\cap\mathcal{A})\cup[(K_{0}\cap\omega)\setminus(\bigcup\\{B:B\in
K_{0}\cap\mathcal{A}\\})]$. Then, $F$ is a finite set of $X$. Moreover, note
that $F\cap V_{i}\neq\emptyset$ $(1\leq i\leq n)$ and $F\cap U=\emptyset$.
Therefore, $\zeta$ is a $\pi_{V}(\mathbb{K}(X))$-network of $X$.
_Case II:_ $K_{0}$ is finite.
For each $m\in K_{0}\cap\omega$ (if $K_{0}\cap\omega\neq\emptyset$), let
$B_{m}\in\mathcal{A}$ such that $m\in B_{m}$. Then, the collection
$\mathcal{B}=(K_{0}\cap\mathcal{A})\cup\\{B_{m}:m\in K_{0}\cap\omega\\}$ is
finite. We enumerate this collection as
$\mathcal{B}=\\{A_{1},\ldots,A_{n}\\}$. We define, for each
$i\in\\{1,\ldots,n\\}$, $V_{i}=\\{A_{i}\\}\cup A_{i}$. Then, each $V_{i}$ is
an open set of $X$ and therefore, $(V_{1},\dots,V_{n})\in\zeta$. Note that
$K_{0}\subseteq\bigcup_{i=1}^{n}V_{i}$. Hence,
$\bigcap_{i=1}^{n}V_{i}^{c}\subseteq U$. Let $F=K_{0}$. Then, $F$ is a finite
set of $X$ such that $F\cap V_{i}\neq\emptyset$ $(1\leq i\leq n)$ and $F\cap
U=\emptyset$. Therefore, $\zeta$ is a $\pi_{V}(\mathbb{K}(X))$-network of $X$.

_Claim 2:_ $\zeta$ is not a $\pi_{V}$-network of $X$.
Let $U=\omega$. Then, $U$ is an open set of $X$ with $U\neq X$. Let
$(V_{1},\dots,V_{n})$ be any element of $\zeta$, where
$V_{i}=\\{A_{i}\\}\cup(A_{i}\setminus D_{i})$ $(1\leq i\leq n)$ for some
$A_{1},\ldots,A_{n}\in\mathcal{A}$, $D_{1}\ldots,D_{n}\in[\omega]^{<\omega}$,
$n\in\mathbb{N}$. Then, $\bigcap_{i=1}^{n}[\\{A_{i}\\}\cup(A_{i}\setminus
D_{i})]^{c}\nsubseteq U$. Since this fact holds for any element of $\zeta$, we
conclude that $\zeta$ is not a $\pi_{V}$-network of $X$.  ∎

From now on, if $J_{n}$ is an element in $\Pi_{V}(\Delta)$, we put:

$J_{n}=\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n}):s\in S_{n}\\}.$

Another notion, defined also in [8], that involves a family $\Delta$ is the
following:

######  Definition 2.3 ([8]).

Let $(X,\tau)$ be a topological space. A family
$\mathcal{U}\subseteq\Delta^{c}$ is called a _$c_{V}(\Delta)$ -cover of $X$_,
if for any open subsets $V_{1},\ldots,V_{m}$ of $X$, there exists
$U\in\mathcal{U}$ and $F\in[X]^{<\omega}$ such that for each
$i\in\\{1,\ldots,m\\}$, $F\cap V_{i}\neq\emptyset$,
$\bigcap_{i=1}^{m}V_{i}^{c}\subseteq U$ and $F\cap U=\emptyset$. The family of
all $c_{V}(\Delta)$-covers of a space $X$ is denoted by
$\mathbb{C}_{V}(\Delta)$.

The following two lemmas will be useful in the proofs of next results in this
section and the proofs of these lemmas can be easily obtained; we refer the
reader to [8] for details. The first lemma says how $c_{V}(\Delta)$-covers on
a space $X$ can be viewed as dense subspaces of certain hyperspaces of $X$.

######  Lemma 2.4 ([8]).

Let $X$ be a topological space and $\mathcal{U}\subseteq\Delta^{c}$. Then
$\mathcal{U}$ is a $c_{V}(\Delta)$-cover of $X$ if and only if
$\mathcal{U}^{c}$ is a dense subset of $(\Delta,\mathbf{V})$.

The next lemma says how $\pi_{V}(\Delta)$-networks of a space $X$ can be
interpreted as open covers of certain hyperspaces of $X$.

######  Lemma 2.5 ([8]).

Let $X$ be a topological space and
$\zeta=\\{(V_{1},\ldots,V_{n}):V_{1},\ldots,V_{n}$ $\text{ are open subsets of
}X,n\in\omega\\}.$ Then $\zeta$ is a $\pi_{V}(\Delta)$-network of $X$ if and
only if the collection $\mathcal{U}=\\{\langle
V_{1},\ldots,V_{n}\rangle:(V_{1},\ldots,V_{n})\in\zeta\\}$ is an open cover of
$(\Delta,\mathbf{V})$.

The following selection principle will help us to characterize the selectively
strongly star-Menger property on hyperspaces with the Vietoris topology.

######  Definition 2.6.

Let $X$ be a topological space. We define:

$\mathbf{SV}_{M}(\Pi_{V}(\Delta),\Pi_{V}(\Delta))$: For each sequence
$\\{J_{n}:n\in\mathbb{N}\\}\subseteq\Pi_{V}(\Delta)$ and each sequence
$\\{\mathcal{C}_{n}:n\in\mathbb{N}\\}\subseteq\mathbb{C}_{V}(\Delta)$, there
are finite subsets $\mathcal{V}_{n}\subseteq\mathcal{C}_{n}$,
$n\in\mathbb{N}$, such that
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\text{ there exists }U_{n}\in\mathcal{V}_{n}\text{ such that }$
$\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{c}\subseteq U_{n},V_{i,s}^{n}\nsubseteq
U_{n}\ (1\leq i\leq m_{s})\\}$ is an element of $\Pi_{V}(\Delta)$.

######  Theorem 2.7.

Given a topological space $X$, the following conditions are equivalent:

  1. (1)

$(\Delta,\mathbf{V})$ is selSSM;

  2. (2)

$X$ satisfies $\mathbf{SV}_{M}(\Pi_{V}(\Delta),\Pi_{V}(\Delta))$.

###### Proof.

$(1)\Rightarrow(2)$: Let $\\{J_{n}:n\in\mathbb{N}\\}$ be a sequence of
$\pi_{V}(\Delta)$-networks of $X$ and let
$\\{\mathcal{C}_{n}:n\in\mathbb{N}\\}$ be a sequence of $c_{V}(\Delta)$-covers
of $X$. By Lemma 2.4, the collections $\mathcal{D}_{n}=\mathcal{C}_{n}^{c}$
are dense subsets of $(\Delta,\mathbf{V})$, for each $n\in\mathbb{N}$.
Furthermore, if we put, for each $n\in\omega$,
$J_{n}=\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n}):s\in S_{n}\\}$, then by Lemma
2.5, the collections $\mathcal{U}_{n}=\\{\langle
V_{1,s}^{n},\ldots,V_{m_{s},s}^{n}\rangle:s\in S_{n}\\}$ are open covers of
$(\Delta,\mathbf{V})$, for each $n\in\mathbb{N}$.

Now, applying (1) to the sequence $\\{\mathcal{U}_{n}:n\in\mathbb{N}\\}$ and
the sequence $\\{\mathcal{D}_{n}:n\in\mathbb{N}\\}$, there is a sequence
$\\{\mathcal{A}_{n}:n\in\mathbb{N}\\}$ of finite sets such that, for each
$n\in\mathbb{N}$, $\mathcal{A}_{n}\subseteq\mathcal{D}_{n}$ and the collection
$\\{St(\mathcal{A}_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{V})$. We put, for each $n\in\mathbb{N}$,
$\mathcal{V}_{n}=\\{A^{c}:A\in\mathcal{A}_{n}\\}$. Then, for each
$n\in\mathbb{N}$, $\mathcal{V}_{n}$ is a finite subset of $\mathcal{C}_{n}$.

Let us show that the collection
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\exists\ U_{n}\in\mathcal{V}_{n}\text{ such that
}\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{c}\subseteq U_{n},\ V_{i,s}^{n}\nsubseteq
U_{n}\ (1\leq i\leq m_{s})\\}$ is a $\pi_{V}(\Delta)$-network of $X$.

Let $U\in\Delta^{c}$. Then $U^{c}\in\Delta$ and therefore, there exists
$n_{0}\in\omega$ such that $U^{c}\in
St(\mathcal{A}_{n_{0}},\mathcal{U}_{n_{0}})$. Then, there are $\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle\in\mathcal{U}_{n_{0}}$
and $A_{n_{0}}\in\mathcal{A}_{n_{0}}$ so that
$\\{U^{c},A_{n_{0}}\\}\subseteq\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle$. Let
$U_{n_{0}}=A^{c}_{n_{0}}$. Then $U_{n_{0}}\in\mathcal{V}_{n_{0}}$. Since
$A_{n_{0}}$ belongs to $\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle$, it follows
that $\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{c}\subseteq U_{n_{0}},\
\ V_{i,s_{0}}^{n_{0}}\nsubseteq U_{n_{0}}(1\leq i\leq m_{s_{0}})$; hence
$(V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})\in\mathcal{J}$. On
the other hand, using the fact that $U^{c}$ also belongs to $\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle$, we can take,
for each $i\in\\{1,\ldots,m_{s_{0}}\\}$, $x_{i}\in U^{c}\cap
V_{i,s_{0}}^{n_{0}}$. We put $F=\\{x_{i}:i\in\\{1,\ldots,m_{s_{0}}\\}\\}$.
Hence, $F\in[X]^{<\omega}$ with $F\cap V_{i,s_{0}}^{n_{0}}\neq\emptyset$ and
$F\cap U=\emptyset$. Moreover, since
$U^{c}\subseteq\bigcup_{i=1}^{m_{s_{0}}}V_{i,s_{0}}^{n_{0}}$, then we obtain
that $\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{c}\subseteq U$. We
conclude that $\mathcal{J}\in\Pi_{V}(\Delta)$.

$(2)\Rightarrow(1)$: Let $\\{\mathcal{U}_{n}:n\in\mathbb{N}\\}$ be a sequence
of open covers of $(\Delta,\mathbf{V})$ and let
$\\{\mathcal{D}_{n}:n\in\mathbb{N}\\}$ be a sequence of dense subsets of
$(\Delta,\mathbf{V})$. We can assume that each open cover $\mathcal{U}_{n}$
consists of basic open subsets in $CL(X)$. Thus, put for each
$n\in\mathbb{N}$, $\mathcal{U}_{n}=\\{\langle
V_{1,s}^{n},\ldots,V_{m_{s},s}^{n}\rangle:s\in S_{n}\\}$, where $V_{i,s}^{n}$
is an open subset of $X$, for every $n\in\mathbb{N}$, $s\in S_{n}$ and
$i\in\\{1,\ldots,m_{s}\\}$. Let
$J_{n}=\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n}):s\in S_{n}\\}$ for every
$n\in\mathbb{N}$. By Lemma 2.5, note that each $J_{n}$ is a
$\pi_{V}(\Delta)$-network of $X$. On the other hand, for each
$n\in\mathbb{N}$, let $\mathcal{C}_{n}=\mathcal{D}_{n}^{c}$. Thus, by Lemma
2.4, each $\mathcal{C}_{n}$ is a $c_{V}(\Delta)$-cover of $X$.

We apply $(2)$ to the sequence of $\pi_{V}(\Delta)$-networks
$\\{J_{n}:n\in\mathbb{N}\\}$ and the sequence of $c_{V}(\Delta)$-covers
$\\{\mathcal{C}_{n}:n\in\mathbb{N}\\}$ to obtain a sequence
$\\{\mathcal{V}_{n}:n\in\mathbb{N}\\}$ such that, for each $n\in\mathbb{N}$,
$\mathcal{V}_{n}\in[\mathcal{C}_{n}]^{<\omega}$, and the collection
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\text{ there exists }U_{n}\in\mathcal{V}_{n}\text{ such that
}\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{c}\subseteq U_{n},\ V_{i,s}^{n}\nsubseteq
U_{n}\ \ (1\leq i\leq m_{s})\\}$ is a $\pi_{V}(\Delta)$-network of $X$. For
each $n\in\omega$, we define $\mathcal{A}_{n}=\mathcal{V}^{c}_{n}$. It follows
that $\mathcal{A}_{n}\in[\mathcal{D}_{n}]^{<\omega}$, for each
$n\in\mathbb{N}$.

Let us show that the collection
$\\{St(\mathcal{A}_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{V})$. Let $A\in\Delta$. Since $\mathcal{J}$ is a
$\pi_{V}(\Delta)$-network of $X$ and $A^{c}\in\Delta^{c}$, there exist
$(V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})\in\mathcal{J}$ (for
some $n_{0}\in\mathbb{N}$ and some $s_{0}\in S_{n_{0}}$) and a finite set
$F\subseteq X$ such that for every $i\in\\{1,\ldots,m_{s_{0}}\\}$, $F\cap
V_{i,s_{0}}^{n_{0}}\neq\emptyset$,
$\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{c}\subseteq A^{c}$ and $F\cap
A^{c}=\emptyset$. Since
$(V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})\in\mathcal{J}$,
there is $U_{n_{0}}\in\mathcal{V}_{n_{0}}$ such that
$\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{c}\subseteq U_{n_{0}}$ and
for each $i\in\\{1,\ldots,m_{s_{0}}\\}$, $V_{i,s_{0}}^{n_{0}}\nsubseteq
U_{n_{0}}$. It means that $\\{A,A_{n_{0}}\\}\subseteq\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle\in\mathcal{U}_{n_{0}}$,
where $A_{n_{0}}=U^{c}_{n_{0}}$. Since $A_{n_{0}}$ is an element of
$\mathcal{A}_{n_{0}}$, we obtain that $A\in
St(\mathcal{A}_{n_{0}},\mathcal{U}_{n_{0}})$. This shows that the collection
$\\{St(\mathcal{A}_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{V})$.  ∎

We obtain the following particular cases by taking different choices of our
family $\Delta$.

######  Corollary 2.8.

Let $X$ be a topological space. Then:

  1. 1.

$(CL(X),\mathbf{V})$ is selSSM if and only if $X$ satisfies
$\mathbf{SV}_{M}(\Pi_{V},\Pi_{V})$;

  2. 2.

$(\mathbb{K}(X),\mathbf{V})$ is selSSM if and only if $X$ satisfies
$\mathbf{SV}_{M}(\Pi_{V}(\mathbb{K}(X)),\Pi_{V}(\mathbb{K}(X)))$;

  3. 3.

$(\mathbb{CS}(X),\mathbf{V})$ is selSSM if and only if $X$ satisfies
$\mathbf{SV}_{M}(\Pi_{V}(\mathbb{CS}(X)),\Pi_{V}(\mathbb{CS}(X)))$;

  4. 4.

$(\mathbb{F}(X),\mathbf{V})$ is selSSM if and only if $X$ satisfies
$\mathbf{SV}_{M}(\Pi_{V}(\mathbb{F}(X)),\Pi_{V}(\mathbb{F}(X)))$.

Let us now define another selection principle to characterize the selectively
strongly star-Rothberger property on hyperspaces with the Vietoris topology.

######  Definition 2.9.

Let $X$ be a topological space. We define:

$\mathbf{SV}_{R}(\Pi_{V}(\Delta),\Pi_{V}(\Delta))$: For each sequence
$\\{J_{n}:n\in\mathbb{N}\\}\subseteq\Pi_{V}(\Delta)$ and each sequence
$\\{\mathcal{C}_{n}:n\in\mathbb{N}\\}\subseteq\mathbb{C}_{V}(\Delta)$, there
is a sequence $\\{C_{n}:n\in\mathbb{N}\\}$ with $C_{n}\in\mathcal{C}_{n}$,
$n\in\mathbb{N}$, such that
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{c}\subseteq C_{n},$
$V_{i,s}^{n}\nsubseteq C_{n}\ (1\leq i\leq m_{s})\\}$ is an element of
$\Pi_{V}(\Delta)$.

######  Theorem 2.10.

Given a topological space $X$, the following conditions are equivalent:

  1. (1)

$(\Delta,\mathbf{V})$ is selSSR;

  2. (2)

$X$ satisfies $\mathbf{SV}_{R}(\Pi_{V}(\Delta),\Pi_{V}(\Delta))$.

###### Proof.

$(1)\Rightarrow(2)$: By mimicking the first part of $(1)\Rightarrow(2)$ in the
proof of Theorem 2.7, we can obtain a sequence $\\{D_{n}:n\in\mathbb{N}\\}$ of
points in $(\Delta,\mathbf{V})$ such that, for each $n\in\mathbb{N}$,
$D_{n}\in\mathcal{D}_{n}$ and the collection
$\\{St(D_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{V})$. We put, for each $n\in\mathbb{N}$, $C_{n}=D_{n}^{c}$.
Then, $C_{n}\in\mathcal{C}_{n}$, for each $n\in\mathbb{N}$.

Let us show that, taking the sequence $\\{C_{n}:n\in\mathbb{N}\\}$, the
collection
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{c}\subseteq C_{n},\
V_{i,s}^{n}\nsubseteq C_{n}\ (1\leq i\leq m_{s})\\}$ is a
$\pi_{V}(\Delta)$-network of $X$.

Let $U\in\Delta^{c}$. Then there exists $n_{0}\in\omega$ such that $U^{c}\in
St(D_{n_{0}},\mathcal{U}_{n_{0}})$. Thus, $U^{c},D_{n_{0}}\in\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle$ for some
$\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle\in\mathcal{U}_{n_{0}}$.
Since $C_{n_{0}}=D_{n_{0}}^{c}$ and $D_{n_{0}}\in\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle$, it follows
that $\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{c}\subseteq C_{n_{0}}$
and $\ V_{i,s_{0}}^{n_{0}}\nsubseteq C_{n_{0}}(1\leq i\leq m_{s_{0}})$; hence
$(V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})\in\mathcal{J}$. In
addition, using that $U^{c}$ is an element of $\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle$, we obtain that
$\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{c}\subseteq U$ and also, we
can define a finite subset $F$ of $X$ such that $F\cap
V_{i,s_{0}}^{n_{0}}\neq\emptyset$ and $F\cap U=\emptyset$. We conclude that
$\mathcal{J}\in\Pi_{V}(\Delta)$.

$(2)\Rightarrow(1)$: In the same way as in the first part of
$(2)\Rightarrow(1)$ in the proof of Theorem 2.7, we obtain a sequence
$\\{C_{n}:n\in\mathbb{N}\\}$ such that, for each $n\in\mathbb{N}$,
$C_{n}\in\mathcal{C}_{n}$, and the collection
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{c}\subseteq C_{n},\
V_{i,s}^{n}\nsubseteq C_{n}\ \ (1\leq i\leq m_{s})\\}$ is a
$\pi_{V}(\Delta)$-network of $X$. We put $D_{n}=C^{c}_{n}$ for each
$n\in\omega$. Then, we have that $\\{D_{n}:n\in\mathbb{N}\\}$ is a sequence of
points in $(\Delta,\mathbf{V})$ with $D_{n}\in\mathcal{D}_{n}$, for each
$n\in\mathbb{N}$.

Let us show that the collection
$\\{St(D_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{V})$. Let $A\in\Delta$. There exists
$(V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})\in\mathcal{J}$ (for
some $n_{0}\in\mathbb{N}$ and some $s_{0}\in S_{n_{0}}$) and a finite set
$F\subseteq X$ such that for every $i\in\\{1,\ldots,m_{s_{0}}\\}$, $F\cap
V_{i,s_{0}}^{n_{0}}\neq\emptyset$,
$\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{c}\subseteq A^{c}$ and $F\cap
A^{c}=\emptyset$. Using the fact that
$(V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})$ is an element of
$\mathcal{J}$, we get that
$\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{c}\subseteq C_{n_{0}}$ and
for each $i\in\\{1,\ldots,m_{s_{0}}\\}$, $V_{i,s_{0}}^{n_{0}}\nsubseteq
C_{n_{0}}$. Thus, $A,D_{n_{0}}$ are in $\langle
V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}}\rangle$ which is an
element of $\mathcal{U}_{n_{0}}$. In other words, $A\in
St(D_{n_{0}},\mathcal{U}_{n_{0}})$. This shows that the collection
$\\{St(D_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{V})$.  ∎

We obtain the following particular cases by taking different choices of our
family $\Delta$.

######  Corollary 2.11.

Let $X$ be a topological space. Then:

  1. 1.

$(CL(X),\mathbf{V})$ is selSSR if and only if $X$ satisfies
$\mathbf{SV}_{R}(\Pi_{V},\Pi_{V})$;

  2. 2.

$(\mathbb{K}(X),\mathbf{V})$ is selSSR if and only if $X$ satisfies
$\mathbf{SV}_{R}(\Pi_{V}(\mathbb{K}(X)),\Pi_{V}(\mathbb{K}(X)))$;

  3. 3.

$(\mathbb{CS}(X),\mathbf{V})$ is selSSR if and only if $X$ satisfies
$\mathbf{SV}_{R}(\Pi_{V}(\mathbb{CS}(X)),\Pi_{V}(\mathbb{CS}(X)))$;

  4. 4.

$(\mathbb{F}(X),\mathbf{V})$ is selSSR if and only if $X$ satisfies
$\mathbf{SV}_{R}(\Pi_{V}(\mathbb{F}(X)),\Pi_{V}(\mathbb{F}(X)))$.

Now, let us deal with the absolute versions of star selection principles for
Menger and Rothberger cases. We start giving the following principle that will
allow us to characterize the absolutely strongly star-Menger property on
different hyperspaces with the Vietoris topology.

######  Definition 2.12.

Let $X$ be a topological space. We define:

$\mathbf{AV}_{M}(\Pi_{V}(\Delta),\Pi_{V}(\Delta))$: For each sequence
$\\{J_{n}:n\in\mathbb{N}\\}\subseteq\Pi_{V}(\Delta)$ and each
$\mathcal{C}\in\mathbb{C}_{V}(\Delta)$, there is a sequence
$\\{\mathcal{V}_{n}:n\in\mathbb{N}\\}\subseteq[\mathcal{C}]^{<\omega}$ such
that
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\text{ there exists }V\in\mathcal{V}_{n}\text{ such that
}\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{c}\subseteq V,\ V_{i,s}^{n}\nsubseteq V\
(1\leq i\leq m_{s})\\}$ is an element of $\Pi_{V}(\Delta)$.

By doing some light modifications to the proof of Theorem 2.7, one can easily
prove the following result.

######  Theorem 2.13.

Given a topological space $X$, the following conditions are equivalent:

  1. (1)

$(\Delta,\mathbf{V})$ is aSSM;

  2. (2)

$X$ satisfies $\mathbf{AV}_{M}(\Pi_{V}(\Delta),\Pi_{V}(\Delta))$.

Again, some immediate consequences of the previous theorem are obtained by
taking different choices of our family $\Delta$.

######  Corollary 2.14.

Let $X$ be a topological space. Then:

  1. 1.

$(CL(X),\mathbf{V})$ is aSSM if and only if $X$ satisfies
$\mathbf{AV}_{M}(\Pi_{V},\Pi_{V})$;

  2. 2.

$(\mathbb{K}(X),\mathbf{V})$ is aSSM if and only if $X$ satisfies
$\mathbf{AV}_{M}(\Pi_{V}(\mathbb{K}(X)),\Pi_{V}(\mathbb{K}(X)))$;

  3. 3.

$(\mathbb{CS}(X),\mathbf{V})$ is aSSM if and only if $X$ satisfies
$\mathbf{AV}_{M}(\Pi_{V}(\mathbb{CS}(X)),\Pi_{V}(\mathbb{CS}(X)))$;

  4. 4.

$(\mathbb{F}(X),\mathbf{V})$ is aSSM if and only if $X$ satisfies
$\mathbf{AV}_{M}(\Pi_{V}(\mathbb{F}(X)),\Pi_{V}(\mathbb{F}(X)))$.

Next principle allows us to characterize the absolutely strongly star-
Rothberger property on hyperspaces considering the Vietoris topology.

######  Definition 2.15.

Let $X$ be a topological space. We define:

$\mathbf{AV}_{R}(\Pi_{V}(\Delta),\Pi_{V}(\Delta))$: For each sequence
$\\{J_{n}:n\in\mathbb{N}\\}\subseteq\Pi_{V}(\Delta)$ and each
$\mathcal{C}\in\mathbb{C}_{V}(\Delta)$, there is a sequence
$\\{C_{n}:n\in\mathbb{N}\\}\subseteq\mathcal{C}$, such that
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})$
$\in J_{n}:\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{c}\subseteq C_{n},\
V_{i,s}^{n}\nsubseteq C_{n}\ (1\leq i\leq m_{s})\\}$ is an element of
$\Pi_{V}(\Delta)$.

Following same idea as in the proof of Theorem 2.10, it is easy to prove the
following result.

######  Theorem 2.16.

Given a topological space $X$, the following conditions are equivalent:

  1. (1)

$(\Delta,\mathbf{V})$ is aSSR;

  2. (2)

$X$ satisfies $\mathbf{AV}_{R}(\Pi_{V}(\Delta),\Pi_{V}(\Delta))$.

By taking different choices of the family $\Delta$, we obtain the following
particular cases.

######  Corollary 2.17.

Let $X$ be a topological space. Then:

  1. 1.

$(CL(X),\mathbf{V})$ is aSSR if and only if $X$ satisfies
$\mathbf{AV}_{R}(\Pi_{V},\Pi_{V})$;

  2. 2.

$(\mathbb{K}(X),\mathbf{V})$ is aSSR if and only if $X$ satisfies
$\mathbf{AV}_{R}(\Pi_{V}(\mathbb{K}(X)),\Pi_{V}(\mathbb{K}(X)))$;

  3. 3.

$(\mathbb{CS}(X),\mathbf{V})$ is aSSR if and only if $X$ satisfies
$\mathbf{AV}_{R}(\Pi_{V}(\mathbb{CS}(X)),\Pi_{V}(\mathbb{CS}(X)))$;

  4. 4.

$(\mathbb{F}(X),\mathbf{V})$ is aSSR if and only if $X$ satisfies
$\mathbf{AV}_{R}(\Pi_{V}(\mathbb{F}(X)),\Pi_{V}(\mathbb{F}(X)))$.

##  3 Absolute and selective versions on hyperspaces with the Fell topology

In this section we now introduce some other technical principles which will
help us to make some characterizations of same variations considered in
previous section on different hyperspaces with the Fell topology. For that
end, we will show a couple of lemmas which will be often used in the proofs of
theorems in this section; it is worth to mention that these lemmas are the
analogous ones to the Lemmas 2.4 and 2.5 mentioned in Section 2.

Let us first recall the definition of a $\pi_{F}(\Delta)$-network of a space
$X$ (see [4]). Let $\xi$ denote the family

$\xi=\\{(K;V_{1},\ldots,V_{n}):K\text{ is a compact subset of }X,\
V_{1},\ldots,V_{n}\text{ are }$ $\text{ open subsets of }X\text{ with
}V_{i}\cap K^{c}\neq\emptyset\ (1\leq i\leq n)\text{ , }n\in\mathbb{N}\\}.\\\
$

######  Definition 3.1 ([4]).

Let $(X,\tau)$ be a topological space. A family $\xi$ is called a
$\pi_{F}(\Delta)$-network of $X$, if for each $U\in\Delta^{c}$, there exist a
$(K;V_{1},\ldots,V_{n})\in\xi$ and a finite set $F$ with $F\cap
V_{i}\neq\emptyset$ $(1\leq i\leq n)$ such that $K\subset U$ and $F\cap
U=\emptyset$. The family of all $\pi_{F}(\Delta)$-network of $X$ is denoted by
$\Pi_{F}(\Delta)$.

Similarly to Definition 2.3, we can define the notion of $k_{F}(\Delta)$-cover
of a space $X$ by doing some light modifications to the notion of
$k_{F}$-cover introduced in [16].

######  Definition 3.2.

Let $(X,\tau)$ be a topological space. A family
$\mathcal{U}\subseteq\Delta^{c}$ is called a _$k_{F}(\Delta)$ -cover of $X$_,
if for any compact subset $K$ of $X$ and open subsets $V_{1},\ldots,V_{m}$ of
$X$, there exists $U\in\mathcal{U}$ and $F\in[X]^{<\omega}$ with $F\cap
V_{i}\neq\emptyset$ ($1\leq i\leq m$) such that $K\subseteq U$ and $F\cap
U=\emptyset$. The family of all $k_{F}(\Delta)$-covers of $X$ is denoted by
$\mathbb{K}_{F}(\Delta)$.

The following lemma says how $k_{F}(\Delta)$-covers on a space $X$ can be
viewed as dense subspaces of certain hyperspaces of $X$ with the Fell
topology.

######  Lemma 3.3.

Let $X$ be a topological space and $\mathcal{U}\subseteq\Delta^{c}$. Then
$\mathcal{U}$ is a $k_{F}(\Delta)$-cover of $X$ if and only if
$\mathcal{U}^{c}$ is a dense subset of $(\Delta,\mathbf{F})$.

###### Proof.

$[\rightarrow]$ Let $(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}$ be a
nonempty basic open set in $(\Delta,\mathbf{F})$. Since $V_{1},\ldots,V_{n}$
are open subsets of $X$, $K$ is a compact subset of $X$ and $\mathcal{U}$ is a
$k_{F}(\Delta)$-cover, then there exist $U\in\mathcal{U}$ and
$F\in[X]^{<\omega}$ with $F\cap V_{i}\neq\emptyset$ ($1\leq i\leq m$) such
that $K\subseteq U$ and $F\cap U=\emptyset$. Put $A=U^{c}\in\Delta$. Then, it
easy to show that $A$ belongs to
$\left[(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}\right]\cap\mathcal{U}^{c}$.
Thus, $\mathcal{U}^{c}$ is dense set in $(\Delta,\mathbf{F})$.

$[\leftarrow]$ Let $K$ be a compact subset of $X$ and let $V_{1},\ldots,V_{m}$
be open subsets of $X$. Using these sets to define the basic open set
$(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}$ of $(\Delta,\mathbf{F})$, we
have that
$\left[(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}\right]\cap\mathcal{U}^{c}\neq\emptyset$.
So, we take
$A\in\left[(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}\right]\cap\mathcal{U}^{c}$.
Using this fact, it is easy to define a finite subset of $X$ so that $F\cap
V_{i}\neq\emptyset$ ($1\leq i\leq m$). Moreover, if we define $U=A^{c}$, then
$K\subseteq U$ and $F$ can be defined so that $F\cap U=\emptyset$. Thus,
$\mathcal{U}$ is a $k_{F}(\Delta)$-cover of $X$.  ∎

The next lemma says how $\pi_{F}(\Delta)$-networks of a space $X$ can be
interpreted as open covers of certain hyperspaces of $X$ considering the Fell
topology. We remark that both directions in the following lemma were used in
some proofs of theorems in [4]; for an easier reference of the reader, we
explicitly state this fact and give the proof here.

######  Lemma 3.4.

Let $X$ be a topological space and $\xi=\\{(K;V_{1},\ldots,V_{n}):\ K\text{ is
a }\newline \text{compact subset of }X,\ V_{1},\ldots,V_{n}\text{ are open
subsets of }X\text{ with }V_{i}\cap K^{c}\neq\emptyset\ (1\leq i\leq n)\text{,
}n\in\mathbb{N}\\}$. Then $\xi$ is a $\pi_{F}(\Delta)$-network of $X$ if and
only if the collection
$\mathcal{U}=\\{(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}:(K;V_{1},\ldots,V_{n})\in\xi\\}$
is an open cover of $(\Delta,\mathbf{F})$.

###### Proof.

$[\rightarrow]$ Let $B\in\Delta$. Since $\xi$ is a $\pi_{F}(\Delta)$-network
of $X$ and $B^{c}\in\Delta^{c}$, then there exists
$(K;V_{1},\ldots,V_{n})\in\xi$ and a finite subset $F$ of $X$ with $F\cap
V_{i}\neq\emptyset$ $(1\leq i\leq n)$ such that $K\subseteq B^{c}$ and $F\cap
B^{c}=\emptyset$. These conditions imply that $B\subseteq K^{c}$ and
$F\subseteq B$. Since $F\cap V_{i}\neq\emptyset$ for each
$i\in\\{1,\dots,n\\}$, then $B\cap V_{i}\neq\emptyset$ for each
$i\in\\{1,\dots,n\\}$. It follows that
$B\in(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}$ with
$(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}$ being an element of
$\mathcal{U}$. Hence, the collection $\mathcal{U}$ is an open cover of
$(\Delta,\mathbf{F})$.

$[\leftarrow]$ Let $U\in\Delta^{c}$. Since $U^{c}\in\Delta$ and the collection
$\mathcal{U}=\\{(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}:(K;V_{1},\ldots,V_{n})\in\xi\\}$
is an open cover of $(\Delta,\mathbf{F})$, there exists
$(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}\in\mathcal{U}$ such that
$U^{c}\in(\bigcap_{i=1}^{n}V_{i}^{-})\cap(K^{c})^{+}$. From this fact, we
obtain that $K\subseteq U$ and $U^{c}\cap V_{i}\neq\emptyset$ for each
$i\in\\{1\ldots,n\\}$. Hence, we can take $x_{i}\in U^{c}\cap V_{i}$ for each
$i\in\\{1,\ldots,n\\}$ and define $F=\\{x_{i}:1\leq i\leq n\\}$. Thus, $F$ is
a finite subset of $X$ with $F\cap U=\emptyset$ and $F\cap V_{i}\neq\emptyset$
$(1\leq i\leq n)$. Since the respective element $(K;V_{1},\ldots,V_{n})$ is an
element of $\xi$, we conclude that $\xi$ is a $\pi_{F}(\Delta)$-network of
$X$.  ∎

The following selection principle will help us to characterize the selectively
strongly star-Menger property on hyperspaces with the Fell topology.

######  Definition 3.5.

Let $X$ be a topological space. We define:

$\mathbf{SF}_{M}(\Pi_{F}(\Delta),\Pi_{F}(\Delta))$: For each sequence
$\\{J_{n}:n\in\mathbb{N}\\}\subseteq\Pi_{F}(\Delta)$ and each sequence
$\\{\mathcal{K}_{n}:n\in\mathbb{N}\\}\subseteq\mathbb{K}_{F}(\Delta)$, there
are finite subsets $\mathcal{W}_{n}\subseteq\mathcal{K}_{n}$,
$n\in\mathbb{N}$, such that
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\text{ there exists }W\in\mathcal{W}_{n}\text{ such that }K_{s}^{n}$
$\subseteq W,\ V_{i,s}^{n}\nsubseteq W\ (1\leq i\leq m_{s})\\}$ is an element
of $\Pi_{F}(\Delta)$.

######  Theorem 3.6.

Given a topological space $X$, the following conditions are equivalent:

  1. (1)

$(\Delta,\mathbf{F})$ is selSSM;

  2. (2)

$X$ satisfies $\mathbf{SF}_{M}(\Pi_{F}(\Delta),\Pi_{F}(\Delta))$.

###### Proof.

$(1)\Rightarrow(2)$: Let $\\{J_{n}:n\in\mathbb{N}\\}$ be a sequence of
$\pi_{F}(\Delta)$-networks of $X$ and let
$\\{\mathcal{K}_{n}:n\in\mathbb{N}\\}$ be a sequence of $k_{F}(\Delta)$-covers
of $X$. For each $n\in\omega$, we denote
$J_{n}=\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n}):s\in S_{n}\\}$.
Applying Lemma 3.4, we obtain that, for each $n\in\omega$, the collection
$\mathcal{U}_{n}=\\{(\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{-})\cap((K_{s}^{n})^{c})^{+}:s\in
S_{n}\\}$ is an open cover of $(\Delta,\mathbf{F})$. Furthermore, by Lemma
3.3, the collection $\mathcal{D}_{n}=\mathcal{K}_{n}^{c}$ is a dense subset of
$(\Delta,\mathbf{F})$, for each $n\in\mathbb{N}$.

Thus, applying (1) to the sequence $\\{\mathcal{U}_{n}:n\in\mathbb{N}\\}$ and
the sequence $\\{\mathcal{D}_{n}:n\in\mathbb{N}\\}$, there is a sequence
$\\{\mathcal{F}_{n}:n\in\mathbb{N}\\}$ of finite sets such that, for each
$n\in\mathbb{N}$, $\mathcal{F}_{n}\subseteq\mathcal{D}_{n}$ and the collection
$\\{St(\mathcal{F}_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{F})$. We put, for each $n\in\mathbb{N}$,
$\mathcal{W}_{n}=\\{F^{c}:F\in\mathcal{F}_{n}\\}$. Then, for each
$n\in\mathbb{N}$, $\mathcal{W}_{n}$ is a finite subset of $\mathcal{K}_{n}$.

Let us show that the collection
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\text{there exists }\ W\in\mathcal{W}_{n}\text{ such that
}K_{s}^{n}\subseteq W,\ V_{i,s}^{n}\nsubseteq W\ (1\leq i\leq m_{s})\\}$ is a
$\pi_{F}(\Delta)$-network of $X$.

Let $U\in\Delta^{c}$. Then $U^{c}\in\Delta$ and therefore, there exists
$n_{0}\in\omega$ such that $U^{c}\in
St(\mathcal{F}_{n_{0}},\mathcal{U}_{n_{0}})$. Thus, there is
$(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}\in\mathcal{U}_{n_{0}}$
such that
$\left[(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}\right]\bigcap\mathcal{F}_{n_{0}}\neq\emptyset$
and
$U^{c}\in(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$.
We fix
$F_{0}\in\left[(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}\right]\bigcap\mathcal{F}_{n_{0}}$
and define $W_{0}=F^{c}_{0}$. Thus $W_{0}\in\mathcal{W}_{n_{0}}$ and, observe
that $K_{s_{0}}^{n_{0}}\subseteq W_{0}$ and $V_{i,s_{0}}^{n_{0}}\nsubseteq
W_{0}(1\leq i\leq m_{s_{0}})$ are obtained from the fact that
$F_{0}\in(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$.
It follows that the corresponding element
$(K_{s_{0}}^{n_{0}};V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})$
belongs to $\mathcal{J}$. In addition, from the fact that
$U^{c}\in\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-}$, we can fix
$x_{i}\in U^{c}\cap V_{i,s_{0}}^{n_{0}}$ ($i\in\\{1,\ldots,m_{s_{0}}\\}$) and
let $M=\\{x_{i}:i\in\\{1,\ldots,m_{s_{0}}\\}\\}$. So, $M$ is a finite subset
of $X$ that satisfies $M\cap V_{i,s_{0}}^{n_{0}}\neq\emptyset$ ($1\leq i\leq
m_{s_{0}}$) and $M\cap U=\emptyset$. On the other hand,
$K_{s_{0}}^{n_{0}}\subseteq U$ follows from
$U^{c}\in((K_{s_{0}}^{n_{0}})^{c})^{+}$. We conclude that
$\mathcal{J}\in\Pi_{F}(\Delta)$.

$(2)\Rightarrow(1)$: Let $\\{\mathcal{U}_{n}:n\in\mathbb{N}\\}$ be a sequence
of open covers of $(\Delta,\mathbf{F})$ and let
$\\{\mathcal{D}_{n}:n\in\mathbb{N}\\}$ be a sequence of dense subsets of
$(\Delta,\mathbf{F})$. Without loss of generality, suppose that the elements
of each open cover $\mathcal{U}_{n}$ are basic open subsets in $CL(X)$.
Therefore, let us denote, for each $n\in\mathbb{N}$,
$\mathcal{U}_{n}=\\{(\bigcap_{i=1}^{m_{s}}(V_{i,s}^{n})^{-})\cap((K_{s}^{n})^{c})^{+}:s\in
S_{n}\\}$, where each $V_{i,s}^{n}$ is an open subset of $X$ and $K_{s}^{n}$
is a compact subset of $X$ (for every $n\in\mathbb{N}$, $s\in S_{n}$ and
$i\in\\{1,\ldots,m_{s}\\}$). For each $n\in\mathbb{N}$, let
$J_{n}=\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n}):s\in S_{n}\\}$. Then,
by Lemma 3.4, we get that each $J_{n}$ is a $\pi_{F}(\Delta)$-network of $X$.
Moreover, if we define, for each $n\in\mathbb{N}$,
$\mathcal{K}_{n}=\mathcal{D}_{n}^{c}$, then each $\mathcal{K}_{n}$ is a
$k_{F}(\Delta)$-cover of $X$ by Lemma 3.3.

Applying $(2)$ to the sequence of $\pi_{F}(\Delta)$-networks
$\\{J_{n}:n\in\mathbb{N}\\}$ and the sequence of $k_{F}(\Delta)$-covers
$\\{\mathcal{K}_{n}:n\in\mathbb{N}\\}$ we obtain a sequence
$\\{\mathcal{W}_{n}:n\in\mathbb{N}\\}$ with $\mathcal{W}_{n}$ a finite subset
of $\mathcal{K}_{n}$, for each $n\in\mathbb{N}$, and so that the collection
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\text{ there exists }W\in\mathcal{W}_{n}\text{ such that
}K_{s}^{n}\subseteq W,\ V_{i,s}^{n}\nsubseteq W\ (1\leq i\leq m_{s})\\}$ is a
$\pi_{F}(\Delta)$-network of $X$. We define, for each $n\in\omega$,
$\mathcal{F}_{n}=\mathcal{W}^{c}_{n}$. Then $\mathcal{F}_{n}$ is a finite
subset of $\mathcal{D}_{n}$, for each $n\in\mathbb{N}$.

Let us show that the collection
$\\{St(\mathcal{F}_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{F})$. Let $A\in\Delta$. Using the fact that $\mathcal{J}$ is
a $\pi_{F}(\Delta)$-network of $X$ with the set $A^{c}\in\Delta^{c}$, we
obtain an element
$(K_{s_{0}}^{n_{0}};V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})$
of $\mathcal{J}$ (for some $n_{0}\in\mathbb{N}$ and some $s_{0}\in S_{n_{0}}$)
and a finite set $F\subseteq X$ such that for every
$i\in\\{1,\ldots,m_{s_{0}}\\}$, $F\cap V_{i,s_{0}}^{n_{0}}\neq\emptyset$,
$K_{s_{0}}^{n_{0}}\subseteq A^{c}$ and $F\cap A^{c}=\emptyset$. It easily
follows from these last conditions that $A$ belongs to
$(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$.
On the other hand, since
$(K_{s_{0}}^{n_{0}};V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})\in\mathcal{J}$,
there exists $W_{0}\in\mathcal{W}_{n_{0}}$ such that
$K_{s_{0}}^{n_{0}}\subseteq W_{0}$ and for each
$i\in\\{1,\ldots,m_{s_{0}}\\}$, $V_{i,s_{0}}^{n_{0}}\nsubseteq W_{0}$. These
conditions imply that the element $F_{0}=W_{0}^{c}$ belongs to
$(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$.
Notice that $F_{0}\in\mathcal{F}_{n_{0}}$ and
$(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}\in\mathcal{U}_{n_{0}}$.
It means that $A\in St(\mathcal{F}_{n_{0}},\mathcal{U}_{n_{0}})$. Therefore,
$\\{St(\mathcal{F}_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{F})$.  ∎

As a consequence of previous theorem, we can characterize the selectively
strongly star-Menger property on different hyperspaces considered with the
Fell topology.

######  Corollary 3.7.

Let $X$ be a topological space. Then:

  1. 1.

$(CL(X),\mathbf{F})$ is selSSM if and only if $X$ satisfies
$\mathbf{SF}_{M}(\Pi_{F},\Pi_{F})$;

  2. 2.

$(\mathbb{K}(X),\mathbf{F})$ is selSSM if and only if $X$ satisfies
$\mathbf{SF}_{M}(\Pi_{F}(\mathbb{K}(X)),\Pi_{F}(\mathbb{K}(X)))$;

  3. 3.

$(\mathbb{CS}(X),\mathbf{F})$ is selSSM if and only if $X$ satisfies
$\mathbf{SF}_{M}(\Pi_{F}(\mathbb{CS}(X)),\Pi_{F}(\mathbb{CS}(X)))$;

  4. 4.

$(\mathbb{F}(X),\mathbf{F})$ is selSSM if and only if $X$ satisfies
$\mathbf{SF}_{M}(\Pi_{F}(\mathbb{F}(X)),\Pi_{F}(\mathbb{F}(X)))$.

For the Rothberger case we define the following selection principle to get, in
the same way, characterizations of the selectively strongly star-Rothberger
property on hyperspaces with the Fell topology.

######  Definition 3.8.

Let $X$ be a topological space. We define:

$\mathbf{SF}_{R}(\Pi_{F}(\Delta),\Pi_{F}(\Delta))$: For each sequence
$\\{J_{n}:n\in\omega\\}\subseteq\Pi_{F}(\Delta)$ and each sequence
$\\{\mathcal{K}_{n}:n\in\omega\\}\subseteq\mathbb{K}_{F}(\Delta)$, there is a
sequence $\\{K_{n}:n\in\omega\\}$ with $K_{n}\in\mathcal{K}_{n}$,
$n\in\omega$, such that
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:K_{s}^{n}\subseteq K_{n},\ V_{i,s}^{n}\nsubseteq K_{n}\ (1\leq i\leq
m_{s})\\}$ is an element of $\Pi_{F}(\Delta)$.

######  Theorem 3.9.

Given a topological space $X$, the following conditions are equivalent:

  1. (1)

$(\Delta,\mathbf{F})$ is selSSR;

  2. (2)

$X$ satisfies $\mathbf{SF}_{R}(\Pi_{F}(\Delta),\Pi_{F}(\Delta))$.

###### Proof.

$(1)\Rightarrow(2)$: By using Lemmas 3.3 and 3.4 and same idea as in the first
part of $(1)\Rightarrow(2)$ in the proof of Theorem 3.6, we can obtain a
sequence $\\{D_{n}:n\in\mathbb{N}\\}$ of points in $(\Delta,\mathbf{F})$ such
that, for each $n\in\mathbb{N}$, $D_{n}\in\mathcal{D}_{n}$ and the collection
$\\{St(D_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{F})$. For each $n\in\mathbb{N}$, we define $K_{n}=D_{n}^{c}$.
Then, $K_{n}\in\mathcal{K}_{n}$, for each $n\in\mathbb{N}$.

Considering the sequence $\\{K_{n}:n\in\mathbb{N}\\}$, let us show that the
collection
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:K_{s}^{n}\subseteq K_{n},\ V_{i,s}^{n}\nsubseteq K_{n}\ (1\leq i\leq
m_{s})\\}$ is a $\pi_{F}(\Delta)$-network of $X$.

Let $U\in\Delta^{c}$. Then, for the element $U^{c}\in\Delta$, there exists
$n_{0}\in\omega$ such that $U^{c}\in St(D_{n_{0}},\mathcal{U}_{n_{0}})$. Thus,
there is
$(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}\in\mathcal{U}_{n_{0}}$
such that
$U^{c},D_{n_{0}}\in(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$.

As $K_{n_{0}}=D_{n_{0}}^{c}$ and
$D_{n_{0}}\in(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$,
it easily follows that $K_{s_{0}}^{n_{0}}\subseteq K_{n_{0}}$ and $\
V_{i,s_{0}}^{n_{0}}\nsubseteq K_{n_{0}}(1\leq i\leq m_{s_{0}})$. This implies
that the corresponding
$(K_{s_{0}}^{n_{0}};V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})$
is an element of $\mathcal{J}$. Furthermore, since $U^{c}$ belongs to
$(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$,
we get that $K_{s_{0}}^{n_{0}}\subseteq U$ and also, we can define a finite
subset $F$ of $X$ such that $F\cap V_{i,s_{0}}^{n_{0}}\neq\emptyset$ and
$F\cap U=\emptyset$. We conclude that $\mathcal{J}\in\Pi_{F}(\Delta)$.

$(2)\Rightarrow(1)$: Following same arguments and employing Lemmas 3.3 and 3.4
as in the first part of $(2)\Rightarrow(1)$ in the proof of Theorem 3.6, we
obtain a sequence $\\{K_{n}:n\in\mathbb{N}\\}$ such that, for each
$n\in\mathbb{N}$, $K_{n}\in\mathcal{K}_{n}$, and the collection
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:K_{s}^{n}\subseteq K_{n},\ V_{i,s}^{n}\nsubseteq K_{n}\ (1\leq i\leq
m_{s})\\}$ is a $\pi_{F}(\Delta)$-network of $X$. We define, for each
$n\in\mathbb{N}$, $D_{n}=K^{c}_{n}$. Then, $\\{D_{n}:n\in\mathbb{N}\\}$ is a
sequence of points in $(\Delta,\mathbf{F})$ with $D_{n}\in\mathcal{D}_{n}$,
for each $n\in\mathbb{N}$.

We claim that the collection $\\{St(D_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$
is an open cover of $(\Delta,\mathbf{F})$. Indeed, let $A\in\Delta$. Since
$\mathcal{J}$ is a $\pi_{F}(\Delta)$-network of $X$ and $A^{c}\in\Delta^{c}$,
there exists
$(K_{s_{0}}^{n_{0}};V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})\in\mathcal{J}$
(for some $n_{0}\in\mathbb{N}$ and some $s_{0}\in S_{n_{0}}$) and a finite set
$F\subseteq X$ such that $F\cap V_{i,s_{0}}^{n_{0}}\neq\emptyset$ ($1\leq
i\leq m_{s_{0}}$), $K_{s_{0}}^{n_{0}}\subseteq A^{c}$ and $F\cap
A^{c}=\emptyset$. The fact
$(K_{s_{0}}^{n_{0}};V_{1,s_{0}}^{n_{0}},\ldots,V_{m_{s_{0}},s_{0}}^{n_{0}})\in\mathcal{J}$
means that $K_{s_{0}}^{n_{0}}\subseteq K_{n_{0}}$ and
$V_{i,s_{0}}^{n_{0}}\nsubseteq K_{n_{0}}$ ( $1\leq i\leq m_{s_{0}}$) and
hence,
$D_{n_{0}}\in(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$.
Moreover, the conditions $F\cap V_{i,s_{0}}^{n_{0}}\neq\emptyset$ ($1\leq
i\leq m_{s_{0}}$), $K_{s_{0}}^{n_{0}}\subseteq A^{c}$ and $F\cap
A^{c}=\emptyset$ mean that
$A\in(\bigcap_{i=1}^{m_{s_{0}}}(V_{i,s_{0}}^{n_{0}})^{-})\cap((K_{s_{0}}^{n_{0}})^{c})^{+}$.
Thus, $A\in St(D_{n_{0}},\mathcal{U}_{n_{0}})$. In conclusion,
$\\{St(D_{n},\mathcal{U}_{n}):n\in\mathbb{N}\\}$ is an open cover of
$(\Delta,\mathbf{F})$.  ∎

Now, we can characterize the selectively strongly star-Rothberger property on
several hyperspaces with the Fell topology.

######  Corollary 3.10.

Let $X$ be a topological space. Then:

  1. 1.

$(CL(X),\mathbf{F})$ is selSSR if and only if $X$ satisfies
$\mathbf{SF}_{R}(\Pi_{F},\Pi_{F})$;

  2. 2.

$(\mathbb{K}(X),\mathbf{F})$ is selSSR if and only if $X$ satisfies
$\mathbf{SF}_{R}(\Pi_{F}(\mathbb{K}(X)),\Pi_{F}(\mathbb{K}(X)))$;

  3. 3.

$(\mathbb{CS}(X),\mathbf{F})$ is selSSR if and only if $X$ satisfies
$\mathbf{SF}_{R}(\Pi_{F}(\mathbb{CS}(X)),\Pi_{F}(\mathbb{CS}(X)))$;

  4. 4.

$(\mathbb{F}(X),\mathbf{F})$ is selSSR if and only if $X$ satisfies
$\mathbf{SF}_{R}(\Pi_{F}(\mathbb{F}(X)),\Pi_{F}(\mathbb{F}(X)))$.

Now we characterize the absolute versions of the Menger-type and Rothberger-
type star selection principles with the Fell topology. The following principle
allows us to characterize the absolutely strongly star-Menger property on
several hyperspaces with the Fell topology.

######  Definition 3.11.

Let $X$ be a topological space. We define:

$\mathbf{AF}_{M}(\Pi_{F}(\Delta),\Pi_{F}(\Delta))$: For each sequence
$\\{J_{n}:n\in\omega\\}\subseteq\Pi_{F}(\Delta)$ and each
$\mathcal{K}\in\mathbb{K}_{F}(\Delta)$, there is a sequence
$\\{\mathcal{W}_{n}:n\in\omega\\}\subseteq[\mathcal{K}]^{<\omega}$ such that
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(K_{s}^{n};V_{1,s}^{n},\ldots,V_{m_{s},s}^{n})\in
J_{n}:\text{ there exists }W\in\mathcal{W}_{n}\text{ such that
}K_{s}^{n}\subseteq W,\ V_{i,s}^{n}$ $\nsubseteq W\ (1\leq i\leq m_{s})\\}$ is
an element of $\Pi_{F}(\Delta)$.

Adapting the proof of Theorem 3.6, one can easily obtain the following result.

######  Theorem 3.12.

Given a topological space $X$, the following conditions are equivalent:

  1. (1)

$(\Delta,\mathbf{F})$ is aSSM;

  2. (2)

$X$ satisfies $\mathbf{AF}_{M}(\Pi_{F}(\Delta),\Pi_{F}(\Delta))$.

We obtain the following particular cases by taking different choices of our
family $\Delta$.

######  Corollary 3.13.

Let $X$ be a topological space. Then:

  1. 1.

$(CL(X),\mathbf{F})$ is aSSM if and only if $X$ satisfies
$\mathbf{AF}_{M}(\Pi_{F},\Pi_{F})$;

  2. 2.

$(\mathbb{K}(X),\mathbf{F})$ is aSSM if and only if $X$ satisfies
$\mathbf{AF}_{M}(\Pi_{F}(\mathbb{K}(X)),\Pi_{F}(\mathbb{K}(X)))$;

  3. 3.

$(\mathbb{CS}(X),\mathbf{F})$ is aSSM if and only if $X$ satisfies
$\mathbf{AF}_{M}(\Pi_{F}(\mathbb{CS}(X)),\Pi_{F}(\mathbb{CS}(X)))$;

  4. 4.

$(\mathbb{F}(X),\mathbf{F})$ is aSSM if and only if $X$ satisfies
$\mathbf{AF}_{M}(\Pi_{F}(\mathbb{F}(X)),\Pi_{F}(\mathbb{F}(X)))$.

The following principle will give us characterizations of the absolutely
strongly star-Rothberger property on hyperspaces considering the Fell
topology.

######  Definition 3.14.

Let $X$ be a topological space. We define:

$\mathbf{AF}_{R}(\Pi_{F}(\Delta),\Pi_{F}(\Delta))$: For each sequence
$\\{J_{n}:n\in\mathbb{N}\\}\subseteq\Pi_{F}(\Delta)$ and each
$\mathcal{K}\in\mathbb{K}_{F}(\Delta)$, there is a sequence
$\\{K_{n}:n\in\mathbb{N}\\}\subseteq\mathcal{K}$, such that
$\mathcal{J}=\bigcup_{n\in\mathbb{N}}\\{(K_{s}^{n};V_{1,s}^{n},\ldots,$
$V_{m_{s},s}^{n})\in J_{n}:K_{s}^{n}\subseteq K_{n},\ V_{i,s}^{n}\nsubseteq
K_{n}\ (1\leq i\leq m_{s})\\}$ is an element of $\Pi_{F}(\Delta)$.

Similarly to the proof of Theorem 3.9, one can prove the following result.

######  Theorem 3.15.

Given a topological space $X$, the following conditions are equivalent:

  1. (1)

$(\Delta,\mathbf{F})$ is aSSR;

  2. (2)

$X$ satisfies $\mathbf{AF}_{R}(\Pi_{F}(\Delta),\Pi_{F}(\Delta))$.

The following particular cases are obtained by choosing different families
$\Delta$.

######  Corollary 3.16.

Let $X$ be a topological space. Then:

  1. 1.

$(CL(X),\mathbf{F})$ is aSSR if and only if $X$ satisfies
$\mathbf{AF}_{R}(\Pi_{F},\Pi_{F})$;

  2. 2.

$(\mathbb{K}(X),\mathbf{F})$ is aSSR if and only if $X$ satisfies
$\mathbf{AF}_{R}(\Pi_{F}(\mathbb{K}(X)),\Pi_{F}(\mathbb{K}(X)))$;

  3. 3.

$(\mathbb{CS}(X),\mathbf{F})$ is aSSR if and only if $X$ satisfies
$\mathbf{AF}_{R}(\Pi_{F}(\mathbb{CS}(X)),\Pi_{F}(\mathbb{CS}(X)))$;

  4. 4.

$(\mathbb{F}(X),\mathbf{F})$ is aSSR if and only if $X$ satisfies
$\mathbf{AF}_{R}(\Pi_{F}(\mathbb{F}(X)),\Pi_{F}(\mathbb{F}(X)))$.

Concluding remark. It worth remarking that, by doing some slightly
modifications to definitions given in Sections 2 and 3, it is possible to
easily obtain hit-and-miss generalizations that involve both the Vietoris
topology and the Fell topology (similar to how it was done in [6] and [7]).
However, for convenience of the reader, the author decided to establish
simpler notation results depending of which topology is being considered.

## References

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Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito Exterior
S/N, Ciudad Universitaria, CP 04510, Ciudad de México, México.

Email address: J. Casas-de la Rosa<EMAIL_ADDRESS>


# Heavy resonances ($\textrm{W}^{\prime}$, $\textrm{Z}^{\prime}$, jets) in
ATLAS and CMS in Run 2

A. Zucchetta on behalf of the ATLAS and CMS Collaborations  Universität
Zürich, Physik-Institut, Winterthurerstrasse 190, 8057 Zürich

###### Abstract

An overview of the searches for heavy resonances that decay to leptons or
quarks is presented. The results are based on the data collected by the ATLAS
and CMS experiments in proton-proton collisions at
$\sqrt{s}=13\leavevmode\nobreak\ \textrm{TeV}$, and include for the first time
the entire LHC Run 2 data set.

00footnotetext: © 2019 CERN for the benefit of the ATLAS and CMS
Collaborations. CC-BY-4.0 license.

##  1 Introduction

New resonances are predicted by several theories addressing the open issues of
the standard model (SM). Depending on the mass and the couplings to the SM
quarks and leptons of the heavy particles, these new states could be
accessible to the LHC and observable by the ATLAS [1] and CMS [2] experiments.
Spin-1 heavy resonances are predicted in minimal extensions of the SM, as in
the sequential standard model (SSM) $\textrm{W}^{\prime}$ [3] and
$\textrm{Z}^{\prime}$ [4]. Massive spin-2 gravitons (G) may represent the
first Kaluza-Klein excitation in Randall-Sundrum warped extra dimension models
[5]. Exotic models also include vector-like quarks (VLQ) [6], and excited
leptons in compositeness models [7].

As the most powerful proton-proton collider currently in operation, the LHC
has the possibility to make such a discovery. At the end of 2018, the LHC Run
2 has finished its operations, and thus a large data set, accounting for
approximately $140\leavevmode\nobreak\ \textrm{fb}^{-1}$, is now available for
analysis. In this proceeding, the first results based on the full Run 2 data
set are presented.

##  2 Dilepton searches

The dilepton (ee or $\mu\mu$) final state is considered the golden channel for
the discovery of a heavy resonance, thanks to the minimal backgrounds from the
SM, and the electron and muon reconstruction efficiency and energy resolution.
The ATLAS experiment presents the first search for dilepton resonances based
on the full Run 2 data set [8], which amounts to 139$\leavevmode\nobreak\
\textrm{fb}^{-1}$. The CMS experiment has updated the dielectron search to
$77\leavevmode\nobreak\ \textrm{fb}^{-1}$ collected during 2016 and 2017 [9],
while the dimuon channel is based on the 2016 data set consisting of
$36\leavevmode\nobreak\ \textrm{fb}^{-1}$ of data [10].

These searches select events with two same-flavor leptons, with a large
transverse momentum (${p}_{T}$), and isolated from other hadronic activity in
the event. In ATLAS, the electron channel dominates the sensitivity, because
of the larger selection efficiency with respect to muons (70–75% and 50–40%,
respectively) and the far better dilepton resolution (a flat $\approx 1\%$ for
electrons, and 3–27% for muons). In CMS, the dielectron reconstruction
efficiency is 70%, and the dielectron invariant mass resolution is between 1.3
and 2.0%. The dimuon selection efficiency is larger at 93%, and the momentum
resolution is about 7% for muons with ${p}_{T}$ of 1 TeV.

A heavy resonance would emerge as a localized excess of events above the
smoothly falling SM background processes. The SM background, represented
mostly by off-shell Z bosons, is estimated in ATLAS by performing a combined
fit of the signal model and a smooth power-law function that is used to
parametrize the background distribution. The signal model is composed by a
variable-width non-relativistic Breit-Wigner function modeling the natural
width of the resonance, and a sum of a Gaussian and a Crystal Ball function to
describe the detector response. The background estimation of the CMS search is
instead based on Monte Carlo simulations of the background processes. The
dielectron invariant mass spectra are presented in Fig. 1.

Figure 1: Dielectron invariant mass observed by ATLAS [8] with
$139\leavevmode\nobreak\ \textrm{fb}^{-1}$ (left) and by CMS [9] with
$77\leavevmode\nobreak\ \textrm{fb}^{-1}$ (right).

Both experiments report no significant excesses above the SM background
expectation. The most significant local excess observed by ATLAS corresponding
to the dielectron invariant mass of 774 GeV amounts to 2.9 standard
deviations, which become 0.1 when the _look-elsewhere_ effect is taken into
account. Upper limits at 95% confidence level (CL) are set on the cross
section of heavy resonances decaying to a pair of leptons, shown in Fig. 2.
The ATLAS search also performs a scan of the resonance width up to 10% of its
mass. The results reported by ATLAS exclude a SSM $\textrm{Z}^{\prime}$ with
mass lower than 5.1 TeV, and CMS, in spite of the smaller data set, is able to
exclude up to 4.7 TeV.

Figure 2: Exclusion limits from ATLAS [8] (left) and CMS [9] (right) on heavy
resonances decaying to dilepton final states. The result represent the
combination of the ee and $\mu\mu$ channels.

##  3 Dilepton combination

Even though the minimal extensions of the SM predict only one heavy resonance,
other theories, described in an effective approach by the heavy vector triplet
(HVT) model [11], foresee the simultaneous presence of mass-degenerate triplet
of heavy resonances: one $\textrm{Z}^{\prime}$ and two electrically charged
$\textrm{W}^{\prime}$±. In the scenarios with SM-like couplings, the
$\textrm{W}^{\prime}\to\ell\nu$ and $\textrm{Z}^{\prime}\to\ell\ell$ decays
represent the privileged channels for discovery. The ATLAS and CMS
Collaborations recently published a statistical combination of these final
states based on the 2016 data set [12] [13], which is complemented by the
$\textrm{W}^{\prime}$ and $\textrm{Z}^{\prime}$ decays to a pair of SM bosons
in the scenarios where the couplings to bosons are enhanced with respect to
those to fermions. The dilepton and diboson searches provide complementary
sensitivity, as dilepton searches constrain the fermionic couplings in regions
that are not accessible to diboson final states alone, as depicted in Fig. 3.

Figure 3: Exclusion limits from ATLAS in the plane of the couplings to
fermions and SM bosons of a triplet of heavy resonances [12] (left). Exclusion
limit from CMS by combining the $\textrm{W}^{\prime}\to\ell\nu$ and
$\textrm{Z}^{\prime}\to\ell\ell$ searches in a scenario where the couplings of
the heavy resonances are SM-like [13] (right).

##  4 Dijet searches

Heavy resonances decaying to a pair of quarks or gluons represent one of the
most natural searches for the LHC programme, as the initial and final states
are the same. The ATLAS experiment recently published the results for dijet
resonances based on the full Run 2 data set [14], accounting for
139$\leavevmode\nobreak\ \textrm{fb}^{-1}$. This search is favored by the
dijet invariant mass resolution, which is stable between 2.4–2.9% in the
1.2–8.1 TeV range probed in this analysis. On the other hand, the QCD multijet
production background is very large; its $t$-channel dijet production mode is
suppressed by requiring a minimal separation in the rapidity angle between the
two jets, $\frac{1}{2}|y_{j1}-y_{j2}|<0.6$. The SM background is estimated by
parametrizing its distribution with a power-law function with 4 free
parameters, as reported in Fig. 4. The results are expressed as upper limits
on the product of the cross section, acceptance, and selection efficiency as a
function of the resonance mass and width, the latter varying from 3 to 15%. In
the context of excited quarks, the exclusion limit is increased by 700 GeV
with respect to the previous publications.

The corresponding CMS search is based on $77\leavevmode\nobreak\
\textrm{fb}^{-1}$ of data [15]. In the region with dijet invariant mass larger
than 2.4 TeV, the analysis introduces a novel background estimation technique
based on three regions defined depending on the separation $\Delta\eta$
between the two jets. The background in the signal region ($\Delta\eta<1.1$)
is estimated from a control region ($\Delta\eta>1.5$) and a transfer function
based on simulation. The intermediate validation region is used to derive a
further correction from the data in the signal region. This technique yields
results consistent with the previous function fit method but is more robust
against non-narrow resonances that are reconstructed as broad excesses of
events. Additionally, in order to increase the dijet resolution and recover
particles emitted from final state radiation, jets are merged to the two
leading jets if the angular separation between them is small enough ($\Delta
R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}<1.1$). No significant excess is seen,
as reported in Fig. 4, and upper limits are set on the cross section of the
resonances predicted by several theories beyond the SM.

Figure 4: Dijet invariant mass spectrum observed by ATLAS with the full Run 2
data set [14] (left); comparison between the CMS dijet invariant mass
distribution in data and the two background models [15] (center); ATLAS
exclusion limits on a graviton with a natural width comprised between 3 and
15% (right).

##  5 Low mass dijet searches

The sensitivity of high-mass dijet searches is inferiorly limited by the
trigger thresholds, which prevent the exploration of the sub-TeV regime. This
limitation can be overcome by requiring more complex final states involving
additional particles, such as an initial state radiation (ISR) jet or a
photon, or alternatively, if the resonance decays to b quarks, by tagging its
heavy flavor content. The changes in the topology imply that the resonance
recoils against the ISR particles and is produced with a large boost.

The ATLAS Collaboration presents a search based on 80$\leavevmode\nobreak\
\textrm{fb}^{-1}$ for light resonances decaying to a pair of quarks or b
quarks in association with a photon [16]. Two triggers based on high energy
photons are used to collect data, and two categories (flavor inclusive and b
tagged) enhance the sensitivity to
$\textrm{Z}^{\prime}\to\textrm{b}\bar{\textrm{b}}$ decays. These selections
lower the probed $\textrm{Z}^{\prime}$ masses down to 225 GeV.

Both ATLAS and CMS are looking for light resonances decaying to b quarks in
region of the phase space with large resonance boost [17] [18]. Both
experiments use large-cone jets (with radius parameter 1.0 for ATLAS, and CMS
uses either 0.8 or 1.5) and the investigation of the jet substructure to
simultaneously identify the two b quarks within the same jet. The events are
required to have at least one additional high-${p}_{T}$ jet, and the exotic
resonance (a $\textrm{Z}^{\prime}$ or a scalar $\Phi$) is sought in the large-
cone jet mass spectrum. The usage of jet substructure extends the resonances
masses down to approximately the W/Z boson masses.

A significant improvement in this field is presented by CMS exploring low mass
resonances decaying to a pair of light quarks [19]. The events are collected
by triggering on an ISR photon, and the dominating non-resonant background
from $\gamma$+jets is estimated using a novel technique based on the Energy
Correlation Functions [20] decorrelated from the jet mass. Upper limits on the
cross section of light $\textrm{Z}^{\prime}$ resonance or on its couplings to
quarks can be set for masses as low as 10 GeV. The corresponding ATLAS search
[21] uses a N-subjettiness $\tau_{21}$ tagger [22], also decorrelated from the
jet mass, but stops at 100 GeV.

These searches allow to extend the reach of the resonance searches at LHC to
masses that would be naturally inaccessible by ATLAS and CMS, and cover the
range from 10 GeV to 8 TeV, as reported in Fig. 5. However, no convincing
excess is observed.

Figure 5: Left: large-cone jet mass spectrum for events passing the analysis
selections of the CMS search [19]. Right: summary of the ATLAS and CMS dijet
searches [23], from 10 GeV to 8 TeV.

##  6 Vector-like Quarks

Vector-like quarks (VLQ) are hypothetical heavy fermions that decay to a
third-generation quark (t, b) and a SM boson (W, Z, H). The large mass of the
VLQ implies that these particle have large momenta, and can be reconstructed
as single large-cone jets if they decay to quarks. Since VLQ are produced in
pairs at the LHC, there is a large multiplicity of channels with a large
number of possible final states. The CMS search [24] is based on a novel
tagger (Boosted Event Shape Tagger, or BEST) based on a deep neural network
that is able to classify W, Z, H bosons, top and bottom quarks that decay to
hadrons. Depending on the possible final states, 126 search regions are
defined according to the BEST output. In each region, a fit to the scalar sum
of the ${p}_{T}$ of the jets in the event is performed. The search, based on
36$\leavevmode\nobreak\ \textrm{fb}^{-1}$ of data, extends the excluded mass
for VLQ by approximately 100 GeV with respect to the previous analyses,
depending on the channel. The ATLAS experiment published a statistical
combination of 6 different searches looking for VLQ in both fully-hadronic,
semi-leptonic, and fully-leptonic final states [25], as depicted in Fig. 6.
The SM backgrounds are found to be consistent with the background expectation,
and no significant excess is found.

##  7 Excited leptons

Excited leptons ($\ell^{*}$) are included in compositeness models, which
postulate that leptons are not fundamental particles. Heavy excited leptons
are thus produced from $\textrm{q}\bar{\textrm{q}}$ annihilation through
contact interactions with scale $\Lambda$ and in association with a SM lepton.
In the CMS search, which analyzes 77$\leavevmode\nobreak\ \textrm{fb}^{-1}$ of
data, the excited lepton also decays through contact interaction to an other
SM lepton and a pair of quarks [26]. The final state thus consists of two
same-flavor leptons and to resolved jets. The main Z+jets background is
estimated in the region where the Z boson is on-shell, while the search
regions considers a dilepton invariant mass larger than 500 GeV. A fit is then
performed on the four-body invariant mass, shown in Fig. 6. Assuming
$\Lambda=M_{\ell^{*}}$, excited electrons and muon with mass smaller than 5.6
and 5.7 TeV are excluded, respectively.

Figure 6: Left: ATLAS combination of VLQ searches [25]. Right: four-body
invariant mass spectrum of the CMS excited muon search [26].

##  8 Summary

The data set provided by the LHC in the past four years of running is now
being analyzed, and the first analyses looking for heavy dijet and dilepton
resonances and exploiting the full integrated luminosity have been presented
at the Moriond EW conference. However, including new data is not the only
improvement from the ATLAS and CMS experiments. Searches are being refined by
including more complex and robust background estimation methods, and designing
bleeding edge taggers based on the latest machine learning developments. More
searches will follow in the near future based on the new data and analysis
techniques, significantly improving the reach of the LHC in the search for
heavy resonances.

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