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\begin{center}{\footnotesize Khayyam J. Math. 1 (2015), no. 1, 82--106}\\\end{center}
\noindent\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=0.24]{KJM.jpg}}
\vspace{0.5cm}

\title[Star selection principles]{Star selection principles: A survey}

\author[Lj.D.R. Ko\v cinac]{Ljubi\v sa D.R. Ko\v cinac}

\address{University of Ni\v s, Faculty of Sciences and Mathematics\\
18000 Ni\v s, Serbia} \email{lkocinac@gmail.com}

\dedicatory{{\rm Communicated by H.R. Ebrahimi Vishki}}

\subjclass[2010]{Primary 54D20; Secondary 54A35, 54B20, 54E15,
54H10, 91A44.}

\keywords{Star selection principles, {\sf ASSM}, selectively
$(a)$, uniform selection principles.}

\date{Received: 29 November 2014;  Accepted: 30 December 2014.}
%\newline \indent $^{*}$ Corresponding author}

\begin{abstract}
We review selected results obtained in the last fifteen years on
star selection principles in topology, an important subfield of
the field of selection principles theory. The results which we
discuss concern also uniform structures and, in particular,
topological groups and their generalizations.
\end{abstract}

\maketitle

%%%%%%%%%%%%% 11111 %%%%%%%%%%%%%%%

\section{Introduction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There are many results in the literature which show that a number
of topological properties can be characterized by using the method
of stars. In particular it is the case with many covering
properties of topological spaces.  The method of stars has been
used to study the problem of metrization of topological spaces,
and for definitions of several important classical topological
notions. More information on star covering properties can be found
in \cite{vDRRT}, \cite{misha-survey}. We use here such a method in
investigation of selection principles for topological and uniform
spaces.

Although Selection Principles Theory is a field of mathematics
having a rich history going back to the papers by Borel, Menger,
Hurewicz, Rothberger, Seirpi\'nski in 1920--1930's, a systematic
investigation in this area rapidly increased and attracted a big
number of mathematicians in the last two-three decades after
Scheeper's paper \cite{coc1}. Nowadays, this theory has deep
connections with many branches of mathematics such as Set theory
and General topology, Game theory, Ramsey theory, Function spaces
and hyperspaces, Cardinal invariants, Dimension theory, Uniform
structures, Topological groups and relatives, Karamata theory.
Researchers working in this area have organized four international
mathematical forums called ``Workshop on Coverings, Selections and
Games in Topology". There are several survey papers about
selection principles theory (see, for example,
\cite{koc-iransurvey, koc-proc-steklov, sakai-marion} and the
paper \cite{boaz-problems} for open problems).

\medskip
Two basic ideas in this theory are simple and may be described by
the following two schemes:

\smallskip
\noindent {\bf Scheme 1:} To a topological property $\mathcal P$
associate selectively $\mathcal P$ as follows:

$\mathcal P$: for each $A$ there is a $B$ such that ...

${\rm selectively} \mathcal P$: For each sequence $\langle
A_n:n\in \naturals \rangle$ there is a sequence $\langle B_n:n\in
\naturals\rangle$ such that ...

\medskip
\noindent {\bf Scheme 2:} $\mathcal A$ and $\mathcal B$ are given
collections, $\pi$ is a procedure of selection. Apply $\pi$ to
$\mathcal A$ to arrive to $\mathcal B$.

\medskip
For example, if $\mathcal P$ is compactness (for each open cover
$\mathcal U$ of a space $X$ there is a finite subcover $\mathcal
V$), then selectively $\mathcal P$ is defined as follows: for each
sequence $\langle \mathcal U_n;n\in \naturals\rangle$ of open
covers of $X$ there is a sequence $\langle \mathcal V_n:n\in
\naturals\rangle$ of finite sets with $\mathcal V_n \subset
\mathcal U_n$, $n\in\mathbb N$, and $\bigcup_{n\in\mathbb
N}\mathcal V_n$ covers $X$. This property is called the Menger
property (see below).

Many other selective versions of classical topological concepts
have been defined in this way.

\medskip
Three classical selection principles defined in general forms in
\cite{coc1} are:

\medskip
Let ${\mathcal A}$ and ${\mathcal B}$ be sets consisting of
families of subsets of an infinite set $X$. Then the following
selection hypothesis are defined:

\smallskip
\noindent $\sfin({\mathcal A},{\mathcal B})$:  for each sequence
$\langle A_n:n\in\naturals\rangle$ of elements of ${\mathcal A}$
there is a sequence $\langle B_n:n\in\naturals\rangle$  of finite
sets such that for each $n$, $B_n\subset A_n$, and
$\bigcup_{n\in\naturals}B_n\in {\mathcal B}$.

\smallskip
\noindent  $\sone({\mathcal A},{\mathcal B})$: for each sequence
$\langle A_n:n\in\naturals\rangle$ of elements of ${\mathcal A}$
there is a sequence $\langle b_n:n\in\naturals\rangle $ such that
for each $n$, $b_n\in A_n$, and $\{b_n:n\in\naturals\}$ is an
element of ${\mathcal B}$.

\smallskip
\noindent $\ufin({\mathcal A},{\mathcal B})$: for each sequence
$\langle A_n:n\in\naturals\rangle$  of elements of ${\mathcal A}$
there is a sequence $\langle B_n:n\in\naturals\rangle $ such that
for each $n$, $B_n$ is a finite subset of $A_n$ and $\{\bigcup
B_n:n\in\naturals\} \in \mathcal B$.

\medskip
In this paper we use the following notation for collections of
covers of a topological space $X$:
\begin{itemize}
\item $\mathcal O$ is the collection of all open covers of $X$;

\item $\Omega$ is the collection of $\omega$-covers of $X$. An
open cover $\mathcal U$ of $X$ is said to be an
\emph{$\omega$-cover} if each finite subset of $X$ is contained in
a member of $\mathcal U$ and $X\notin \mathcal U$;

\item $\Gamma$ denotes the collection of $\gamma$-covers of $X$.
An open cover $\mathcal U$ of $X$ is said to be a
\emph{$\gamma$-cover} if each point of $X$ does not belong to at
most finitely many elements of $\mathcal U$.
\end{itemize}

Then:
\begin{itemize}
\item[{\sf M}:] $\sfin(\mathcal O,\mathcal O)$ is the \emph{Menger
property} \cite{menger}, \cite{hurewicz};

\item[{\sf R}:]  $\sone(\mathcal O,\mathcal O)$ is the
\emph{Rothberger property} \cite{rothberger};

\item[{\sf H}:] $\ufin(\Gamma,\Gamma)$ is the \emph{Hurewicz
property} \cite{hurewicz}
\end{itemize}

\medskip
The paper is organized in the following way. Immediately after
this introduction in Section 2 we give information about
terminology and notation, and also about known topological
constructions we use in this paper. In Section 3 we discuss in
details star selection principles in topological spaces. The next
two sections are devoted to neighbourhood and absolute star
selection properties, two variations of the properties we
considered in Section 3. In particular, in Subsection 5.2 we
report results on selectively $(a)$ spaces.  In the second part of
the paper we turn attention to appearance of star selection
properties in special classes of topological structures: uniform
and quasi-uniform spaces, and, especially, in topological and
paratopological groups. Each section contains some open problems
which can motivate new researches for work in this field.


%%%%%%% 22222 %%%%%%%%%%%%%%%%%%%%%%%

\section{Definitions and terminology}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip
Throughout the paper ``space" means ``topological space". By
$\mathbb N$, $\mathbb Z$, and $\mathbb R$ we denote the set of
natural numbers, integers, and real numbers, respectively. The
symbol $\omega$ denotes the set of nonnegative integers and also
the first infinite ordinal, while $\omega_1$ is the first
uncountable ordinal. The cardinality of continuum is denoted by
$\mathfrak c$, and {\sf CH} denotes the Continuum Hypothesis. Most
of undefined notations and terminology are as in \cite{engelking}.

If $X$ is a space, $\mathcal K$ a collection of subsets of $X$,
$A$ a subset of $X$, and $x\in X$, then  ${\rm St}(A,\mathcal K)$
is the union of all elements in $\mathcal K$ which meet $A$. We
write ${\rm St}(x,\mathcal K)$ instead of ${\rm St}(\{x\},\mathcal
K)$.

\medskip
We recall known topological constructions which will be used in
next sections without special mention.

\smallskip
{\bf A.} ({\bf The Baire space $^{\omega}\omega$}) \, Let
$^{\omega}\omega$ be the set of all functions $f:\omega \to
\omega$ (in fact, the countable Tychonoff power of the discrete
space $D(\omega)$).  A natural pre-order $\prec\sp *$ on
$^{\omega}\omega$ is defined by $f\prec\sp * g$ if and only if
$f(n) \le g(n)$ for all but finitely many $n$. A subset $F$ of
$^{\omega}\omega$ is said to be \emph{dominating} if for each
$g\in\,^{\omega}\omega$ there is a function $f\in F$ such that
$g\prec\sp * f$. A subset $F$ of $^{\omega}\omega$ is called
\emph{bounded} if there is an $g\in\,^{\omega}\omega$ such that
$f\prec\sp * g$ for each $f\in F$. The symbol $\mathfrak b$ (resp.
$\mathfrak d$) denotes the least cardinality of an unbounded
(resp. dominating) subset of $^{\omega}\omega$. Another
uncountable small cardinal characterized (by Bartoszy\'nski in
1987) in terms of subsets of $^{\omega}\omega$ is the cardinal
${\sf cov}(\mathcal M)$, the \emph{covering number of the ideal of
meager subsets of $\mathbb R$}:
\[
{\sf cov}(\mathcal M) = \min\{|F|:F\subset {^\omega}{\omega}
\mbox{ such that } \forall g\in {^\omega}{\omega} \ \exists f\in F
\mbox{ with } f(n)\neq g(n) \forall n\in\omega\}.
\]

Recommended literature concerning uncountable small cardinals is
\cite{vanDouwen-handbook} and \cite{vaughan}.

\medskip
{\bf B.} ({\bf $\Psi$-spaces}) \, A family $\mathcal A$ of
infinite subsets of $\naturals$ is called \emph{almost disjoint}
if the intersection of any two distinct sets in $\mathcal A$ is
finite.

Let $\mathcal A$ be an almost disjoint family. The symbol $\Psi
(\mathcal A)$ denotes the space $\naturals \cup \mathcal A$ with
the following topology: all points of $\naturals$ are isolated; a
basic neighborhood of a point $A$ in $\mathcal A$ is of the form
$\{A\} \cup (\naturals \setminus F)$, where $F$ is a finite subset
of $\naturals$.


\medskip
{\bf C.} ({\bf Pixley-Roy space}) \, For a space $X$, let ${\sf
PR}(X)$ be the space of all nonempty finite subsets of $X$ with
the Pixley-Roy topology \cite{vanDouwen-PR}: for $A \in �{\sf
PR}(X)$ and an open set $U \subset X$, let $[A,U] = \{B \in {\sf
PR}(X) : A \subset B \subset U\}$; the family $\{[A,U] : A
\in�{\sf PR}(X), U \mbox{ open  in } X\}$ is a base for the
Pixley-Roy topology.

Obviously $\{\{x\} : x \in X\}$ is closed and discrete in ${\sf
PR}(X)$. Therefore, ${\sf PR}(X)$ is Lindel\"of if and only if $X$
is countable. It is known that (1) for a $T_1$-space $X$, ${\sf
PR}(X)$ is always zero-dimensional, Tychonoff and hereditarily
metacompact, and (2) ${\sf PR}(X)$ is developable if and only if
$X$ is first-countable (see \cite{vanDouwen-PR}).

\medskip
{\bf D.} ({\bf Alexandroff duplicate}) \, Let $(X,\tau)$ be a
topological space. The Alexandroff duplicate of $X$ (see
\cite{engelking}, \cite{agata-watson}) is the set ${\sf AD}(X):=
X\times \{0,1\}$ equipped with the following topology. For each
$U\in \tau$ let $\widehat{U} = U\times \{0,1\}$. Define a base for
a topology on ${\sf AD(X)}$ by $\mathcal B = \mathcal B_0 \cup
\mathcal B_1$, where $\mathcal B_0$ is the family of all sets
$\widehat{U} \setminus (F\times \{1\}) \subset {\sf AD}(X)$, with
$U\in \tau$ and $F$ a finite subset of $X$, and $\mathcal B_1
=\{\langle x,1\rangle :x\in X\}$. For every $x\in X$ put $\tau_x=
\{U\in \tau: x\in U\}$ and $\mathcal B_{\langle x,0\rangle } =
\{\widehat{U} \setminus \{\langle x,1\rangle \}:U\in \tau_x\}$,
and $\mathcal B_{\langle x,1\rangle} = \{\{\langle x,1\rangle
\}\}$. Then, if $X$ is a $T_1$-space, $\mathcal B_{\langle
x,0\rangle }$ is a local base at each $\langle x,0\rangle \in {\sf
AD}(X)$, and $\mathcal B^{\prime} = \bigcup_{x\in X}(\mathcal
B_{\langle x,0\rangle } \cup B_{\langle x,1\rangle })$ is a base
in ${\sf AD}(X)$ such that $\mathcal B^{\prime} \subset \mathcal
B$. If $\mathcal U$ is a family of open sets in $X$, then we say
that the family $\mathcal U^{\ast}:= \{\widehat{U} \setminus
(F\times \{1\}):U\in \mathcal U, F \mbox{ a finite subset of }
X\}$ of open subsets of ${\sf AD}(X)$ is associated to $\mathcal
U$ and vice versa.

\smallskip
For many topological properties $\mathcal P$ the space ${\sf
AD}(X)$ has $\mathcal P$ if $X$ has $\mathcal P$ (see, for
example, \cite{agata-watson}). Such properties are, for instance,
complete regularity, normality, compactness, Lindel\"ofness,
(hereditary) paracompactness.

\smallskip
Recall also the definition of subspaces (called lines) of ${\sf
AD}(X)$. Let $A$ and $B$ be disjoint subspaces of $X$. The
subspace $Z= (A\times \{1\}) \cup (B\times \{0\})$ of ${\sf
AD}(X)$ is called a \emph{Michael-type line} (see \cite[Definition
3.14]{agata-watson}).


%%%%%%%%%%%%%%%%%%%%%% 33333 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Star selection principles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In \cite{fle} it was proved that a Hausdorff space $X$ is
countably compact if and only if for every open cover $\mathcal U$
of $X$ there exists a finite subset $F\subset X$ such that ${\rm
St}(F,\mathcal U)=X$.

This result was a motivation for the following two definitions
that appeared in \cite{vDRRT}.

A space $X$ is \emph{starcompact} if for every open cover
$\mathcal U$ of $X$ there exists a finite subset $\mathcal V$ of
$\mathcal U$ such that ${\rm St}(\cup \mathcal V,\mathcal U)=X$.

A space $X$ is \emph{strongly starcompact} if for every open cover
$\mathcal U$ of $X$ there exists a finite subset $F\subset X$ such
that ${\rm St}(F,\mathcal U)=X$.

\medskip
Applying now Schemes 1 and 2 we define selective versions of these
notions, and modifying them we obtain the following star selection
principles introduced by the author of this article in
\cite{koc-starmenger} (see also \cite{koc-starmenger2}).

\medskip
Let ${\mathcal O}$ be the collection of all open covers of a space
$X$, ${\mathcal B}$ a subcollection of $\mathcal O$, and $\mathcal
K$ a family of subsets of $X$. Then:

\smallskip
\noindent {\bf 1.} The symbol $\ssfin(\mathcal O,\mathcal B)$
denotes the selection hypothesis:
\begin{quote}
For each sequence $\langle \mathcal U_n:n\in\naturals\rangle $ of
elements of $\mathcal O$ there is a sequence $\langle \mathcal
V_n:n \in \naturals\rangle $ such that for each $n\in \naturals$,
$\mathcal V_n$ is a finite subset of $\mathcal U_n$, and $\{{\rm
St}(\cup \mathcal V_n,\mathcal U_n):n \in \naturals\} \in \mathcal
B$;
\end{quote}

\noindent {\bf 2.} The symbol $\ssone(\mathcal O,\mathcal B)$
denotes the selection hypothesis:
\begin{quote}
For each sequence $\langle \mathcal U_n:n\in\naturals\rangle $ of
elements of $\mathcal O$ there is a sequence $\langle U_n:n \in
\naturals\rangle $ such that for each $n\in \naturals$, $U_n
\in\mathcal U_n$ and $\{{\rm St}(U_n,\mathcal U_n):n\in\naturals\}
\in \mathcal B$;
\end{quote}

\noindent {\bf 3.} ${\sf SS}_{\mathcal K}^{*}(\mathcal O,\mathcal
B)$ denotes the selection hypothesis:
\begin{quote}
For each sequence $\langle \mathcal U_n:n\in \naturals\rangle $ of
elements of $\mathcal O$ there exists a sequence $\langle K_n:n
\in \naturals\rangle $ of elements of $\mathcal K$ such that
$\{{\rm St}(K_n, \mathcal U_n):n \in \naturals\} \in\mathcal B$.
\end{quote}
\noindent When $\mathcal K$ is the collection of all finite (resp.
one-point, compact) subspaces of $X$ we write $\sssfin(\mathcal
O,\mathcal B)$ (resp., $\sssone(\mathcal O,\mathcal B)$, ${\sf
SS}_{{\rm K}}^{*}(\mathcal O,\mathcal B)$) instead of ${\sf
SS}_{\mathcal K}^{*}(\mathcal O,\mathcal B)$.

\medskip
The following terminology we borrow from \cite{koc-starmenger}.
\smallskip
\noindent For a space $X$ we have:

\noindent \ \ {\sf SM}: the \emph{star-Menger property} =
$\ssfin(\mathcal O,\mathcal O)$;

\noindent \ \ {\sf SR}: the \emph{star-Rothberger property}  =
$\ssone(\mathcal O,\mathcal O)$;

\noindent \ {\sf SSM}: the \emph{strongly star-Menger property} =
$\sssfin(\mathcal O,\mathcal O)$;

\noindent \ {\sf SSR}:  the \emph{strongly star-Rothberger
property} = $\sssone(\mathcal O,\mathcal O)$;

\noindent {\sf SS-K-M}: the \emph{star-$K$-Menger property} =
${\sf SS}_{{\rm K}}^{*}(\mathcal O,\mathcal O)$.

\medskip
In \cite{star-hur}, two star versions of the Hurewicz property
were studied:

\smallskip
\noindent \ \ {\sf SH}: the \emph{star-Hurewicz property} =
$\ssfin(\mathcal O,\Gamma)$;

\noindent \ {\sf SSH}: the \emph{strongly star-Hurewicz property}
= $\sssfin(\mathcal O,\Gamma)$.

\medskip
It is clear that each of properties {\sf SM}, {\sf SH}, {\sf SR}
can be viewed as a selective version of starcompactnes, while the
properties {\sf SSM}, {\sf SSH}, {\sf SSR}, {\sf SS-K-M} can be
viewed as selective versions of strong starcompactness.
Starcompctness implies {\sf SH}, hence also {\sf SM}, and strong
starcompactness implies {\sf SSH} and thus {\sf SSM}. In
\cite[Example 2.3]{koc-starmenger} we have shown that the
Tychonoff Plank $[0,\omega_1]\times [0,\omega] \setminus
\{\langle\omega_1,\omega\rangle \}$ is {\sf SSM} but not strongly
starcompact. On the other hand, in \cite[Example 2.1]{song-polish}
it is proved that the Tychonoff Plank is {\sf SSH} but not
starcompact (thus not strongly starcompact). It is worth to
mention that for each ordinal $\alpha$, the space $[0,\alpha)$
with the order topology is ${\sf SSR}$.

Of course, Menger spaces are {\sf SSM}, and every {\sf SSM} space
is {\sf SM}. Similarly for the Hurewicz and Rothberger properties.

The simplest example which shows that the converse need not be
true is the ordinal space $[0,\omega_1)$ which is ${\sf SSH}$
(hence {\sf SSM}, {\sf SH}, {\sf SM}) but not {\sf M} (thus not
{\sf H}) (see \cite{koc-starmenger} and \cite{star-hur}).

By results in \cite{koc-starmenger} and \cite{star-hur} we have
that every metacompact (every open cover $\mathcal U$ has a
point-finite open refinement $\mathcal V$) strongly star-Menger
space is Menger, and that for paracompact Hausdorff spaces the
three Menger-type properties, ${\sf SM}$, ${\sf SSM}$ and ${\sf
M}$ are equivalent \cite{koc-starmenger}. The same situation is
with the classes ${\sf SSR}$, ${\sf SR}$ and ${\sf R}$
\cite{koc-starmenger} and ${\sf SSH}$, ${\sf SH}$ and ${\sf H}$
\cite{star-hur}.

Let us mention the following

\begin{example} \rm (\cite[Example 2.2]{song-polish})\label{sh-not-ssh} There is a Tychonoff ${\sf SH}$
space which is not ${\sf SSH}$.

\smallskip
Such a space is $\alpha D(\mathfrak{c}) \times [0,\mathfrak{c}^+]
\setminus \{\langle\infty, \mathfrak{c}^+\rangle\}$ of the product
$\alpha D(\mathfrak{c}) \times [0,\mathfrak{c}^+]$, where $\alpha
D(\mathfrak{c}) = D(\mathfrak{c})\cup\{\infty\}$ is the one-point
compactification of the discrete space $D(\mathfrak{c})$ of
cardinality $\mathfrak{c}$.
\end{example}

\medskip
Following the general definition of ${\sf SS}^{*}_{\mathcal
K}(O,O)$ (the beginning of this section) and taking $\mathcal K$
to be the collection of countably compact spaces Song defined
star-$C$-Menger spaces in \cite{song-c-menger} (he also studied
star-$K$-Menger spaces in \cite{song-star-k-menger}). He proved:

\begin{example} \rm (\cite[Example 2.2]{song-c-menger}) There exists a Tychonoff
star-$C$-Menger space which is not star-$K$-Menger.
\end{example}



\smallskip Now we are going to see how above mentioned star
selection properties are related to $\Psi$-spaces and Pixley-Roy
spaces. In fact, in $\Psi$-spaces $\Psi (\mathcal A) = \omega \cup
\mathcal A$ star selection properties strongly depend on the
cardinality of the almost disjoint family $\mathcal A$ and are
related to small infinite cardinals. The first results of this
kind appeared in the preprint/draft \cite{matveev-preprint} sent
me by the author in July 1998 (see \cite[Example
2.2]{koc-starmenger} and \cite{star-hur}), and then included in
the paper \cite{misha-milena-psi}. By combining the results from
\cite{matveev-preprint} and \cite{misha-milena-psi} we can
formulate the following

\begin{theorem} The following hold for a $\Psi$-space
$\Psi(\mathcal A)$:
\begin{itemize}
\item[$(1)$] $\Psi(\mathcal A)$ is {\sf SSM} if and only if
$|\mathcal A| < \mathfrak d$. If $|\mathcal A|=\mathfrak c$, then
$\Psi(\mathcal A)$ is not {\sf SM}, and if $\mathcal A| <
\aleph_{\omega}$, then $\Psi(\mathcal A)$ is {\sf SM} if and only
if it is {\sf SSM};

\item[$(2)$] $\Psi(\mathcal A)$ is {\sf SSH} if and only if
$|\mathcal A| < \mathfrak b$;

\item[$(3)$] If $|\mathcal A| < {\sf cov}(\mathcal M)$, then
$\Psi(\mathcal A)$ is {\sf SSR}. There is an almost disjoint
family $\mathcal A$ of cardinality  ${\sf cov}(\mathcal M)$ such
that $\Psi(\mathcal A)$ is not {\sf SSR}.
\end{itemize}
\end{theorem}

In \cite{sakai-SM}, Sakai investigated star-Mengerness in the
Pixley-Roy space. He established the following:

\begin{theorem} $(1)$ If ${\sf PR}(X)$ is star-Menger, then $|X| < \mathfrak{c}$
holds. Hence, under ${\sf CH}$, ${\sf PR}(X)$ is star-Menger if
and only if $X$ is countable;

$(2)$  If ${\sf PR}(X)$ is star-Menger, then every finite power of
$X$ is Menger.

$(3)$ If $X$ is a cosmic space of cardinality less than
$\mathfrak{d}$, then every finite power of ${\sf PR}(X)$ is
star-Menger;

$(4)$ Let $X$ be a semi-stratifiable space {\rm \cite{creede}}. If
\ ${\sf PR}(X)$ is star-Menger, then ${\sf PR}(X)^{\kappa}$ is
weakly Menger for any cardinal $\kappa$;

$(5)$ If $X$ is first-countable and ${\sf PR}(X)$ is star-Menger,
then ${\sf PR}(X)$ is weakly Menger.
\end{theorem}

A space $X$ is said to be \emph{weakly Menger} \cite{daniels} if
for each sequence $\langle \mathcal U_n:n\in\naturals\rangle$ of
open covers of $X$ there is a sequence $\langle \mathcal
V_n:n\in\naturals\rangle$ of finite sets such that for each $n$,
$\mathcal V_n\subset \mathcal U_n$ and
$\overline{\bigcup_{n\in\naturals}\bigcup \mathcal V_n} = X$.

\bigskip
Since the very beginning of the theory of star selection
principles one the following question was one of the most
interesting: how large the extent of {\sf SM} or {\sf SSM} spaces
can be. Recall that the \emph{extent} $e(X)$ of a space $X$ is the
supremum of cardinalities of closed discrete subspaces of $X$.
Recently, some results in this connection have been obtained by
Y.-K. Song \cite{song-cmuc} and M. Sakai \cite{sakai-SM}, and also
by B. Tsaban \cite{boaz-arxiv}.

Song \cite[Example 2.4]{song-cmuc} observed that the extent of a
$T_1$ strongly star-Menger space can be arbitrarily large, and
asked whether there is a Tychonoff strongly star-Menger space $X$
such that $e(X) \ge \mathfrak{c}$. Answering this question, Sakai
proved in \cite[Corollaries 2.2, 2.6]{sakai-SM}:

\begin{theorem} $(1)$ The extent of a regular strongly star-Menger space
cannot exceed $\mathfrak{c}$;

$(2)$ If $X$ is a star-Menger space with $w(X) = \mathfrak{c}$,
then every closed and discrete subspace of X has cardinality less
than $\mathfrak{c}$;

$(3)$ Let $X$ be a normal star-Menger space. Then $e(X) \le
\mathfrak{d}$;

$(4)$ The assertion every developable strongly star-Menger space
is separable and metrizable is equivalent to $\omega_1 =
\mathfrak{d}$;

$(5)$ The statements $\omega_1 = \mathfrak{d}$ is equivalent to
the statement that for every strongly star-Menger space $X$, $e(X)
\le \omega$ holds.
\end{theorem}


The following problem was posed by Sakai.

\begin{problem} {\rm (\cite[Question 3.3]{sakai-SM})} Can the extent of a metacompact
(or, subparacompact) star-Menger space be arbitrarily large
\end{problem}

\medskip
Another interesting question regarding star selection principles
is their relations with the Alexandroff double. Some of results in
this direction are listed below.

1. (\cite[Corollary 2.9]{song-cmuc}) If $X$ is an {\sf SSM}
$T_1$-space, then ${\sf AD}(X)$ is {\sf SSM} if and only if $e(X)
< \omega_1$.

2.  It was observed in \cite{song-polish} that the Alexandroff
double of the {\sf SH} space in Example \ref{sh-not-ssh} is not
{\sf SH}.

3. (\cite[Theorem 2.4]{song-ssh})  If $X$ is a $T_1$-space and
${\sf AD}(X)$ is an {\sf SSH} space, then $e(X) < \omega_1$.

The last result suggests the following problem.

\begin{problem} Is the Alexandorff
duplicate ${\sf AD}(X)$ of an {\sf SSH} space $X$ with $e(X) <
\omega_1$ also {\sf SSH}
\end{problem}

\subsection{Operations}

Most of star selection properties are not hereditary. Even more,
they are not preserved by nice subspaces such as (regular) closed.
It was proved for {\sf SM} and {\sf SSM} spaces in
\cite{song-houston-sm}, for {\sf SH} and {\sf SSH} spaces in
\cite{song-li}, and for {\sf SR} and {\sf SSR} spaces in
\cite{song-sroth}.

Let us formulate once again a still open question from
\cite{koc-starmenger}.

\begin{problem} Characterize hereditarily {\sf SM} ({\sf SSM}, {\sf SR}, {\sf SSR}, {\sf SH}, {\sf SSH})
spaces.
\end{problem}

There are some partial answers to this question. For example, {\sf
SSM} and {\sf SSH} spaces are preserved by open $F_{\sigma}$-sets
(see \cite{song-houston-sm} and  \cite{song-li}, respectively),
while {\sf SSR} property is preserved by clopen subspaces
\cite{song-sroth}.

\medskip
It is known and easy to prove that continuous mappings preserve
{\sf SSM}, {\sf SH}, and {\sf SSH}) spaces (see \cite{song-cmuc},
\cite{song-polish}, \cite{song-ssh}, respectively).

Open and closed finite-to-one mappings preserve {\sf SSM} and {\sf
SSH} spaces (\cite{song-cmuc} and \cite{song-ssh}) in the preimage
direction, while open, perfect mappings preserve {\sf SH} spaces
in the preimage direction \cite{song-polish}. On the other hand,
it was proved in \cite{song-ssh} that assuming $\mathfrak b =
\mathfrak c$ and �$\neg {\sf CH}$, there exists a closed 2-to-1
continuous mapping $f : X �\to Y$ such that $Y$ is {\sf SSH}, but
$X$ is not.

\medskip
The product of two {\sf SM} (resp. {\sf SH}) spaces need not be in
the same class. For {\sf SSH} spaces, for instance, it was shown
in \cite{song-ssh}. But if one factor is compact, then the product
is in the same class \cite{koc-starmenger}, \cite{star-hur}.
Similarly, the product a star-$C$-Menger space and a compact space
is also star-$C$-Menger \cite{song-c-menger}. However, under
$\mathfrak b = \mathfrak c$ and $\neg CH$, there exist an {\sf
SSH} space $X$ and a compact space $Y$ such that $X \times Y$ is
not {\sf SSH} \cite{song-ssh}.

Let us observe that a Lindel\"of space is not a preserving factor
for the classes ${\sf SSM}$ and ${\sf SSH}$ \cite{koc-starmenger,
star-hur}.

The following question is an open problem.

\begin{problem} {\rm (\cite{song-ssh})} Do there exist a {\sf ZFC}
example of an {\sf SSH} space $X$ and a compact space $Y$ such
that $X\times Y$ is not {\sf SSH}
\end{problem}

In \cite{koc-starmenger} we posed the following still open
problem.

\begin{problem} Characterize spaces $X$ which are
{\sf SM} ({\sf SSM}, {\sf SR}, {\sf SSR}) in all finite powers.
\end{problem}

A partial solution of this problem was given in \cite{star-hur}.

\begin{theorem}\label{smpowers} The following statements hold:
\begin{itemize}
\item[$(1)$] If each finite power of a space $X$ is ${\sf SM}$,
then $X$ satisfies $\ssfin(\mathcal O,\Omega)$;

\item[$(2)$] If all finite powers of a space $X$ are {\sf SSM},
then $X$ satisfies $\sssfin(\mathcal O,\Omega)$.
\end{itemize}
\end{theorem}

In the same paper we have the following two assertions. (We remind
the reader that the symbol $\mathcal O^{wgp}$ denotes the
collection of weakly groupable covers of a space.  A countable
open cover $\mathcal U$ of a space $X$ is said to be \emph{weakly
groupable} if there is a partition $\mathcal U =
\bigcup_{n\in\mathbb N}\mathcal U_n$ of $\mathcal U_n$ into
finite, pairwise disjoint subcollections, so that for each finite
subset $F$ of $X$ there is $n\in \mathbb N$ with $F \subset
\bigcup\mathcal U_n$.)

\begin{theorem}\label{ufin-wgp} For a space $X$ the following are equivalent:
\begin{itemize}
\item[$(1)$] $X$ satisfies $\ssfin(\mathcal O,\Omega)$;
\item[$(2)$] $X$ satisfies $\ssfin(\mathcal O,\mathcal O^{wgp})$.
\end{itemize}
\end{theorem}

\begin{theorem}\label{ssfin-wgr} For a space $X$ the following are equivalent:
\begin{enumerate}
\item[$(1)$] {$X$ satisfies $\sssfin(\mathcal O,\Omega);$}
\item[$(2)$] {$X$ satisfies $\sssfin(\mathcal O,\mathcal
O^{wgp})$.}
\end{enumerate}
\end{theorem}

So, we have actually the following problem.

\begin{problem}
Does $X \in \ssfin(\mathcal O,\mathcal O^{wgp})$ imply that all
finite powers of $X$ are star-Menger Is it true that
$\ssfin(\mathcal O,\Omega) = \ssfin(\mathcal O,\mathcal O^{wgp})$
Does $X \in \sssfin(\mathcal O,\mathcal O^{wgp})$ imply that each
finite power of $X$ is {\sf SSM}
\end{problem}

The following result was proved in \cite{star-hur}. First we
recall that $\mathcal O^{gp}$ denotes the collection of groupable
covers of a space. A countable open cover $\mathcal U$ of a space
$X$ is said to be \emph{groupable} if there is a partition
$\mathcal U = \bigcup_{n\in\mathbb N}\mathcal U_n$ of $\mathcal
U_n$ into finite, pairwise disjoint subcollections, so that each
$x\in X$ belongs to all but finitely many $\bigcup \mathcal U_n$.


\begin{theorem}\label{sshgroup} For a space $X$ the following are
equivalent:
\begin{enumerate}
\item[$(1)$]{$X$ has the strongly star-Hurewicz property;}
\item[$(2)$]{$X$ satisfies the selection principle
$\sssfin(\mathcal O, \mathcal O^{gp})$.}
\end{enumerate}
\end{theorem}

This result naturally suggests the following

\begin{problem} Is it true that  $\ssfin(\mathcal O,\Gamma)=
\ssfin(\mathcal O,\mathcal O^{gp})$
\end{problem}


Let us end this section by some comments.

\medskip
1. In this paper we did not consider connections between star
selection properties and games naturally associated to them.

[For example, the \emph{strongly star-Hurewicz game} illustrates
this situation; it is defined as follows. Let $X$ be a space. Two
players, ONE and TWO, play a round per each natural number $n$. In
the $n$--th round ONE chooses an open cover $\mathcal U_n$ of $X$
and TWO responds by choosing a finite set $A_n\subset X$. A play
$\mathcal U_1,A_1;\cdots;\mathcal U_n,A_n;\cdots$ is won by TWO if
$\{{\rm St}(A_n,\mathcal U_n):n \in \naturals\}$ is a
$\gamma$-cover of $X$; otherwise, ONE wins.

Evidently, if ONE has no winning strategy in the strongly
star-Hurewicz game, then $X$ is an {\sf SSH} space. But the
converse need not be true.]

It would be interesting to study these connections for all classes
we discussed in this section.

\medskip
2. We also did not discuss relative versions of star selection
principles (initiated by the author) that can be found in the
literature (see, for instance, \cite{star-hur},
\cite{bon-pans-relative}).

\medskip
3. Recently, I introduced  \emph{selection principles in relator
spaces} as generalizations of uniform selection principles. My PhD
student Kocev studied these selection properties in
\cite{darko-relator-amh}, \cite{darko-relator-filomat},
\cite{darko-doct}. We did not include these results in this survey
although there are many interesting results and open questions in
this connection.

4. Selection properties of fuzzy metric spaces \cite{koc-fuzzy}
are a kind of star selection properties.

%%%%%%%%%%%%%%%%%%%  44444 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Neighbourhood star selection principles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In this section we investigate star selection principles which are
very close to the already considered star selection properties,
but defined by neighbourhoods and stars. Selection properties
defined in this way are weaker than the Menger, Rothberger and
Hurewicz properties and are between strong star versions and star
versions of the corresponding properties. The definitions of these
selection principles were given in \cite[Def.
0.2]{koc-starmenger2}, and studied in details in \cite{bckm-mv}.
Our exposition here mainly follows the last mentioned paper.

\begin{definition} \rm
Let $\mathcal O$ and $\mathcal B$ be as in the previous section. A
space $X$ satisfies:

\item[${\sf NSM}({\mathcal O},{\mathcal B})$] if for every
sequence $\langle {\mathcal U}_n : n\in\mathbb N\rangle $ of
elements of $\mathcal A$ one can choose finite $A_n \subset X$,
$n\in\Bbb N$, so that for every open $O_n \supset A_n$,
$n\in\mathbb N$, $\{{\rm St}(O_n,{\mathcal U}_n) : n\in \mathbb
N\}\in\mathcal B$;

\item [${\sf NSR}({\mathcal O},{\mathcal B})$] if for every
sequence $\langle {\mathcal U}_n : n\in\mathbb N\rangle $ of
elements of $\mathcal A$ one can choose $x_n \in X$, $n\in\mathbb
N$, so that for every open $O_n \ni x_n$, $n\in\mathbb N$, $\{{\rm
St}(O_n,{\mathcal U}_n) : n\in \mathbb N\}\in\mathcal B$;

\item[${\sf NSH}({\mathcal O},{\mathcal B})$]if if for every
sequence $\langle {\mathcal U}_n : n\in\mathbb N\rangle $ of
elements of $\mathcal A$ one can choose finite $A_n \subset X$,
$n\in\mathbb N$, so that for every open $O_n \supset A_n$,
$n\in\mathbb N$, and for every $x\in X$, $x\in {\rm
St}(O_n,{\mathcal U}_n)$ for all but finitely many $n$.
\end{definition}

In particular we have the following definitions:

\begin{definition} \rm A space $X$ is:
\begin{itemize}
\item[${\sf NSM}$:] (\emph{neighbourhood star-Menger}) if the
selection hypothesis ${\sf NSM}(\mathcal O,\mathcal O)$ is true
for $X$;

\item[${\sf NSR}$:] (\emph{neighbourhood star-Rothberger}) if the
property ${\sf NSR}(\mathcal O,\mathcal O)$ is true for $X$;

\item[${\sf NSH}$:] (\emph{neighbourhood star-Hurewicz}) if the
selection hypothesis ${\sf NSH}({\mathcal O},{\Gamma})$ is true
for $X$.
\end{itemize}
\end{definition}

\medskip
\noindent{\bf Note.} ${\sf NSR}$ and ${\sf NSM}$ spaces (as well
as neighbourhood star-$K$-spaces) were defined in
\cite{koc-starmenger2} under different names (nearly strongly
star-Rothberger and nearly strongly star-Menger spaces).

\begin{remark}\rm
Since in the class of paracompact Hausdorff we have that  ${\sf R}
\Leftrightarrow {\sf SR}$, ${\sf M} \Leftrightarrow {\sf SM}$ (see
\cite{koc-starmenger}) and ${\sf H} \Leftrightarrow {\sf SH}$ (see
\cite{star-hur}), we have that in the class of paracompact
Hausdorff spaces all Rothberger-type properties, all Menger-type
properties and all Hurewicz-type properties considered are
equivalent respectively (see Diagram 1).
\end{remark}

The implications ${\sf NSM}\Rightarrow {\sf SM}$, ${\sf NSH}
\Rightarrow {\sf SH}$ and ${\sf NSR} \Rightarrow {\sf SR}$ can not
be reversed as the following example shows.

\begin{example} \rm (\cite[Example 3.7]{bckm-mv}) A Tychonoff space which is ${\sf SR}$ and ${\sf SH}$
(and thus {\sf SM}), but is neither of ${\sf NSR}$, ${\sf NSH}$,
${\sf NSM}$.
\end{example}

Such a space $X$ is constructed in the following way. Let $\kappa$
be an uncountable cardinal and $\alpha(D(\kappa)) = D(\kappa) \cup
\{\infty\}$ the one point compactification of the discrete space
$D(\kappa)$. Set $X_0= \alpha D(\kappa) \times [0,\kappa^+)$,
$X_1=D(\kappa) \times\{\kappa^+\}$, $X=X_0\cup X_1$. Endow $X$
with the topology inherited from the product $\alpha
D(\kappa)\times [0,\kappa^{+}]$.

\medskip
We show now that consistently, ${\sf NSM}$, ${\sf NSH}$ and {\sf
NSR} do not imply ${\sf SSM}$,  ${\sf SSH}$ and {\sf SSR},
respectively.

\begin{example}\rm (\cite[Examples 3.1--3.3]{bckm-mv}) \label{3.1-3.3} Let $S$ be a subset of
$\mathbb R$ such that $|S|=\omega_1$ and for every nonempty open
$U\subset \mathbb R$, $|S\cap U| = \omega_1$. Set $X_S =
S\times[0,\omega]$ topologized in the following way: (i) a basic
neighbourhood of a point $\langle x,n\rangle \in X_S$ has the form
$((U\cap S)\setminus A)\times \{n\}$, where $U$ is a neighbourhood
of $x$ in the usual topology of $\mathbb R$ and $A$ is a countable
set not containing $x$; (ii) a point $\langle x,\omega\rangle$,
$x\in S$, has basic neighbourhoods of the form $((U\cap
S)\setminus A)\times (n,\omega) \cup \{\langle x,\omega\rangle\}$,
where $U$ is a neighbourhood of $x$ in the usual topology of
$\mathbb R$, $A$ is a countable subset of $S$, and $n\in\omega$.
Then $X_S$ is a Urysohn space and:

\smallskip
(1) Under $\omega_1 < \mathfrak d$ the space $X_S$ is  an ${\sf
NSM}$ space which is not ${\sf SSM}$.

\smallskip
(2) Under $\omega_1 < \mathfrak b$, $X_S$ is an ${\sf NSH}$ space
which is not ${\sf SSH}$.

\smallskip
(3) Under $\omega_1 < {\sf cov}(\mathcal M)$, $X_S$ is an {\sf
NSR} space which is not {\sf SSR}.
\end{example}

The following problem is still open.

\begin{problem} {\rm (\cite[Problem 3.6]{bckm-mv})} Do there exist {\sf ZFC} examples
of spaces as in Example {\rm \ref{3.1-3.3}}
\end{problem}

%%%%%%%%%%%%%%%% 55555 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Absolute versions of selection principles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In \cite{matveev-acc} Matveev introduced the class of absolutely
countable compact spaces: A space $X$ is \emph{absolutely
countable compact} (shortly {\sf acc})  if for each open cover
$\mathcal U$ of $X$ and each dense subset $D$ of $X$ there is a
finite $A\subset D$ such that ${\rm St}(A,\mathcal U) = X$.

In his subsequent paper \cite{matveev97}, Matveev applied a
similar idea to introduce the following property: a space $X$ is
said to be an \emph{$(a)$-space} if for each open cover $\mathcal
U$ of $X$ and each dense subset $D$ of $X$ there is a closed
discrete (in $X$) set $A\subset D$ such that ${\rm St}(A,\mathcal
U) = X$. He also defined the class of $(wa)$-spaces replacing in
the previous definition ``closed discrete" by ``discrete". These
spaces were studied in a number of papers  \cite{gauldvamana},
\cite{justmatvszept}, \cite{rudinjerry}, \cite{samuel-a},
\cite{szeptvaughan}.

In 2010, we employed Matveev's idea to define selective versions
of several star selection principles in the following general form
(see \cite[p. 1361]{sel-(a)}).

\begin{definition} \rm (\cite{sel-(a)}) Let $\mathcal O$ and $\mathcal B$ be collections
of open covers of a space $X$ as mentioned above, and let
$\mathcal K$ be a collection of subsets of $X$. Then $X$ is said
to be a \emph{selectively $(\mathcal O,\mathcal B)$-$(a)_{\mathcal
K}$-space}, denoted by $X\in {\sf Sel}(\mathcal O,\mathcal
B)$-$(a)_{\mathcal K}$, if for each sequence $\langle \mathcal
U_n:n\in\naturals\rangle $ of elements of $\mathcal O$ and each
dense subset $D$ of $X$ there is a sequence $\langle K_n:n\in
\naturals\rangle$ of elements of $\mathcal K$ such that each $A_n$
is a subset of $D$ and $\{{\rm St}(K_n,\mathcal U_n):n\in
\naturals\} \in \mathcal B$.
\end{definition}

In this definition we have the following classes of spaces:

\smallskip
(1) selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm
finite}}$-spaces are called \emph{absolutely strongly star-Menger
spaces} (shortly {\sf ASSM} spaces), which form a subclass of
${\sf SSM}$-spaces;

(2) selectively $(\mathcal O, \Gamma)$-$(a)_{{\rm finite}}$ spaces
are \emph{absolutely strogly star-Hurewicz spaces} (shortly  =
{\sf ASSH} spaces), which form a subclass of {\sf SSH} spaces;,

(3) ${\sf Sel}(\mathcal O,\mathcal O)$-$(a)_{{\rm singleton}}$ is
the class of \emph{absolutely strongly Rothberger spaces} ({\sf
ASSR} spaces for short), a subclass of the class {\sf SSR};

(4)  For a space $X$ satisfying ${\sf Sel}(\mathcal O,\mathcal
O)$-$(a)_{\rm closed \, discrete}$ we say that $X$ is a
\emph{selectively $(a)$-space}, and this is a direct
generalization of the notion of $(a)$-spaces. This class of spaces
will be discussed in a separate subsection of this section.

\medskip
The following diagram shows relationships among the classes of
spaces that we have defined so far. Let us mention that arrows in
this diagram are not reversible; for some of them it was already
demonstrated by examples in the previous sections, and for some
other it will be done in what follows.

\newpage
\begin{center}
\begin{picture}(260,260)

\put(30,25){{\sf SR}} \put(55,28){\vector(1,0){35}}

\put(107,25){{\sf SM}} \put(170,28){\vector(-1,0){35}}

\put(180,25){{\sf SH}}

\put(30,105){{\sf NSR}} \put(35,100){\vector(0,-1){55}}
\put(55,108){\vector(1,0){35}}

\put(107,105){{\sf NSM}} \put(112,100){\vector(0,-1){55}}
\put(170,108){\vector(-1,0){35}}

\put(180,105){{\sf NSH}} \put(185,100){\vector(0,-1){55}}

\put(30,185){{\sf SSR}} \put(55,188){\vector(1,0){35}}
\put(35,180){\vector(0,-1){55}}

\put(107,185){{\sf SSM}} \put(170,188){\vector(-1,0){35}}
\put(112,180){\vector(0,-1){55}}

\put(180,185){{\sf SSH}} \put(185,180){\vector(0,-1){55}}

\put(233,185){{\sf ASSH}} \put(225,188){\vector(-1,0){25}}
%\put(233,125){{\sf SSC}} \put(238,120){\vector(0,-1){120}}
%\put(233,-20){{\sf SC}}\put(228,145){\vector(-1,1){35}}
%\put(230,-10){\vector(-1,1){30}}

\put(30,245){{\sf R}} \put(35,240){\vector(0,-1){35}}
\put(55,248){\vector(1,0){35}}

\put(107,245){{\sf M}} \put(112,240){\vector(0,-1){35}}
\put(170,248){\vector(-1,0){35}}

\put(180,245){{\sf H}} \put(185,240){\vector(0,-1){35}}

\put(-20,185){{\sf ASSR}} \put(10,188){\vector(1,0){15}}
\put(-17,180){\vector(0,-1){40}}

\put(-20,125){{\sf ASSM}} \put(0,140){\vector(2,1){85}}
%\put(5,120){\vector(0,-1){60}}


%\put(0,45){{\sf NSL}} \put(110,100) {\vector(-2,-1){85}}
%\put(5,40){\vector(0,-1){57}}

%\put(5,-30){{\sf SL}} \put(110,20){\vector(-2,-1){85}}

\end{picture}\\
{\sf Diagram 1: Star selection properties}
\end{center}

%\newpage
\subsection{{\sf ASSM, ASSH, ASSR} spaces}

In this subsection we review very few basic results and examples
concerning {\sf ASSM, ASSH, ASSR} spaces. We begin with some
examples.

\begin{example} \rm (1) The Tychonoff plank is a Tychonoff {\sf
ASSM} which is not {\sf acc}.

\smallskip
(2) (\cite[Examples 2.1]{song-cejm}) There exists a Tychonoff {\sf
ASSH} space $X$ which is not {\sf acc}.

Let $X = [0,\omega]\times [0,\omega] \setminus \{(\omega,\omega\}$
as a subspace of the product $[0, \omega]\times [0,\omega]$.

\smallskip
(3) (\cite[Example 2.2]{song-cejm}) There exists a Tychonoff {\sf
SSH} space $X$ which is not {\sf ASSH}.

Such a space is $X= [0,\omega_1)\times[0,\omega_1]$.
\end{example}

Here are some properties of absolute star selection properties.

First, similarly to other star selection properties, these
properties are not hereditary.

In \cite{song-assm} and \cite{song-cejm}, it is proved that in the
class of Thychonoff spaces  {\sf ASSM} and {\sf ASSH} properties
are not preserved by regular-closed $G_{\delta}$-subspaces.

\medskip
Song noticed also that {\sf ASSM} and {\sf ASSH} properties are
not invariants of continuous mappings. But he proved that these
two properties, similarly to the {\sf acc} property
\cite{matveev-acc}, are preserved by continuous varpseudoopen
mappings. Recall that a continuous mapping $f: X \to Y$ is
\emph{varpseudoopen} provided ${\rm int}_Y (f(U))\neq\emptyset$
for every nonempty open set $U$ of $X$.


\smallskip
Theorem 2.15 in \cite{song-cejm} states that if the the product of
two spaces is {\sf ASSH}, then both spaces are {\sf ASSH}. On the
other hand, in difference of some other star selection properties,
the product of an {\sf ASSM} or {\sf ASSH} space $X$ and a compact
space $Y$ need not be {\sf ASSM} or {\sf ASSH} as it was observed
in  \cite{song-assm} and \cite{song-cejm}. For both cases the
product $[0,\omega_1) \times [0,\omega_1]$ can serve as an
example.

Matveev showed that the product of a Hausdorff {\sf acc} space and
a first countable compact space is {\sf acc} (see \cite[Theorem
2.3]{matveev-acc}. So, it is naturally to ask:

\begin{problem} {\rm (Song)} Is the product of an {\sf ASSH} space
and a first countable compact space also {\sf ASSH}
\end{problem}

Let us finish with the following fact \cite[Theorem
3.8]{song-assm}: If $X$ is an {\sf ASSM} space with $e(X) <
\omega_1$, then ${\sf AD}(X)$ is {\sf ASSM}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Selectively $(a)$ and related spaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The importance of property $(a)$ was established in the
literature: there are strong connections of this property with
countable compactness, normality and metrizability (see the
already mentioned papers \cite{gauldvamana, justmatvszept,
matveev97, rudinjerry, samuel-a, szeptvaughan}).


Evidently, every $(a)$-space is selectively $(a)$. So, every
monotonically normal space, in particular every $GO$-space, is
selectively $(a)$, being an $(a)$-space (see \cite[Theorem
1]{rudinjerry}). For the same reason every selectively paracompact
space is selectively $(a)$. It was observed in \cite{samuel} that
every $T_1$, $\sigma$-compact space is selectively $(a)$. The
Tychonoff plank is an example of a selectively $(a)$-space which
is not an $(a)$-space (\cite[Example 2.6]{song-sel-a}).

\smallskip
Notice that every countably compact selectively $a$-space is ${\sf
SSM}$, and every selectively $(\mathcal O,\Gamma)$-$(a)_{{\rm
closed \, discrete}}$ space is {\sf SSH}.


We will demonstrate similarities and differences between
$(a)$-spaces and selectively $(a)$-spaces; in particular, we will
show that there are many similarities between them.


We begin with the following result which was stated in
\cite{sel-(a)} without proof and which may be obtained by small
changes in the proof of Lemma 1 and its corollary in
\cite{matveev97}.

\begin{theorem} \label{a-e} Let $X$ be a separable space. Then:
\begin{itemize}
\item[$(1)$] If $X$ is selectively $(a)$, then every closed
discrete subset of $X$ has cardinality $< 2^{\omega}$;
\item[$(2)$] If $X$ contains a discrete subspace having
cardinality $\ge 2^{\omega}$, then $X^2$ is not hereditarily
selectively $(a)$.
\end{itemize}
\end{theorem}

In \cite{samuel}, the item (2) of this result was proved for a
general case.

\begin{theorem} {\rm (\cite[Theorem 3.1]{samuel})} If $X$ is a selectively $(a)$-space,
then $X$ cannot contain closed and discrete subsets of size $\ge
2^{d(X)}$.
\end{theorem}

The following theorem is a nice strengthening of a result
established in \cite{matveev97} by Matveev for $(a)$-spaces.

\begin{theorem} {\rm (\cite[Theorem 3.4]{samuel})} Under ${\sf CH}$, separable, Moore,
selectively $(a)$-spaces are metrizable.
\end{theorem}

It is shown in \cite[Theorem 3]{szeptvaughan} that there are
$\Psi$-spaces which are $(a)$-spaces, hence selectively $(a)$, and
those which are not $(a)$-spaces. It was observed in
\cite{sel-(a)} that there are also $\Psi$-spaces which are not
selectively $(a)$.


For $\Psi$-spaces we have the following (Propositions 4.1 and 4.2,
in \cite{samuel}).

\begin{theorem} Let $\mathcal A$ be an almost disjoint family of
subsets of $\mathbb N$. Then:
\begin{itemize}
\item[$(1)$] If $|\mathcal A| < \mathfrak d$, then $\Psi(\mathcal
A)$ is selectively $(a)$;

\item[$(2)$] If $\mathcal A$ is maximal, then $\Psi(\mathcal A)$
is selectively $(a)$ if and only if $|\mathcal A| < \mathfrak{d}$;

\item[$(3)$] If $\mathfrak{p} = \mathfrak{c}$, then a $\Psi$-space
satisfies property $(a)$ if and only if satisfies selectively
$(a)$.
\end{itemize}
\end{theorem}

(Here, $\mathfrak p$ is the pseudointersection number
\cite{vaughan}.)

It follows from this results that it is consistent that there are
$\Psi$-spaces which are selectively $(a)$-spaces but not
$(a)$-spaces.

\begin{problem} $(1)$ {\rm (\cite[Question 5.3]{samuel})} Is it consistent that there is an
almost disjoint family $\mathcal A$ of size $\mathfrak d$ such
that $\Psi(\mathcal A)$ is selectively $(a)$

$(2)$ {\rm (\cite[Question 5.4]{samuel})} If $\Psi(\mathcal A)$ is
normal, is it a selectively $(a)$-space

$(3)$ {\rm (\cite[Question 5.5]{samuel})} If $\Psi(\mathcal A)$ is
countably paracompact, is it a selectively $(a)$- space
\end{problem}

Let us notice that in \cite{song-sel-a} it was proved that
assuming $2^{\aleph_0} = 2^{\aleph_1}$ there exists a normal space
$X$ that is not selectively $(a).$

\medskip
Generalizing a result of Szeptycki and Vaughan regarding
characterization of property $(a)$ in $\Psi$-spaces,  da Silva
gave in \cite{samuel} the following combinatorial characterization
of selectively $(a)$ $\Psi$-spaces.

\begin{theorem}  Let $\mathcal A = \{A_{\alpha}:\alpha <\kappa\} \subset
\omega^{\omega}$ be an almost disjoint family of size $\kappa$.
The corresponding space $\Psi(\mathcal A)$ is selectively $(a)$ if
and only if the following property holds: for every sequence
$\{f_n : n < \omega\}$ in $\omega^{\omega}$ there is a sequence
$\{P_n : n <\omega\}$ of subsets of $\omega$ satisfying the
following two conditions:

(i) $|P_n\cap A_n| < \omega$ for all $n\in\omega$ and all $\alpha
< \kappa$;

(ii) for every $\alpha < \kappa$ there is $n\in\omega$ such that
$P_n cap A_{\alpha} \nsubseteq f_n(\alpha)$.
\end{theorem}

In \cite{morgan-samuel} the authors proved that a certain
effective parametrized weak diamond principle is enough to ensure
countability of the almost disjoint family in this setting.

In \cite[Corollary 3.3]{morgan-samuel} it was observed that
selectively $(a)$-spaces from almost disjoint families are
necessarily countable under some additional set-theoretic
assumptions, and concluded that it follows that the statement ``all
selectively $(a)$-spaces are countable" is consistent with ${\sf
CH}$.

These authors also noticed that there are no selectively $(a)$
almost disjoint families of size $\mathfrak{c}$; on the other
hand, countable almost disjoint families are associated to
metrizable $\Psi$-spaces, so if $\mathcal A$ is countable, then
$\Psi(\mathcal A)$ is paracompact and therefore it is $(a)$ (thus,
selectively $(a)$).

\medskip
The following results show the behaviour of selectively $(a)$-type
spaces under mappings and basic operations with spaces.

It is trivial that the selective $(a)$ property is not a
hereditary property. It is also true in case of some special
subspaces, for example, regular closed subspaces.

\begin{theorem}\label{aimages} {\rm (\cite{sel-(a)})} A closed-and-open image $Y=f(X)$ of a
selectively $(a)$-space $X$ is also selectively $(a)$.
\end{theorem}

The product of two selectively $(a)$-spaces need not be
selectively $(a)$; the Sorgenfrey line $S$ and its square $S^2$
can serve as an example (by Theorem \ref{a-e} $S^2$ is not
selectively $(a)$).

It would be interesting to answer the following question posed in
\cite{sel-(a)} (compare with \cite[Theorem 16]{justmatvszept}):

\begin{problem} Is the product of a selectively $(a)$-space $X$ and
a metrizable compact space $Y$ selectively $(a)$
\end{problem}

We have the following

\begin{theorem}\label{aproduct} {\rm (\cite{sel-(a)})} If the product $X\times Y$ of a space
$X$ and a compact space $Y$ is selectively $(a)$, then $X$ is
selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm closed}}$.
\end{theorem}

Now we consider when ${\sf AD}$ spaces have some of properties
under consideration.

\begin{theorem} \label{ax-sel-a} If $X \in {\sf Sel}(\mathcal O,\mathcal O)$-$(a)_{{\rm discrete}}$
and $e({\sf AD}(X)) < \omega_1$, then ${\sf AD}(X)$ is also in
${\sf Sel}(\mathcal O,\mathcal O)$-$(a)_{{\rm discrete}}$.
\end{theorem}

Another result of the same sort was proved in \cite{song-sel-a}:
If $X$ is a normal selectively $(a)$-space with $e(X) < \omega_1$,
then ${\sf AD}(X)$ is selectively $(a)$.

Similarly, in \cite{sel-(a)} it was proved:

\begin{theorem}
\label{sel-a-ax} If the Alexandroff duplicate ${\sf AD}(X)$ of a
space $X$ is selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm
countable}}$, then $e(X) < \omega_1$.
\end{theorem}

In \cite[Question 2.11]{sel-(a)}, the authors asked if a space $X$
is selectively $(a)$ provided the space ${\sf AD}(X)$ is
selectively $(a)$

This question was answered in \cite{song-sel-a}: there exists a
Tychonoff countably compact space $X$ such that ${\sf AD}(X)$ is
selectively $(a)$, but $X$ is not selectively $(a)$.


We close this subsection by one more natural question of this
kind: when subspaces of the Alexandroff duplicate ${\sf AD}(X)$ of
a space $X$ have properties of selectively $(a)$-type. We have the
following:

\begin{theorem} {\rm (\cite{sel-(a)})} Let $A$ and $B$ be subspaces of a space $X$ such
that $\overline{A} \cap B = \emptyset$ and $Z = (A\times \{1\})
\cup (B\times \{0\})$. If $e(Z) < \omega_1$ and $B$ is selectively
$(\mathcal O,\mathcal O)$-$(a)_{{\rm discrete}}$, then $Z$ is
selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm discrete}}$.
\end{theorem}

%%%%%%%%%%%%%%%%% 66666 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Uniform selection principles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip
In \cite{koc-unif} we have defined selection properties in uniform
spaces and demonstrated that selection principles in uniform
spaces are a good application of star selection principles to
concrete special classes of spaces. The exposition in this section
is based mainly on the paper \cite{koc-unif}, although the
approach in this article is different from (but equivalent to) the
approach in \cite{koc-unif}.

\medskip
Recall two equivalent approaches to the definition of uniform
spaces; one is to define a uniformity on a set $X$ in terms of
uniform covers, and the second to define it by using  entourages
of the diagonal \cite{engelking}. The first approach allows to
define uniform selection principles similarly to definitions of
the usual topological selection principles. By using this way we
showed in \cite{koc-unif} that uniform selection principles are a
kind of star selection properties as well as a kind of strongly
star selection properties. Then we passed to description of
uniform selection principles in terms of entourages of the
diagonal.

\medskip
Recall some definitions and facts about uniform spaces.

\smallskip
A \emph{quasi-uniformity} on a set $X$ is a filter $\mathbb U$ on
$X\times X$ satisfying the following two conditions:
\begin{itemize}
\item[$(QU1)$] $\Delta_X \subset U$ for each $U\in\mathbb U$;
\item[$(QU2)$] For each $U\in\mathbb U$ there is $V\in\mathbb U$
such that $V\circ V \subset U$,
\end{itemize}
where $\Delta_X = \{(x,x):x\in X\}$ is the diagonal of $X$, and
$V\circ V =\{(x,y)\in X\times X: \exists z\in X \mbox{ with }
(x,z) \in V, (z,y)\in V\}$.

The pair $(X,\mathbb U)$ is called a \emph{quasi-uniform space}.

A quasi-uniformity $\mathbb U$ is a \emph{uniformity} on $X$, and
$(X,\mathbb U)$ is a uniform space, if $\mathbb U$ satisfies also
the condition
\begin{itemize}
\item[$(QU3)$] $U\in\mathbb U$ implies $U^{-1}\in\mathbb U$,
\end{itemize}
where $U^{-1} = \{(x,y)\in X\times X:(y,x)\in U\}$.

For a subset $A$ of a (quasi-)uniform space $(X,\mathbb U)$ and
$U\in\mathbb U$ we write
\[
U[A]:= \{y\in X: (x,y)\in U \mbox{ for some } x\in A\}.
\]

We define uniform selection principles as follows. If $(X,\mathbb
U)$ is a uniform space, then it is said to be:
\begin{itemize}
\item[{\sf UM:}] \emph{uniformly Menger} or \emph{{\sf M}-bounded}
if for each sequence $\langle U_n:n\in\naturals\rangle$ of
entourages of the diagonal of $X$ there is a sequence $\langle
A_n:n\in\naturals \rangle $ of finite subsets of $X$ such that
$X=\bigcup_{n\in\naturals}U_n[A_n]$.

\item[{\sf U$\omega$M:}] \emph{$\omega$-{\sf M}-bounded} if for
each sequence $\langle U_n:n\in\naturals\rangle $ of entourages of
the diagonal of $X$ there is a sequence $\langle
A_n:n\in\naturals\rangle$ of finite subsets of $X$ such that each
finite subset of $X$ is contained in some $U_n[A_n]$.

\item[{\sf UH:}] \emph{uniformly Hurewicz} or \emph{{\sf
H}-bounded} if for each sequence $\langle U_n:n\in
\naturals\rangle $ of elements of $\mathbb U$ there is a sequence
$\langle A_n:n\in\naturals\rangle $ of finite subsets of $X$ such
that each $x\in X$ belongs to all but finitely many sets
$U_n[A_n]$.

\item[{\sf UR:}] \emph{uniformly Rothberger} or \emph{{\sf
R}-bounded} (resp. \emph{$\omega$-{\sf R}-bounded}) if for each
sequence $\langle U_n:n\in\naturals\langle $ of entourages of the
diagonal of $X$ there is a sequence $\langle
x_n:n\in\naturals\langle$ of points in $X$ such that
$X=\bigcup_{n\in\naturals}U_n[x_n]$ (resp. each finite subset of
$X$ is contained in some $U_n[x_n]$.
\end{itemize}

\medskip
\begin{remark} \rm It is evident that if a uniform space $X$ has the Menger
property with respect to topology generated by the uniformity,
then $X$ is {\sf M}-bounded. However, the converse need not be
true: a non-Lindel\"of Tychonoff space is an example of {\sf
M}-bounded space (with respect to the generated uniformity) which
has no the Menger property. (Similar remarks hold for the {\sf
R}-boundedness and {\sf H}-boundedness.) But a regular topological
space $X$ has the Menger (Hurewicz, Rothberger) property if and
only if its fine uniformity is {\sf M}-bounded ({\sf H}-bounded,
{\sf R}-bounded).

{\sf M}-bounded and especially {\sf H}-bounded uniform spaces have
some properties which are similar to the corresponding properties
of totally bounded uniform spaces.
\end{remark}

Recall that a uniform space $(X,\mathbb U)$ is said to be
\emph{totally bounded} or \emph{precompact} (resp.
\emph{pre-Lindel\"of} or \emph{$\omega$-bounded} if for each
$U\in\mathbb U$ there is a finite (resp. countable) $A\subset X$
such that $U[A] = X$. It is understood that totally bounded
uniform spaces are {\sf H}-bounded and thus {\sf M}-bounded and
that {\sf M}-boundedness implies pre-Lindel\"ofness.

\medskip
The difference between uniform and topological selection
principles is big enough \cite{koc-unif}. Here we point out some
of differences on the example of Hurewicz properties.

\medskip
(1) Every subspace of an {\sf H}-bounded uniform space is {\sf
H}-bounded. ({\sf M}- boundedness is also a hereditary property.)

(2) A uniform space $X$ is {\sf H}-bounded if and only if its
completion $\tilde{X}$ is {\sf H}-bounded.

(3) The product of two {\sf H}-bounded uniform spaces is also {\sf
H}-bounded.

\medskip
Let us mention that the product of two {\sf M}-bounded uniform
spaces need not be {\sf M}-bounded (see the case of topological
groups in the next subsection).

\medskip
We states the following two results from \cite{koc-unif}

\begin{theorem} For a uniform space $(X,\mathbb U)$ the following
are equivalent:
\begin{itemize}
\item[$(1)$] $X$ is $\omega$-{\sf M}-bounded;

\item[$(2)$] For each sequence $\langle U_n:n\in\mathbb N\rangle$
of elements of $\mathbb U$ there is a sequence $\langle
F_n:n\in\mathbb N\rangle$ of finite subsets of $X$ such that there
is a sequence $n_1 < n_2 < \cdots$ such that each finite $A\subset
X$ is contained in $\bigcup\{U_i[F_i]:n_k\le i <n_{k+1}\}$ for
some $k\in\naturals$.
\end{itemize}
\end{theorem}

\begin{theorem} For a uniform space $(X,\mathbb U)$ the following
are equivalent:
\begin{itemize}
\item[$(1)$] $X$ is ${\sf H}$-bounded;

\item[$(2)$] For each sequence $\langle U_n:n\in\mathbb N\rangle$
of elements of $\mathbb U$ there is a sequence $\langle
F_n:n\in\mathbb N\rangle$ of finite subsets of $X$ such that there
is a sequence $n_1 < n_2 < \cdots$ such that each $x\in X$ belongs
$\bigcup\{U_i[F_i]:n_k\le i <n_{k+1}\}$ for all but finitely many
$k\in\naturals$.
\end{itemize}
\end{theorem}

%%%%%%%%%%%%%%%%%%%%% 6.1 6.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Topological groups}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this subsection we discuss selection principles in topological
groups to illustrate the general theory of uniform selection
properties on a specific topological structure. The book
\cite{arh-tkachenko} is an excellent source concerning topological
groups.

Definitions of selection properties in topological groups are as
follows.

\begin{definition} \rm A topological group $(G,\cdot)$ is said to be
\begin{itemize}
\item[$(1)$] \emph{Menger-bounded} (shortly, \emph{{\sf
M}-bounded}) if for each sequence $\langle
U_n:n\in\naturals\rangle$ of neighborhoods of the neutral element
$e\in G$ there is a sequence $\langle A_n:n\in\naturals\rangle $
of finite subsets of $G$ such that
$X=\bigcup_{n\in\naturals}A_n\cdot U_n$;

\item[$(2)$] \emph{$\omega$-Menger-bounded} (shortly,
\emph{$\omega$-{\sf M}-bounded}), called also
\emph{Scheepers-bounded}, if for each sequence $\langle
U_n:n\in\naturals\rangle$ of neighborhoods of the neutral element
$e\in G$ there is a sequence $\langle A_n:n\in\naturals\rangle $
of finite subsets of $G$ such that each finite subset of $G$ is
contained in some  $A_n\cdot U_n$;

\item[$(3)$] \emph{Hurewicz-bounded} (shortly, \emph{{\sf
H}-bounded}) if for each sequence $\langle
U_n:n\in\naturals\rangle$ of neighborhoods of the neutral element
$e\in G$ there is a sequence $\langle A_n:n\in\naturals\rangle $
of finite subsets of $G$ such that each $x\in G$ belongs to all
but finitely many  $A_n\cdot U_n$;

\item[$(4)$] \emph{Rothberger-bounded} (shortly, \emph{{\sf
R}-bounded}) if for each sequence $\langle
U_n:n\in\naturals\rangle$ of neighborhoods of the neutral element
$e\in G$ there is a sequence $\langle x_n:n\in\naturals\rangle $
of elements of $G$ such that $X=\bigcup_{n\in\naturals}x_n\cdot
U_n$;
\end{itemize}
\end{definition}

These classes of groups have been introduced by the author of this
article in 1998 (see \cite[p. 1269]{coc11}), and the class of {\sf
M}-bounded groups was introduced independently by Okunev and
Tkachenko under the name \emph{$o$-bounded groups}.

The class of ${\sf M}$-bounded groups is the most investigated and
there is a big list of papers on this topic. More information on
{\sf M}-bounded topological groups the reader can find in
\cite{hernandez}, \cite{tkachenko-TA}, \cite{coc11},
\cite{banakh-cmuc}, \cite{banakh-ta},
\cite{banakh-repovs-zdomski}, \cite{Machura-Boaz},
\cite{machura-shelah-tsaban}, \cite{boaz-groups}, \cite{weiss},
\cite{zdomsky}; see also \cite{guran}.

\medskip
 There are two-person infinite games naturally associated
to each of mentioned classes of groups. For example, the game
associated to {\sf M}-bounded groups was introduced in
\cite{tkachenko-TA} as follows. Two players, ONE and TWO, play a
round for each $n\in \naturals$. In the $n$-th round ONE chooses a
neighborhood $U_n$ of the neutral element of $G$ and then TWO
chooses a finite set $F_n\subset G$. Two wins a play $U_1,F_1;
U_2,F_2; ...$ if and only if $\{F_n\cdot U_n:n \in \naturals\}$
covers $G$. A topological group $G$ is called \emph{strictly
$o$-bounded} or \emph{strictly {\sf M}-bounded} if TWO has a
winning strategy in the above game. It is easy to see that each
strictly {\sf M}-bounded group is {\sf M}-bounded. Also, each
group having the Menger property is ${\sf M}$-bounded. Every
subgroup of a $\sigma$-compact group is strictly {\sf M}-bounded
\cite{hernandez}.

In \cite{babin1} it is proved that in metrizable case strictly
{\sf M}-bounded groups are exactly {\sf H}-bounded groups.

\begin{theorem} {\rm (\cite[Theorem 5]{babin1})} For a metrizable
group $G$ the following statements are equivalent:
\begin{itemize}
\item[$(1)$] $G$ is strictly {\sf M}-bounded;

\item[$(2)$] $G$ is {\sf H}-bounded.
\end{itemize}
\end{theorem}


\medskip
Many selection principles in topological spaces can be
characterized game-theoretically. For example, it is a classical
result by Hurewicz that a topological space $X$ has the Menger
property if and only if ONE does not have a winning strategy in
the corresponding game (see \cite{coc1}). However, for topological
groups (and, more general, for star selection principles) it is
not the case.

\medskip
In \cite{hernandez}, Hernandez has constructed an ${\sf
M}$-bounded subgroup $G$ of $\mathbb R^{\omega}$ that is not strictly ${\sf
M}$-bounded.  In \cite[Theorem
8.5]{Machura-Boaz} it is proved that assuming ${\sf cov}(\mathcal
M) = \mathfrak d = \mathfrak b$ there is a group $G$ (a subgroup
of $\mathbb Z^{\mathbb N}$) which is {\sf R}-bounded and {\sf
H}-bounded (in all finite powers) but $G$ does not have the Menger
property $\sfin(\mathcal O,\mathcal O)$. Tsaban
\cite{boaz-groups}, Tsaban constructed strictly ${\sf M}$-bounded
groups which have the Menger and Hurewicz covering properties, but
are not $\sigma$-compact.


However, in \cite{coc11} the following game-theoretic
characterization for metrizable {\sf R}-bounded groups has been
obtained:

\begin{theorem} {\rm (\cite[Theorem 22]{coc11})} Let $(G, \cdot)$ be a $\sigma$-compact metrizable group.
The following are equivalent:
\begin{itemize}
\item[$(1)$] $G$ is {\sf R}-bounded;

\item[$(2)$] {\rm ONE} has no winning strategy in the game
naturally corresponded to {\sf R}-boundedness.
\end{itemize}
\end{theorem}

Metrizable {\sf M}-bounded groups and {\sf R}-bounded groups can
be also characterized measure-theoretically. Recall that a metric
space $(X,d)$ has \emph{strong measure zero} if for each sequence
$\langle \varepsilon_n:n\in\naturals\rangle$ of positive real
numbers there is a sequence $\langle A_n:n\in \naturals\rangle$ of
subsets of $X$ such that for each $n$, ${\rm diam}_d(A_n)<
\varepsilon_n$ and $X=\bigcup_{n\in\naturals}A_n$. $(X,d)$ has
\emph{${\sf M}$-measure zero} if for each sequence $\langle
\varepsilon_n:n\in\naturals\rangle$ of positive real numbers there
is a sequence $\langle \mathcal A_n:n\in \naturals\rangle$ of such
that for each $n$, $\mathcal A_n$ is a finite family of subsets of
$X$, diam$_d(A)< \varepsilon_n$ for each $A\in\mathcal A_n$ and
$\bigcup_{n\in\naturals}\mathcal A_n$ is an open cover of $X$.

\begin{theorem} {\rm (\cite[Theorem 12]{coc11})} For a metrizable
group $G$ the following are equivalent :
\begin{itemize}
\item[$(1)$] $G$ is {\sf M}-bounded;

\item[$(2)$] $G$ has {\sf M}-measure zero in each left-invariant
metrization of $G$.
\end{itemize}
\end{theorem}

\begin{theorem} {\rm (\cite[Theorem 19]{coc11})} For a metrizable
group $G$ the following are equivalent :
\begin{itemize}
\item[$(1)$] $G$ is {\sf R}-bounded;

\item[$(2)$] $G$ has strong measure zero in each left-invariant
metrization of $G$.
\end{itemize}
\end{theorem}

Let us mention that {\sf H}-bounded metrizable groups can be
characterize measure-theoretically (see \cite{babin1}; H.
Michalewski has obtained independently a similar result in his PhD
dissertation in 2003).

An interesting result proved by Scheepers in \cite[Th. 3, Cor.
4]{marion-roth-ramsey} states that $\sigma$-compact topological
groups can be characterized Ramsey-theoretically. (Many selection
(covering) properties in topological spaces can be characterized
in this manner.)


\medskip
Machura and Tsaban estimated minimal cardinalities of subgroups
$G$ of $\mathbb Z^{\mathbb N}$ which does not have boundedness
properties: for {\sf M}-boundedness and $\omega$-{\sf
M}- boundedness it is $\mathfrak d$, for {\sf H}-boundedness it is
$\mathfrak b$, and for {\sf R}-boundedness it is ${\sf
cov}(\mathcal M)$.

\medskip
We will discuss now preservation of boundedness properties of
groups by products of groups. Tkachenko \cite{tkachenko-TA} and
Hernandez \cite{hernandez} asked if the product of two {\sf
M}-bounded groups also {\sf M}-bounded.

Several authors answered this question in negative. In
\cite{krawczyk-mich} (see also \cite{krawczyk-mich-TA}), it was
given (assuming {\sf CH}) an example of two linear metric spaces
with Menger property such that their product is not ${\sf
M}$-bounded. Tsaban proved in \cite{boaz-groups} that there are
{\sf M}-bounded subgroups of ${\mathbb R}^{\mathbb N}$ whose
product is not ${\sf M}$-bounded. The paper
\cite{machura-shelah-tsaban} contains the result stating that
under {\sf CH} there is a Menger-bounded group $G\le \mathbb
Z^{\mathbb N}$ whose square is not Menger-bounded.

Another question was asked in \cite{hernandez-robbie-tkachenko}:
(i) is the product of two strictly {\sf M}-bounded groups $G$ and
$H$ also strictly {\sf M}-bounded; (ii) is the product of an {\sf
M}-bounded group with a strictly {\sf M}-bounded group again an
{\sf M}-bounded group. In \cite{babin1} these questions were
answered by the following: (a) the product of two metrizable
strictly {\sf M}-bounded groups is strictly {\sf M}-bounded; (b)
If $G$ an {\sf M}-bounded group and $H$ is a metrizable strictly
{\sf M}-bounded group, then $G\times H$  is an {\sf M}4o-bounded
group.

\medskip
At the end of this subsection we list  some other interesting
results concerning boundedness properties of products of groups.

\medskip
1. (\cite{hernandez}) The product of a $\sigma$-compact and an
{\sf M}-bounded group is {\sf M}-bounded.

2. (\cite{coc11}) A group $G$ is $\omega$-{\sf M}-bounded if and
only if $G^n$ is {\sf M}-bounded for all $n\in\mathbb N$.

3. (\cite{machura-shelah-tsaban}) Under some additional
assumptions (weaker than {\sf CH}) there is for each $k\in \mathbb
N$ a metrizable group $G$ such that $G^k$ is Menger-bounded but
$G^{k+1}$ is not.

4. (\cite{heike}) Under some additional cardinal restrictions
there are subgroups of $\mathbb Z^{\mathbb N}$ whose $k$th power
is Menger-bounded and whose $(k +1)$st power is not.

5.  (Mildenberger-Shelah and, independently, Banakh-Zdomskyy)
Consistently, every topological group $G$ such that $G^2$ is
Menger-bounded has Menger-bounded all finite powers.


%%%%%%%%%%%%%%%%%% 6.2 6.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Asymmetric cases}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this short subsection, reporting some results from
\cite{koc-kunzi}, we demonstrate that for quasi-uniform spaces the
situation with boundedness properties may be quite different from
ones in uniform spaces. Necessary information about quasi-uniform
spaces the interested reader can find in \cite{fletcher-lindgren}
and \cite{kunzi}.

Here are basic fact we need in the sequel.

If $(X,\mathbb U)$ is a quasi-uniform space, then $(X,\mathbb
U^{-1})$ is also a quasi-uniform space. Here
\begin{center}
$\mathbb U^{-1} = \{U^{-1}:U\in\mathbb U\}$
\end{center}
is called the \emph{conjugate} of $\mathbb U$.

The supremum of $\mathbb U$ and  $\mathbb U^{-1}$, denoted by
$\mathbb U^s$, is a uniformity on $X$ called the
\emph{symmetrization} of $\mathbb U$.

Recall that a quasi-uniform space $(X,\mathbb U)$ is said to be:
\begin{itemize}
\item[$(1)$] \emph{precompact} (resp. \emph{pre-Lindel\"of}) if
for each $U\in\mathbb U$ there is a finite (resp. countable) set
$F\subset X$ such that $U[F] = X$; \item[$(2)$] \emph{totally
bounded} if for each $U\in \mathbb U$ there is a finite cover
$\mathcal C$ of $X$ such that $C\times C \subset U$ for each $C\in
\mathcal C$.
\end{itemize}

\noindent In uniform spaces precompactness and total boundedness
coincide. Evidently, total boundedness of a quasi-uniform space
implies its precompactness. It is known that there are precompact
(even compact) quasi-uniform spaces which are not totally bounded.

\medskip
Having in mind the previous note we define now selective versions
of precompactness and total boundedness in quasi-uniform spaces.

\begin{definition} \label{defpre} \rm  A quasi-uniform space $(X,\mathbb
U)$ is:
\begin{itemize}
\item[{\sf (pre-M)}] \emph{pre-Menger} if for each sequence
$\langle U_n:n\in\naturals\rangle$  of elements of $\mathbb U$
there is a sequence $\langle F_n:n\in\naturals\rangle$ of finite
subsets of $X$ such that $X=\bigcup_{n\in\naturals}U_n[F_n]$;

\item[{\sf (pre-$\omega$M)}] \emph{pre-$\omega$-Menger} if for
each sequence $\langle U_n:n\in\naturals\rangle $ of elements of
$\mathbb U$ there is a sequence $\langle F_n:n\in\naturals\rangle
$ of finite subsets of $X$ such that each finite subset $A\subset
X$ is contained in $U_n[F_n]$ for some $n\in\naturals$;

\item[{\sf (pre-H)}] \emph{pre-Hurewicz} if for each sequence
$\langle U_n:n\in\naturals\rangle $ of elements of $\mathbb U$
there is a sequence $\langle F_n:n\in\naturals\rangle $ of finite
subsets of $X$ such that each $x\in X$ belongs to all but finitely
many sets $U_n[F_n]$;

\item[{\sf (pre-R)}] \emph{pre-Rothberger} if for each sequence
$\langle U_n:n\in\naturals\rangle $ of elements of $\mathbb U$
there is a sequence $\langle x_n:n\in\naturals\rangle $ of
elements of $X$ such that $X=\bigcup_{n\in\naturals}U_n[x_n]$;

\item[{\sf (pre-GN)}] \emph{pre-Gerlits-Nagy} if for each sequence
$\langle U_n:n\in\naturals\rangle $ of elements of $\mathbb U$
there is a sequence $\langle x_n:n\in \naturals\rangle $ of
elements of $X$ such that each $x\in X$ belongs to all but
finitely many $U_n[x_n]$.
\end{itemize}
\end{definition}

These selection properties can analogously defined for
quasi-metric spaces (see \cite{kunzi}) by replacing entourages of
the diagonal for quasi-uniform spaces by open balls for
quasi-metric spaces.

To each selection property of a quasi-uniform space $(X,\mathbb
U)$ defined above one can correspond an infinitely long game
similarly to definitions in uniform spaces and topological groups,
but we do not consider this.


\begin{definition} \rm Let $(X,\mathbb U)$ be a
quasi-uniform space and let $\mathcal P$ be  an element of  $\{M, \omega M, H, R,
GN\}$. $X$ is said to be \emph{$\mathcal P$-bounded} if the
uniform space $(X,\mathbb U^s)$ is $\mathcal P$-bounded.
\end{definition}

It would be interesting to know that a quasi-uniform space
$(X,\mathbb U)$ is Menger-bounded if and only if for each sequence
$(U_n:n\in \naturals)$ there is a sequence $(\mathcal C_n:n\in
\naturals)$ of finite collections of subsets of $X$ such that
$\bigcup_{n\in\naturals}\mathcal C_n$ covers $X$ and  for each
$n\in\naturals$, $C\times C \subset U_n$ for each $C\in \mathcal
C_n$.

\medskip
\begin{remark} \rm Let $(X,\mathbb U)$ be a quasi-uniform space.
A cover $\mathcal C$ of $X$ is a \emph{quasi-uniform cover} of $X$
if there is $U\in\mathbb U$ such that for each $x\in X$ there
exists $C\in\mathcal C$ with $U[x] \subset C$. $\mathbb U$ is a
\emph{Lebesgue quasi-uniformity} if each open cover of
$(X,\tau_{\mathbb U})$ is a quasi-uniform cover of $(X,\mathbb U)$
(see \cite[p. 97]{fletcher-lindgren}). It is easy to check that
pre-Menger Lebesgue quasi-uniformities are Menger (i.e. the
topological space $(X,\tau_{\mathbb U})$ is Menger). Observe also
that a $\sigma$-precompact quasi-uniform space is pre-Hurewicz.
\end{remark}

\medskip
In the diagram below we give relationships among the covering
properties of quasi-uniform spaces. The Menger (Hurewicz,
Rothberger) property concerns the topology $\tau_{\mathbb U}$
generated by $\mathbb U$. In \cite{koc-kunzi} we showed that the
arrows in this diagram are not reversible.

\medskip
\footnotesize
\begin{center}
${\sf compact} \, \, \, \, \, \, \, \Rightarrow \, \, \, \, \, \,
\, {\sf Hurewicz} \, \, \, \, \, \, \, \Rightarrow \, \, \, \, \,
\, \, {\sf Menger} \, \, \, \, \, \, \, \Leftarrow \, \, \, \, \,
\, \, {\sf Rothberger} \, \, \, \, \, \, \, \Leftarrow \, \, \, \,
\, \, {\sf GN}$
\end{center}

 \hskip1.3cm $ \, \Downarrow \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
 \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \Downarrow  \, \, \, \, \, \, \, \,\,
 \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
 \,\, \, \, \, \, \, \, \,
 \Downarrow \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,  \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,\Downarrow
 \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,  \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \Downarrow$

\begin{center}
\hskip2mm ${\sf pre\!-\!compact}\,   \Rightarrow \,  {\sf
pre\!-\!Hurewicz} \, \, \Rightarrow \, {\sf pre\!-\!Menger}\,
\Leftarrow  \, {\sf pre\!-\!Rothberger}\, \Leftarrow \, {\sf
pre\!-\!GN}$
\end{center}

\hskip1.4cm $\Uparrow \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \,  \Uparrow \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \, \, \, \,  \Uparrow  \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \, \, \, \, \Uparrow  \, \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,
\, \, \, \, \Uparrow$

\begin{center}
\hskip-3mm${\sf totally \, bounded}\,   \Rightarrow \,   {\sf
H\!-\!bounded} \, \, \, \Rightarrow \, \, \, {\sf M\!-\!bounded}
\, \,  \Leftarrow  \, \, {\sf R\!-\!bounded} \, \, \Leftarrow \,
\,  {\sf GN\!-\!bounded}$
\end{center}

\bigskip
\begin{center}
{\sf Diagram 2: Quasi-uniform case}
\end{center}

%%%%%%%%%%%%% 6.3 6.3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\normalsize
\subsection{Paratopological groups}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Here we give a specific illustration for nonsymmetric cases in
paratopological groups. A group $(G,\cdot)$ with a topology $\tau$
is a \emph{paratopological group} if the group operation is
jointly continuous mapping from $G\times G$ to $G$. For more
details on paratopological groups see \cite{tkachenko-book}.

Let $\eta(e_G)$ denote the system of neighbourhoods of the
identity element $e_G$ of $G$. Then $(G,\cdot,\tau^{-1})$ denotes
the paratopological group such that $\{U^{-1}:U\in\eta(e_G)\}$ is
a neighbourhood system at $e_G$, and $(G^*,\cdot,\tau^*)$ is the
topological group $(G,\cdot,\tau \vee \tau^{-1})$.

%\newpage
A paratopological group $(G,\cdot,\tau)$ is \emph{pre-Menger} if
for each sequence $\langle U_n:n\in\mathbb N\rangle$ in
$\eta(e_G)$ there are finite sets $A_n\subset G$, $n\in \mathbb
N$, such that $G= \bigcup_{n\in\mathbb N}A_nU_n$. $G$ is
\emph{totally Menger} if the group $(G^*,\cdot,\tau^*)$ is ${\sf
M}$-bounded. Similarly we define pre-Rothberger, pre-Hurewicz,
pre-Gerlits-Nagy paratopological groups.

The Sorgenfrey line $\mathbb S$ is an example of a pre-Menger
paratopological group which is not Menger. This group is not
pre-Rothberger, too.

We saw that subgroups of ${\sf M}$-bounded topological groups are
also ${\sf M}$-bounded. But it is not the case in paratopological
groups.

We quote only four results, without proofs, following
\cite{koc-paratop}, to illustrate differences between topological
and paratopological case.

\begin{theorem} If a paratopological group
$(G,\cdot,\tau)$ is pre-Menger and $H$ is a dense subgroup of
$(G,\cdot,\tau^{-1})$, then $H$ is pre-Menger.
\end{theorem}


\begin{theorem} For a paratopological group $(G,\cdot, \tau)$ the
following are equivalent:
\begin{itemize}
\item[$(1)$] All finite powers of $G$ are pre-Menger;

\item[$(2)$] $G$ is pre-$\omega$-Menger.
\end{itemize}
\end{theorem}

\begin{theorem} Let $(G,\cdot, \tau)$ be a pre-Menger paratopological group
and $(H,\sigma)$ a precompact paratopological group. Then $G
\times H$ is a paratopological group.
\end{theorem}

\begin{theorem} If $(G,\cdot, \tau^*)$ is a pre-Menger topological group, and
$(H,\sigma)$ a hereditarily precompact paratopological group, then
the product $(G\times H, \tau^*\times \sigma)$ is hereditarily
pre-Menger.
\end{theorem}


\begin{problem} If paratopological groups $G$ and $H$
are such that  $G$ is hereditarily pre-Menger  and $H$ is
hereditarily precompact, is then the product $G\times H$
hereditarily pre-Menger
\end{problem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
\noindent {\bf Acknowledgement:}  The author is grateful to the
journal Director-in-Chief, Professor Mohammad Sal Moslehian, who
invited me to write this survey paper.  I also thank
Editor-in-Chief, Professor Hamid Reza Ebrahimi Vishki for his kind
cooperation during preparation of this article.

\bibliographystyle{amsplain}
\begin{thebibliography}{99}
\bibitem{arh-tkachenko} A.V. Arhangel�skii, M. Tkachenko, \emph{Topological
Groups and Related Structures}, Atlantis Studies in Mathematics,
Vol. 1, Atlantis Press/World Scientific, Amsterdam�Paris, 2008.

\bibitem{babin1} L. Babinkostova, \emph{Metrizable groups and strict
$o$-boundedness}, Mat. Vesnik {\bf 58} (2006), no. 3-4, 131--138.

\bibitem{coc11} L. Babinkostova, Lj.D.R. Ko\v cinac, M. Scheepers,
\emph{Combinatorics of open covers} (XI): \emph{Menger- and
Rothberger-bounded groups}, Topology Appl. {\bf 154} (2007), no.
7, 1269--1280.

\bibitem{banakh-cmuc} T. Banakh, \emph{Locally minimal topological groups and their embeddings
into products of $o$-bounded groups}, Comment. Math. Univ.
Carolin. {\bf 41} (2000), no. 4, 811-�815.

\bibitem{banakh-ta} T. Banakh, \emph{On index of total boundedness of (strictly) $o$-bounded groups},
Topology Appl. {\bf 120} (2002), no. 3, 427�-439.

\bibitem{banakh-repovs-zdomski} T. Banakh, D. Repov\v s, L. Zdomskyy,
\emph{$o$-boundedness of free topological groups}, Topology Appl.
{\bf 157} (2010), no. 2, 466--481.

\bibitem{star-hur} M. Bonanzinga, F. Cammaroto, Lj.D.R. Ko\v cinac,
\emph{Star-Hurewicz and related properties}, Appl. Gen. Topol.
{\bf 5} (2004), no. 1, 79--89.

\bibitem{bckm-mv} M. Bonanzinga, F. Cammaroto, Lj.D.R. Ko\v cinac,
M.V. Matveev, \emph{On weaker forms of Menger, Rothberger and
Hurewicz properties}, Mat. Vesnik {\bf 61} (2009), no. 1, 13--23.

\bibitem{misha-milena-psi} M. Bonanzinga, M. Matveev, \emph{Some covering properties for
$\Psi$-spaces}, Mat. Vesnik {\bf 61} (2009), no. 1, 3--11.

\bibitem{bon-pans-relative} M. Bonanzinga, B.A. Pansera, \emph{Relative versions of some
star-selection principles}, Acta Math. Hungar. {\bf 117} (2007),
no. 3, 231--243.

\bibitem{sel-(a)} A. Caserta, G. Di Maio, Lj.D.R. Ko\v cinac,
\emph{Versions of properties $(a)$ and $(pp)$}, Topology Appl.
{\bf 158} (2011), no. 12, 1630--1638.

\bibitem{agata-watson} A. Caserta, S. Watson, \emph{Alexandroff duplicate and its subspaces},
Appl. Gen. Topol. {\bf 8} (2007), no. 2, 187�-205.

\bibitem{creede} G.D. Creede, \emph{Concerning semi-stratifiable spaces}, Pacific J.
Math. {\bf 32} (1970), no. 1, 47�-54.

\bibitem{daniels} P. Daniels, \emph{Pixley-Roy spaces over subsets of the reals},
Topology Appl. {\bf 29} (1988), no. 1, 93--106.

\bibitem{vanDouwen-PR} E.K. van Douwen, \emph{The Pixley-Roy topology on spaces of
subsets}, In: Set Theoretic Topology, Academic Press, 1977, pp.
111--134.

\bibitem{vanDouwen-handbook} E.K. van Douwen, \emph{The integers and topology},
In: K. Kunen, J. Vaughan (eds.), Handbook of Set-theoretic
Topology,  North-Holland, Amsterdam, 1984, pp. 111�-167.

\bibitem{vDRRT}
E.K. van Douwen, G.M. Reed, A.W. Roscoe and I.J. Tree, \emph{Star
covering properties}, Topology Appl. {\bf 39} (1991), no. 1,
71--103.

\bibitem{engelking} R. Engelking, \emph{General Topology} (revised and completed
edition), Heldermann, 1989.

\bibitem{fle}
W.M. Fleishman, \emph{A new extension of countable compactness},
Fund. Math. {\bf 67} (1970), no. 1, 1--9.

\bibitem{fletcher-lindgren} P. Fletcher, W.F. Lindgren, \emph{Quasi-Uniform Spaces}, Lecture Notes
Pure Appl. Math., vol. 77, Marcel�Dekker, 1982.

\bibitem{gauldvamana} D. Gauld, M.K. Vamanamurthy, \emph{Covering properties and
metrization of manifolds} 2, Topology Proc. {\bf 24} (1999),
173�-186.

\bibitem{guran} I.I. Guran, \emph{On topological groups close to being Lindel\"of},
Soviet Math. Dokl. {\bf 23} (1981), 173�-175.

\bibitem{hernandez} C. Hern\'andez, \emph{Topological groups close to being
$\sigma$-compact}, Topology Appl. {\bf 102} (2000), no. 1,
101�-111.

\bibitem{hernandez-robbie-tkachenko} C. Hern\'andez, D. Robbie, M. Tkachenko, \emph{Some
properties of $o$-bounded and strictly $o$-bounded groups}, Appl.
Gen. Topol. {\bf 1} (2000), no. 1, 29�-43.

\bibitem{hurewicz} W. Hurewicz, \emph{\"Uber eine Verallgemeinerung
des Borelschen Theorems}, Math. Z. {\bf 24} (1925), 401--421.

\bibitem{justmatvszept} W. Just, M.V. Matveev, P.J. Szeptycki, \emph{Some results on property
$(a)$}, Topology Appl. {\bf 101} (2000), no. 1, 103�-111.

\bibitem{darko-relator-amh} D. Kocev, \emph{Selection principles in relator spaces},
Acta Math. Hungar. {\bf 126} (2010), no. 1-2, 78--93.

\bibitem{darko-relator-filomat} D. Kocev, \emph{On weaker forms of relator Menger, relator
Rothberger and relator Hurewicz properties}, Filomat {\bf 26}
(2012), no. 3, 427--437.

\bibitem{darko-doct} D. Kocev, \emph{Some selection properties of topological
spaces and their generalizations}, Doctoral Dissertation,
University of Ni\v s, 2012, pp. v+65 (in Serbian).

\bibitem{koc-starmenger}  Lj.D.R. Ko\v cinac, \emph{Star-Menger and related
spaces}, Publ. Math. Debrecen  {\bf 55} (1999), no. 3--4,
421--431.

\bibitem{koc-starmenger2}  Lj.D.R. Ko\v cinac, \emph{Star-Menger and related spaces},
II, Filomat {\bf 13} (1999), 129--140.

\bibitem{koc-unif} Lj.D.R. Ko\v cinac, \emph{Selection principles in
uniform spaces}, Note  Mat. {\bf 22} (2003/2004), no. 2, 127--139.

\bibitem{koc-iransurvey} Lj.D.R. Ko\v cinac, \emph{Selected results on selection
principles}, In: Proc. Third Seminar Geom. Topol., July 15--17,
2004, Tabriz, Iran, pp. 71--104.

\bibitem{koc-proc-steklov} Lj.D.R. Ko\v cinac, \emph{Some covering properties
in topological and uniform spaces}, Proc. Steklov Inst. Math. {\bf
252} (2006), no. 1, 122--137.

\bibitem{koc-fuzzy} Lj.D.R. Ko\v cinac, \emph{Selection properties in
fuzzy metric spaces}, Filomat {\bf 26} (2012), no. 2, 305--312.

\bibitem{koc-paratop} Lj.D.R. Ko\v cinac, \emph{Remarks on
selection properties of paratopological groups}, in preparation.

\bibitem{koc-kunzi} Lj.D.R. Ko\v cinac, H.-P.A. K\"unzi,
\emph{Selection properties of uniform and related structures},
Topology  Appl. {\bf 160} (2013), no. 18, 2495--2504.

\bibitem{krawczyk-mich} A. Krawczyk, H. Michalewski, \emph{Linear metric spaces close
to being $\sigma$-compact}, preprint, 2002, 10 pages.

\bibitem{krawczyk-mich-TA} A. Krawczyk, H. Michalewski, \emph{An
example of a topological group}, Topology Appl. {\bf 127} (2003),
no. 3, 325--330.

\bibitem{kunzi} H.-P.A. K\"unzi, \emph{An introduction to quasi-uniform spaces},
Contemp. Math. {\bf 486} (2009), 239�-304.

\bibitem{machura-shelah-tsaban} M. Machura, S. Shelah, B. Tsaban, \emph{Squares of
Menger-bounded groups}, Trans. Amer. Math. Soc. {\bf 362} (2010),
no. 4, 1751--1764.

\bibitem{Machura-Boaz}  M. Machura, B. Tsaban, \emph{The combinatorics of the
Baer-Specker group}, Israel J. Math. {\bf 168} (2008), 125--152.

\bibitem{matveev-acc} M.V. Matveev, \emph{Absoutely countably
compact spaces}, Topology Appl. {\bf 58} (1994), no. 1, 81--92.

\bibitem{matveev97} M.V. Matveev, \emph{Some questions on property $(a)$},
Questions Answers Gen. Topology {\bf 15} (1997), no. 2, 103--111.

\bibitem{misha-survey} M.V. Matveev, \emph{A survey on star covering properties},
Topology Atlas, Preprint No. 330 (1998).

\bibitem{matveev-preprint} M.V. Matveev, \emph{On the extent of {\sf SSM} spaces}, preprint, July 1998.

\bibitem{menger} K. Menger, \emph{Einige \"Uberdeckungss\"atze der Punktmengenlehre},
Stzungsberischte Abt. 3a, Mathematik, Astronomie, Pysik,
Meteorologie und Mechanik (Wiener Akademie, Wien) {\bf 133}
(1924), 421--444.

\bibitem{heike} H. Mildenberger, \emph{Cardinal characteristics for
Menger-bounded subgroups}, Topology Appl. {\bf 156} (2008), no. 1,
130--137.

\bibitem{morgan-samuel} C.J.G. Morgan, S.G. da Silva, \emph{Selectively $(a)$-spaces
from almost disjoint families are necessarily countable under a
certain parametrized weak diamond principle}, Houston J. Math., to
appear.

\bibitem{rothberger} F. Rothberger, \emph{Eine Versch\"arfung der Eigenschaft
${\sf C}$}, Fund. Math. {\bf 30} (1938), 50--55.

\bibitem{rudinjerry} M.E. Rudin, I.S. Stares, J.E. Vaughan, \emph{From countable
compactness to absolute countable compactness}, Proc. Amer. Math.
Soc. {\bf 125} (1997), no. 3, 927�-934.

\bibitem{sakai-SM} M. Sakai, \emph{Star covering versions of
the Menger propperty}, Topology Appl. {\bf 176} (2014), 22--34.

\bibitem{sakai-marion} M. Sakai, M. Scheepers,
\emph{The combinatorics of open covers}, In: K.P. Hart, J. van
Mill, P. Simon (eds.), Recent Progress in General Topology III,
Atlantis Press, 2014, pp. 751--800.

\bibitem{coc1} M. Scheepers, \emph{Combinatorics of open covers} (I):
\emph{Ramsey theory}, Topology Appl. {\bf 69} (1996), no. 1,
31--62.

\bibitem{marion-roth-ramsey} M. Scheepers, \emph{Rothberger bounded groups and Ramsey
theory}, Topology Appl. {\bf 158} (2011), no. 13, 1575--1583.

\bibitem{samuel-a} S.G. da Silva, \emph{On the presence of countable
paracompactness, normality and property $(a)$ in spaces from
almost disjoint families}, Questions Answers  Gen. Topology {\bf
25} (2007), no. 1,  1--18.

\bibitem{samuel} S.G. da Silva, \emph{$(a)$-spaces and selectively
$(a)$-spaces from almost disjoint families}, Acta Math. Hungar.
{\bf 142} (2014), no. 2, 420--432.

\bibitem{song-cejm} Y.-K. Song, \emph{Absolutely strongly star-Hurewicz spaces},
Open Math.  {\bf 13} (2015), no. 1, 26--32.

\bibitem{song-assm} Y.-K. Song, \emph{Absolutely strongly star-Menger spaces},
Topology Appl. {\bf 160} (2013), no. 3,  475--481.

\bibitem{song-ssh} Y.-K. Song, \emph{Remarks on strongly star-Hurewicz spaces},
Filomat {\bf 27} (2013), no. 6, 1127--1131.

\bibitem{song-polish} Y.-K. Song, \emph{Remarks on star-Hurewicz spaces},
Bull. Polish Acad. Sci. Math. {\bf 61} (2013), no. 3--4, 247--255.

\bibitem{song-cmuc} Y.-K. Song, \emph{Remarks on strongly star-Menger spaces},
Comment.  Math. Univ. Carolin. {\bf 54} (2013), no. 1, 97--104.

\bibitem{song-sel-a} Y.-K. Song, \emph{Remarks on selectively $(a)$-spaces},
Topology  Appl. {\bf 160} (2013), no. 6, 806--811.

\bibitem{song-sroth} Y.-K. Song, \emph{A note on star-Rothberger spaces},
Houston J. Math. {\bf 40} (2014), no. 2, 645--653.

\bibitem{song-star-k-menger} Y.-K. Song, \emph{On star-$K$-Menger spaces}, Hacet.
J. Math. Stat. {\bf 43} (2014), no. 5, 769--776.

\bibitem{song-houston-sm} Y.-K. Song, \emph{Remarks on star-Menger spaces}, Houston
J. Math. {\bf 40} (2014), no. 3,  917--925.

\bibitem{song-li} Y.-K. Song, R. Li, \emph{A note on star-Hurewicz spaces},
Filomat {\bf 27} (2013), no. 6, 1091--1095.

\bibitem{song-c-menger} Y.-K. Song, Z. Yin, \emph{On star-$C$-Menger spaces},
Quaest. Math. {\bf 37} (2014), no. 3, 337--347..

\bibitem{szeptvaughan} P.J. Szeptycki, J.E. Vaughan, \emph{Almost disjoint families and
property $(a)$}, Fund. Math. {\bf 158} (1998), no. 3,  229�-240.

\bibitem{tkachenko-TA} M. Tkachenko, \emph{Introduction to topological groups},
Topology Appl. {\bf 86} (1998), no. 3, 179-�231.

\bibitem{tkachenko-book} M. Tkachenko, \emph{Paratopological and Semitopological Groups
Versus Topological Groups}, In: K.P. Hart, J. van Mill, P. Simon
(eds.), Recent Progress in General Topology III, Atlantis Press,
2014, pp. 825--882.

\bibitem{boaz-groups} B. Tsaban, \emph{$o$-bounded groups and other topological
groups with strong combinatorial properties}, Proc. Amer. Math.
Soc. {\bf 134} (2006), no. 3, 881--891.

\bibitem{boaz-problems} B. Tsaban, \emph{Selection principles and special sets of
reals}, In: E. Pearl (ed.),  Open Problems in Topology II,
Elsevier Science, 2007, pp. 91--108.

\bibitem{boaz-arxiv} B. Tsaban, \emph{Combinatorial aspects of selective star covering properties in $Psi-$spaces}, Topology Appl. to appear.

\bibitem{vaughan} J.E. Vaughan, \emph{Small uncountable cardinals and topology},
In: J. van Mill, G.M. Reed (eds.), Open Problems in Topology,
North-Holland, Amsterdam, 1990, pp. 195�-216.

\bibitem{weiss} T. Weiss, \emph{A note on unbounded strongly measure zero
subgroups of the Baer-Specker group}, Topology Appl. {\bf 156}
(2008), no. 1, 138--141.

\bibitem{zdomsky} L.S. Zdomskyy, \emph{$o$-boundedness of free objects over a Tychonoff space}, \
Mat. Stud. {\bf 25} (2006), no. 1, 10-28.

\end{thebibliography}

\end{document}

.
