% $o$-bounded groups and other topological groups with strong combinatorial properties
Boaz Tsaban
We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of $\R$ (thus strictly $o$-bounded) which have the Menger and Hurewicz properties but are not $\sigma$-compact, and show that the product of two $o$-bounded subgroups of $\NR$ may fail to be $o$-bounded, even when they satisfy the stronger property $\sone(\BO,\BO)$. This solves a problem of Tka\v{c}enko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups $G$ of size continuum such that every countable Borel $\w$-cover of $G$ contains a $\gamma$-cover of $G$.
math0307225.tex
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%I) Selection principles:
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\author{Boaz Tsaban}
\address{Department of Applied Mathematics and Computer Science,
Weizmann Institute of Science,
Rehovot 76100,
Israel}
\email{boaz.tsaban@weizmann.ac.il}
\urladdr{http://www.cs.biu.ac.il/\~{}tsaban}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%HEADER END%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title[$o$-bounded and combinatorial groups]{%
$o$-bounded groups and other topological groups with strong combinatorial properties}

\begin{document}

\begin{abstract}
We construct several topological groups with very strong combinatorial properties.
In particular, we give simple examples of subgroups of $\R$ (thus strictly $o$-bounded)
which have the Menger and Hurewicz properties but are not $\sigma$-compact,
and show that the product of two
$o$-bounded subgroups of $\NR$ may fail to be $o$-bounded,
even when they satisfy the stronger property $\sone(\BO,\BO)$.
This solves a problem of Tka\v{c}enko and Hernandez,
and extends independent solutions of Krawczyk and Michalewski
and of Banakh, Nickolas, and Sanchis.
We also construct separable metrizable groups $G$ of size continuum
such that every countable Borel
$\w$-cover of $G$ contains a $\gamma$-cover of $G$.
\end{abstract}

\keywords{%
$o$-bounded groups, $\gamma$-sets, Luzin sets, selection principles.
}
\subjclass{%
Primary: 54H11; %Topological groups
Secondary: 37F20. %Combinatorics and topology
}

\maketitle

\vspace{-0.6cm}

\section{Introduction}

In \cite{coc1, coc2}, a unified framework for topological diagonalizations
is established, which turns out closely related to several notions which
appear in a recently flourishing study of topological groups in terms of
their covering properties (see, e.g., \cite{TkaIntro, Hernandez, HRT, KMexample, KMlinear}
and references therein).
A comprehensive study of these interrelations is currently being carried by
Babinkostova, Ko\v{c}inac, and Scheepers \cite{coc11}.
The purpose of this paper is to adopt several recent construction
techniques from the general theory of topological diagonalizations
to the theory of topological groups.

\subsection{Topological diagonalizations}
We briefly describe the general framework.
Let $X$ be a topological space.
An open cover $\cU$ of $X$ is
an \emph{$\omega$-cover} of $X$ if $X$ is not in $\cU$ and for
       each finite subset $F$ of $X$, there is
       a set $U\in\cU$ such that $F\subseteq U$.
$\cU$ is a \emph{$\gamma$-cover} of $X$ if it is infinite and for each $x$ in
       $X$, $x\in U$ for all but finitely many $U\in\cU$.
Let $\O$, $\Omega$, and $\Gamma$ denote the collections of all countable open
covers, $\omega$-covers, and $\gamma$-covers of $X$, respectively.
Let $\scrA$ and $\scrB$ be collections of covers of a space $X$.
Following are selection hypotheses which $X$ might satisfy or not
satisfy.
\begin{itemize}
\item[$\sone(\scrA,\scrB)$:]{
For each sequence $\seq{\cU_n}$ of members of $\scrA$,
there exist members $U_n\in\cU_n$, $n\in\N$, such that $\seq{U_n}\in\scrB$.
}
\item[$\sfin(\scrA,\scrB)$:]{
For each sequence $\seq{\cU_n}$
of members of $\scrA$, there exist finite (possibly empty)
subsets $\cF_n\sbst\cU_n$, $n\in\N$, such that $\Union_{n\in\N}\cF_n\in\scrB$.
}
\item[$\ufin(\scrA,\scrB)$:]{
For each sequence $\seq{\cU_n}$ of members of $\scrA$
which do not contain a finite subcover,
there exist finite (possibly empty) subsets $\cF_n\sbst\cU_n$, $n\in\N$,
such that $\seq{\cup\cF_n}\in\scrB$.
}
\end{itemize}

Some of the properties defined in this manner
were studied earlier by Hurewicz ($\ufin(\O,\Gamma)$), Menger ($\sfin(\O, \O)$),
Rothberger ($\sone(\O, \O)$, traditionally known as the $C''$ property),
Gerlits and Nagy ($\sone(\Omega,\Gamma)$, traditionally known as the $\gamma$-property),
and others.
Many equivalences hold among these properties, and the surviving ones
appear in Figure \ref{SchDiagram} (where an arrow denotes implication),
to which no arrow can be added except perhaps from
$\ufin(\O,\Gamma)$ or $\ufin(\O,\Omega)$ to $\sfin(\Gamma,\Omega)$
\cite{coc2}.

\begin{figure}[!ht]
{%\scriptsize
\begin{changemargin}{-2cm}{-1cm}
\begin{center}
$\xymatrix@R=10pt{%@C=-2pt@R=10pt{%@=7pt{
%1
&
&
& \ufin(\O,\Gamma)\ar[r]
& \ufin(\O,\Omega)\ar[rr]
& & \sfin(\O,\O)
\\
%2
&
&
& \sfin(\Gamma,\Omega)\ar[ur]
\\
%3
& \sone(\Gamma,\Gamma)\ar[r]\ar[uurr]
& \sone(\Gamma,\Omega)\ar[rr]\ar[ur]
& & \sone(\Gamma,\O)\ar[uurr]
\\
%4
&
&
& \sfin(\Omega,\Omega)\ar'[u][uu]
\\
%5
& \sone(\Omega,\Gamma)\ar[r]\ar[uu]
& \sone(\Omega,\Omega)\ar[uu]\ar[rr]\ar[ur]
& & \sone(\O,\O)\ar[uu]
}$
\caption{The Scheepers Diagram}\label{SchDiagram}
\end{center}
\end{changemargin}
}
\end{figure}
Each selection principle has a naturally associated game,
but we will restrict attention to the game
$\gfin(\scrA,\scrB)$, which is played as follows:
In the $n$th inning, ONE chooses an element
$\cU_n$ of $\scrA$ and then TWO responds by choosing
a finite subset $\cF_n$ of $\cU_n$.
They play an inning per natural number.
A play $(\cU_0, \cF_0, \cU_1, \cF_1\dots)$ is won by TWO if
$\Union_n\cF_n\in\scrB$; otherwise ONE wins.
We will write $\mbox{ONE}\uparrow\gfin(\scrA,\scrB)$ (respectively, $\mbox{TWO}\uparrow\gfin(\scrA,\scrB)$) for
``ONE (respectively, TWO) has a winning strategy in $\gfin(\scrA,\scrB)$''.

\subsection{$o$-bounded groups}
Okunev introduced the following notion as an approximation of $\sigma$-compact
groups: A topological group $G$ is \emph{$o$-bounded} if for each sequence
$\seq{U_n}$ of neighborhoods of the unit element of $G$, there exists
a sequence $\seq{F_n}$ of finite subsets of $G$ such that
$G=\Union_n F_n\cdot U_n$.
It is possible to state this definition in the language of selection principles.
Let $\Onbd$ denote the covers of $G$ of the form $\{g\cdot U : g\in G\}$
where $U$ is a neighborhood of the unit element of $G$.
Then $G$ is $o$-bounded if, and only if, $G$ satisfies $\sfin(\Onbd,\O)$.

According to Tka\v{c}enko, a topological group $G$ is \emph{strictly $o$-bounded}
if TWO has a winning strategy in the game $\gfin(\Onbd,\O)$.
Clearly, every subgroup of a $\sigma$-compact group is strictly
$o$-bounded, but the converse does not hold \cite{Hernandez}.

\subsubsection*{Notational convention}
For sets $X,Y$, ${{}^XY}$ denotes the collection of all functions from $X$ to $Y$.
If $Y$ is a topological space, then the topology on ${{}^XY}$ is the Tychonoff product
topology.

\section{Two almost $\sigma$-compact subgroups of $\R$}
The \emph{Baire space} $\NN$ %is equipped with the product topology and
is (quasi)ordered by eventual dominance: $f\le^* g$ if
$f(n)\le g(n)$ for all but finitely many $n$.
A subset of $\NN$ is \emph{dominating} if it is cofinal in $\NN$ with respect to $\le^*$.
If a subset of $\NN$ is unbounded with respect to $\le^*$ then we simply say that it is
\emph{unbounded}.
Let $\b$ (respectively, $\d$) denote the minimal cardinality of an unbounded
(respectively, dominating) subset of $\NN$.

We use the following setting from \cite{ideals}.
Let $\compactN=\N\cup\{\infty\}$ be the one point compactification of $\N$. (A subset
$A\sbst\compactN$ is open if: $A\sbst\N$, or $\infty\in A$ and $A$
is cofinite.) Let $\Increasing \sbst \NcompactN$ consist of the
\emph{nondecreasing} functions $f\in\NcompactN$.
$\Increasing$ is homeomorphic to the Cantor set of reals.
For each increasing finite sequence
$s$ of natural numbers, let $q_{s} \in \Increasing$ be defined by
$q_{s}(k) = s(k)$ if $k <|s|$, and $q_{s}(k) = \infty$ otherwise.
Let $Q$ be the collection of all these elements $q_s$.

\begin{thm}\label{ZFCgp}
There exists a non $\sigma$-compact subgroup $G_H$ of $\R$ of
cardinality $\b$ such that all finite powers of $G_H$
satisfy $\ufin(\O,\Gamma)$ (in particular, they satisfy $\sfin(\Omega,\Omega)$).
\end{thm}
\begin{proof}
Let $B=\{f_\alpha : \alpha < \b\}\sbst\NN$ be a $\le^*$-unbounded
set of strictly increasing elements of $\NN$ which forms a \emph{$\b$-scale}
(that is, for each $\alpha<\beta$, $f_\alpha \leq^* f_\beta$),
and set $H = B\cup Q$.
In \cite{ideals} it is proved that
all finite powers of $H$ satisfy $\ufin(\O,\Gamma)$.

Think of $H$ as a set of real numbers.
For each $n$, the set
$$G_n = \{\alpha_1g_1+\dots+\alpha_ng_n : \alpha_1,\dots,\alpha_n\in\Z,\ g_1,\dots,g_n\in H\}$$
is a union of countably many continuous images of $H^n$, thus for each $k$, $(G_n)^k$
is a union of countably many continuous images of $H^{nk}$.
As the property $\ufin(\O,\Gamma)$ is preserved under taking continuous images
and countable unions \cite{coc2, AddQuad}, we have that each set $(G_n)^k$ satisfies
$\ufin(\O,\Gamma)$.

Take $G_H=\<H\>$.
Then $G_H=\Union_{n\in\N}G_n$ and for each $n$, $G_n\sbst G_{n+1}$.
Thus, $(G_H)^k = \Union_{n\in\N} (G_n)^k$ for each $k$;
therefore $(G_H)^k$ satisfies $\ufin(\O,\Gamma)$ for each $k$.
Now, satisfying $\ufin(\O,\O)$ in all finite powers
implies $\sfin(\Omega,\Omega)$ \cite{coc2}.

It was observed by Pol and Zdomskyy, that one can make sure that
$G_H$ is not $\sigma$-compact by embedding $\Increasing$ in a Cantor set of reals, $C$,
that is \emph{linearly independent} over $\Q$. This way, $H=\<H\>\cap C$ is a closed
subset of $\<H\>=G_H$. Since $H$ is not $\sigma$-compact \cite{ideals}, we get that $G_H$
cannot be $\sigma$-compact.
\end{proof}

\begin{prob}
Does $G_H$ satisfy $\sone(\Gamma,\Gamma)$?
\end{prob}

\begin{cor}
$\mbox{TWO}\uparrow\gfin(\O,\O)$ is strictly stronger than
$\mbox{TWO}\uparrow\gfin(\Onbd,\O)$ (strict $o$-boundedness).
\end{cor}
\begin{proof}
By a well known theorem of Telg\'arsky, $\mbox{TWO}\uparrow\gfin(\O,\O)$
if, and only if, the space $G$ is $\sigma$-compact.
Contrast this with Theorem \ref{ZFCgp}.
\end{proof}

Theorem \ref{ZFCgp} has a group-theoretic consequence.

\begin{cor}
Assume that for each $m$, $\seq{g_{m,n}+U_m}$ is a cover of $G_H$.
Then there exists a sequence
$\seq{m_n}$ such that
$$G_H=\Union_k\bigcap_{n>k}(\{g_{n,1},\dots,g_{n,m_n}\} + U_n).$$
In fact, this property is satisfied by all finite powers of $G_H$.
\end{cor}

Let $D =\{g_\alpha : \alpha < \d\}$ be a dominating subset
of $\NN$ where each $g_\alpha$ is increasing, and
for each $f\in\NN$ there exists $\alpha_0<\d$ such that for any
finite set $F\sbst\d\setminus\alpha_0$,
$f(n) < \min\{g_\beta(n) : \beta\in F\}$
for infinitely many $n$.
Such a set was constructed in \cite{ideals}.

A subset $F$ of $\NN$ is \emph{finitely-dominating}
if for each $g\in\NN$ there exist $k$ and $f_1,\dots,f_k\in\NN$
such that $g(n)\le^* \max\{f_1(n),\dots,f_k(n)\}$.
For conciseness, we use the following shortened notation.
\bi
\i[$\Ax$:]
Either a union of less than $\d$ many not dominating sets is not
dominating (in other words, $\b=\d$), or else a union of less
than $\d$ many not finitely-dominating sets
of increasing functions is not finitely-dominating.
\ei
In \cite{ideals} it is shown that
$\Ax$ implies that $M=D\cup Q$ satisfies $\sfin(\Omega,\Omega)$.
(Observe that $\sfin(\Omega,\Omega)$ is preserved under taking finite powers \cite{coc2}.)
Assuming $\Ax$, one shows as in Theorem \ref{ZFCgp} that all finite powers of $G_M=\<M\>$
satisfy $\ufin(\O,\O)$ and gets the following.
\begin{thm}\label{mengrp}
Assume that $\Ax$ holds. Then:
\be
\i There exists a non $\sigma$-compact subgroup $G_M$ of $\R$
of cardinality $\d$ such that $G_M$ satisfies $\sfin(\Omega,\Omega)$.
\i Assume that for each $m$, $\seq{g_{m,n}+U_m}$ is a cover of $G_M$.
Then there exists a sequence
$\seq{m_n}$ such that for each finite subset $F$ of $G_M$, there exists
$n$ such that $F\sbst \{g_{n,1},\dots,g_{n,m_n}\} + U_n$.
Moreover, this holds for all finite powers of $G_M$.
\ee
\end{thm}

It follows from the next section that the hypothesis $\Ax$ is not necessary to prove
Theorem \ref{mengrp} (namely, it also follows from the incomparable assumption $\cov(\M)=\c$).

\begin{prob}
Is Theorem \ref{mengrp} provable in ZFC?
\end{prob}

\section{Products of $o$-bounded groups}

In Problem 3.2 of \cite{TkaIntro} and Problem 5.2 of \cite{Hernandez}
it is asked whether the (Tychonoff) product of two $o$-bounded groups is $o$-bounded.
We give a negative answer.
A negative answer was independently given by Krawczyk and Michalewski \cite{KMlinear},
but our result is stronger in the following sense:
Let $G$ be a topological group such that all finite powers of $G$ are Lindel\"of.
Then each open $\w$-cover of $G$ contains a \emph{countable} $\w$-cover of $G$.
Let $\BO$ denote the collection of all \emph{countable Borel} $\w$-covers of $G$.
In this case,
$$\sone(\BO,\BO) \impl \sone(\Omega,\Omega)\impl \sfin(\O,\O)$$
where no implication can be reversed \cite{coc2, CBC}, and the last property
(the Menger property) implies $\sfin(\Onbd,\O)$ ($o$-boundedness).
In \cite{KMlinear} it is proved that, assuming $\cov(\M)=\c$ (this is a small portion of
\CH), there exist groups $G_1$ and $G_2$ satisfying the Menger property $\sfin(\O,\O)$,
such that $G_1\x G_2$ is not $o$-bounded.
We use the same hypothesis to show that there exist such
groups satisfying $\sone(\BO,\BO)$.

\begin{fact}
Assume that $G$ is a $\le^*$-dominating subgroup of $\NR$.
Then $G$ is not $o$-bounded.
\end{fact}
\begin{proof}
As $\N$ can be partitioned into infinitely many infinite sets,
the following holds.
\begin{lem}
For each $o$-bounded group $G$ and sequence
$\seq{U_n}$ of neighborhoods of the identity of $G$,
there exists a sequence $\seq{F_n}$ of finite subsets of $G$
with $G=\bigcap_m\Union_{n>m}F_n\cdot G$.
\end{lem}
Consider the open sets $U_n = \{f\in\NR : |f(n)|<1\}\sbst\NR$.
For each sequence $\seq{F_n}$ of finite subsets of
$\NR$, $\Union_nF_n+U_n$ is $\le^*$-bounded in $\NR$.
Let $h\in\NR$ be a witness for that.
As $G$ is dominating, there exists $g\in G$ such that $h\le^* g$.
Then $g\nin\bigcap_m\Union_{n>m}F_n\cdot G$.
\end{proof}

Let $\c=|\R|$. The assertion $\cov(\M)=\c$ means that $\R$ (or any complete, separable,
metric space) is not the union of less than continuum many meager (=first category) sets.
We say that $L$ is a \emph{$\kappa$-Luzin group} if $|L|\ge\kappa$, and
for each meager set $M$ in $L$, $|L\cap M|<\kappa$.
\begin{thm}\label{prods}
Assume that $\cov(\M)=\c$.
Then there exist $\c$-Luzin subgroups $L_1,L_2$ of $\NR$ satisfying $\sone(\BO,\BO)$,
such that the group $L_1\x L_2$ is not $o$-bounded.
(These Luzin groups are, in fact, linear vector spaces over $\Q$.)
\end{thm}
\begin{proof}
We extend the technique of \cite{CBC, AddQuad}.
We stress that there exists a much easier proof if we only require
that $L_1$ and $L_2$ satisfy $\sone(\Omega,\Omega)$; however we
do not supply this easier proof to avoid repetitions.

A cover $\cU$ of $X$ is \emph{$\w$-fat} if for each finite $F\sbst X$
and each finite family $\cF$ of
nonempty open sets, there exists $U\in\cU$
such that $F\sbst U$ and for each $O\in\cF$, $U\cap O$ is not meager.
Let $\M$ denote the meager subsets of $\NR$.
\begin{lem}[\cite{huremen2}]\label{fatlemma}
%\be
%\i Assume that $\cU$ is a countable collection of Borel sets.
%Then $\cup\cU$ is comeager if, and only if,
%for each nonempty basic open set $O$ there exists $U\in\cU$
%such that $U\cap O$ is not meager.
%\i
Assume that $\cU$ is an $\w$-fat cover of a set $X\sbst\NR$.
Then:
\be
\i $\cup\cU$ is comeager,
\i For each finite $F\sbst G$ and finite family $\cF$ of
nonempty open sets,
$$\cU_{F,\cF} := \{U\in\cU : F\sbst U\mbox{ and for each $O\in\cF$, }U\cap O\nin\M\}$$
is an $\w$-fat cover of $G$.
Consequently, $\cup\cU_{F,\cF}$ is comeager.
\ee
\end{lem}

For shortness, we will say that a cover $\cU$ of $X$ is \emph{good} if:
For each finite $A\sbst\Q\sm\{0\}$ and finite $B\sbst G$,
the family
$$\cU^{A,B} := \left\{\bigcap_{q\in A, g\in B} q(U-g) : U\in\cU\right\}$$
is an $\w$-fat cover of $X$.
These covers allow the inductive construction hinted in the following lemma.

\begin{lem}\label{addelement}
Assume that $\cU$ is a good cover of a group $G\sbst\NR$.
%, and $\Q\cdot G\sbst G$.
Then for each element $x$ in the intersection of all sets of the form
$\cup(\cU^{A,B})_{F,\cF}$
where the members of $\cF$ are \emph{basic} open sets,
$\cU$ is a good cover of the group $G+\Q x$.
\end{lem}
\begin{proof}
Fix finite sets $A\sbst\Q\sm\{0\}$ and $B\sbst G$.
We may assume that $1\in A$.
We must show that $\cU^{A,B}$ is an $\w$-fat cover of $G+\Q x$.
Let $F$ be a finite subset of $G+\Q x$, and $\cF$ be a finite family of
nonempty open sets. By moving to subsets we may assume that all members of
$\cF$ are basic open sets.

Choose finite sets $\tilde A\sbst\Q\sm\{0\}$, $\tilde B\sbst G$, and $\tilde F\sbst G$,
such that $F\sbst (\tilde B + \tilde A x)\cup \tilde F$, $1\in \tilde A$, and $0\in\tilde B$.
As $x\in\cup(\cU^{\tilde A\inv A, B+A\inv\tilde B})_{\tilde F,\cF}$,
there exists $U\in\cU$ such that
$x\in \tilde V:=\bigcap_{q\in \tilde A\inv A, g\in B+A\inv\tilde B}q(U-g)$,
$\tilde F\sbst\tilde V$, and for each $O\in\cF$, $\tilde V\cap O$ is not meager.
Take $V=\bigcap_{q\in A, g\in B}q(U-g)$. Then $\tilde V\sbst V$, thus
$\tilde F\sbst V$ and for each $O\in\cF$, $V\cap O$ is not meager.
Now, $x\in\tilde V$, thus for each $\tilde a\in\tilde A$ and $\tilde b\in \tilde B$,
$x\in\bigcap_{q\in \tilde a\inv A, g\in B+A\inv\tilde b}q(U-g)$, thus
$x\in\bigcap_{q\in A, g\in B}\tilde a\inv q(U-(g+q\inv\tilde b))$, therefore
$x\in\bigcap_{q\in A, g\in B}\tilde a\inv (q(U-g)-\tilde b)$, thus
$\tilde a x+\tilde b\in V$. This shows that
$F\sbst (\tilde A x + \tilde B)\cup\tilde F\sbst V\in\cU^{A,B}$,
and we are done.
\end{proof}

Since we are going to construct Luzin groups, the following lemma
tells that we need not consider covers which are not good.
\begin{lem}\label{denselusin}
Assume that $L$ is a subgroup of $\NR$ such that $\Q\cdot L\sbst L$ and
for each nonempty basic open set $O$, $L\cap O$ is not meager.
Then every countable Borel $\w$-cover $\cU$ of $L$ is a good cover of $L$.
\end{lem}
\begin{proof}
Assume that $\cU$ is a countable collection of Borel sets which is
not a good cover of $L$.
Then there exist finite sets $A\sbst\Q\sm\{0\}$, $B\sbst L$,
$F\sbst L$, and $\cF$ of nonempty open sets
such that for each $V\in\cU^{A,B}$ containing $F$, $V\cap O$ is meager for some $O\in\cF$.
For each $O\in\cF$ let
$$M_O = \cup\{V\in\cU^{A,B} : F\sbst V\mbox{ and }V\cap O\in\M\}.$$
Then $M_O\cap O$ is meager,
thus there exists $x_O\in (L\cap O)\sm M_O$.
Then $F\cup\{x_O : O\in\cF\}$ is not covered by any $V\in\cU^{A,B}$.
We will show that this cannot be the case.

Put $F' = A\inv F + B$.
As $\Q\cdot L\sbst L$, $F'$ is a finite subset of $L$.
Thus, there exists $U\in\cU$ such that $F'\sbst U$.
Consequently, for each $q\in A$ and $g\in B$,
$x\in q(U-g)$ for each $x\in F$, that is,
$F\sbst \bigcap_{q\in A, g\in B} q(U-g)\in\cU^{A,B}$.
\end{proof}

We need one more lemma. Denote the collection of countable Borel
good covers of a set $X$ by $\Bgood$.
\begin{lem}\label{s1goodgood}
If $|X|<\cov(\M)$, then $X$ satisfies $\sone(\Bgood,\Bgood)$.
\end{lem}
\begin{proof}
Assume that $|X|<\cov(\M)$, and let $\seq{\cU_n}$ be a sequence of
countable Borel good covers of $X$.
Enumerate each cover $\cU_n$ by $\{U^n_k\}_{k\in\N}$.
Let $\seq{Y_n}$ be a partition of $\N$ into infinitely many
infinite sets. For each $m$, let $y_m\in\NN$ be an increasing
enumeration of $Y_m$.
Let $\seq{\cF_n}$ be an enumeration of all finite families of
nonempty basic open sets.

For finite sets $F,B\sbst X$ and $A\sbst\Q\sm\{0\}$,
and each $m$ define a function
$\Psi^{A,B}_{F,m}\in\NN$ by
$$\Psi^{A,B}_{F,m}(n)=\min\{k : F\sbst V:= \bigcap_{q\in A,g\in B}q(U^{y_m(n)}_k-g)\mbox{ and }(\forall O\in\cF_m)\
V\cap O\nin\M\}.$$
Since there are less than $\cov(\M)$ many functions $\Psi^{A,B}_{F,m}$,
there exists by \cite{covM} a function $f\in\NN$ such that
for each $m$, $F$, $A$, and $B$, $\Psi^{A,B}_{F,m}(n)=f(n)$ for infinitely many $n$.
Consequently, $\cV=\{U^{y_m(n)}_{f(n)} : m,n\in\N\}$ is a good cover of $X$.
\end{proof}

We are now ready to carry out the construction.
Let $\NR=\{y_{\alpha}:\alpha<\c\}$,
$\{M_{\alpha}:\alpha<\c\}$ be all $F_\sigma$ meager subsets of $\NR$, and
$\{\seq{\cU^\alpha_n} : \alpha<\c\}$ be all sequences of countable families of
Borel sets.
Let $\{O_k : k\in\N\}$ and $\{\cF_m: m\in\N\}$ be all nonempty basic open sets
and all finite families of nonempty basic open sets, respectively, in $\NR$.

We construct $L_1$ and $L_2$ by induction on $\alpha<\c$ as follows.
At stage $\alpha \ge 0$ set
$L^i_\alpha  = \Union_{\beta<\alpha}L^i_\beta$
and consider the sequence $\seq{\cU^\alpha_n}$.
Say that $\alpha$ is $i$-good if for each $n$
$\cU^\alpha_n$ is a good cover of $L^i_\alpha$.
In this case, by Lemma \ref{s1goodgood} there exist elements
$U^{\alpha,i}_n\in\cU^\alpha_n$ such that $\cU^{\alpha,i} = \seq{U^{\alpha,i}_n}$ is
a good cover of $L^i_\alpha$.
We make the inductive hypothesis that
for each $i$-good $\beta<\alpha$,
$\cU^{\beta,i}$ is a good cover of $L^i_\alpha$.
For finite sets $F,B\sbst L^i_\alpha$ and $A\sbst\Q\sm\{0\}$,
each $i$-good $\beta\le\alpha$, and each $m$ define
$$G^{\beta,i}_{A,B,F,m}=\cup((\cU^{\beta,i})^{A,B})_{F,\cF_m}.$$
As $\cU^{\beta,i}$ is a good cover of $X^i_\alpha$, $(\cU^{\beta,i})^{A,B}$
is $\w$-fat cover of $L^i_\alpha$, and
by Lemma \ref{fatlemma}, $G^{\beta,i}_{A,B,F,m}$ is comeager in $\NR$.
Set
$$Y_\alpha=\Union_{\beta<\alpha}M_\beta\cup
\Union\left\{\NR\sm G^{\beta,i}_{A,B,F,m} :
\txt{
$i<2,\mbox{ $i$-good }\beta\le\alpha, m\in\N,$\\
finite sets $F,B\sbst L^i_\alpha, A\sbst\Q\sm\{0\}$
}
\right\},$$
and $Y_\alpha^* = \{x\in\NR : (\E y\in
Y_\alpha)\ x=^* y\}$ (where $x =^* y$ means that $x(n)=y(n)$ for
all but finitely many $n$.) Then $Y_\alpha^*$ is a union of less
than $\cov(\M)$ many meager sets.
\begin{lem}[\cite{huremen2}]\label{x+y=z}
If $X$ is a union of less than $\cov(\M)$ many
meager sets in $\NR$,
then for each $x\in\NR$ there exist $y,z\in\NR\sm X$ such that $y+z=x$.
\end{lem}
Use Lemma \ref{x+y=z} to pick
$x^0_\alpha,x^1_\alpha\in\NR\sm Y_\alpha^*$ such that
$x^0_\alpha+x^1_\alpha=y_\alpha$. Let $k = \alpha \bmod \omega$,
and change a finite initial segment of $x^0_\alpha$ and
$x^1_\alpha$ so that they both become members of $O_k$. Then
$x^0_\alpha,x^1_\alpha\in O_k\sm Y_\alpha$, and
$x^0_\alpha+x^1_\alpha =^* y_\alpha$.
Finally, define $X^i_{\alpha+1} = L^i_\alpha + \Q\cdot x^i_\alpha$.
By Lemma \ref{addelement}, the inductive hypothesis is preserved.
This completes the construction.

Take $L_i = \Union_{\alpha<\c}L^i_\alpha$, $i=1,2$.
By Lemma \ref{s1goodgood},
each $L_i$ satisfies $\sone(\Bgood,\Bgood)$,
and by the construction,
its intersection with each nonempty basic open set has size $\c$.
By Lemma \ref{denselusin}, $\Bgood = \BO$ for $L_i$.
Finally, $L_1+L_2$ (a homomorphic image of $L_1\x L_2$)
is a dominating subset of $\NR$, thus $L_1\x L_2$ is not $o$-bounded.
\end{proof}

\begin{rem}
None of the $o$-bounded groups $L_1$ and $L_2$ in Theorem \ref{prods} is
strictly $o$-bounded. By an unpublished result of Michalewski,
every metrizable strictly $o$-bounded group is a subgroup of a $\sigma$-compact
group, and in \cite[Theorem 5.3]{Hernandez} it is proved that a product of
such a group with an $o$-bounded group is $o$-bounded.
\end{rem}

The following consequence of Theorem \ref{prods} seems nontrivial.
Say that a subset $S$ of a topological group $G$ is
$\aleph_0$-bounding if there exists a countable
set $F\sbst G$ such that $F\cdot S=G$.
For example, $G$ is \emph{$\aleph_0$-bounded}
if each nonempty open set in $G$
is $\aleph_0$-bounding.
The first property in the following corollary may be called
\emph{Borel $o$-boundedness}.
This property is more interesting when the group in question
is $\aleph_0$-bounded, in which case it is stronger than
$o$-boundedness.

\begin{cor}
Assume that $\cov(\M)=\c$.
Then there exists an $\aleph_0$-bounded topological group $G$
of size continuum such that:
\be
\i For each sequence $\seq{B_n}$ of
$\aleph_0$-bounding Borel sets in $G$,
there exits a sequence $\seq{F_n}$ of finite
subsets of $G$ such that $G=\Union_n F_n\cdot B_n$.
\i Moreover, the sequence in (1) will have the
property that for each finite $F\sbst G$ there exists
$n$ such that $F\sbst F_n\cdot U_n$.
\ee
\end{cor}

\begin{rem}
Banakh, Nickolas, and Sanchis have also, independently, proved the consistency of
$o$-bounded groups not being closed under taking finite prodocts
(however, they do not consider stronger combinatorial properties as done in
\cite{KMlinear} and here).
Their construction uses ultrafilters which are not nearly coherent --
see \cite{BNS}.
\end{rem}

\section{Groups satisfying $\sone(\BO,\BG)$ or $\sone(\BG,\BG)$}

One may wonder whether Theorem \ref{prods} can be strengthened further
so that $L_1$ and $L_2$ will satisfy a stronger property.
By inspection of the Scheepers Diagram \ref{SchDiagram}, the only
candidate for a stronger property (among the ones considered in
this paper) is $\sone(\BO,\BG)$. This is far from possible:
A result of \cite{coc11} (see \cite[Theorem 32]{LecceSurvey})
implies that whenever $G_1$ is a topological group satisfying $\sone(\Omega,\Gamma)$,
and $G_2$ is $o$-bounded, the group $G_1\x G_2$ is $o$-bounded.
However strong, though, the notion of a topological group satisfying
$\sone(\BO,\BG)$ is not trivial.
\begin{thm}
For each cardinal $\kappa$ with $\cf(\kappa)>\aleph_0$,
it is consistent that $\c=\kappa$ and there exists a topological
subgroup of $\R$ of size $\c$, which satisfies $\sone(\BO,\BG)$.
\end{thm}
\begin{proof}
This is really a theorem of Miller:
Let $M$ be a countable standard model of ZFC
satisfying $\c=\kappa$.
In \cite{MilNonGamma} it is proved that
there exists a ccc poset $\P$ in $M$ of size continuum
(so that forcing with $\P$ does not change the size of the continuum)
such that the old reals $M\cap\R$ satisfy $\sone(\BO,\BG)$ in $V^\P$.

But observe that, as the operations of addition and substraction in $\R$
are absolute, $G=M\cap\R$ is a group in $V^\P$.
\end{proof}

%We do not know to translate this result into a standard mathematical
%theorem.

\begin{prob}\label{gammagp}
Does \CH{} imply the existence of a separable metrizable
group of size $\c$ which satisfies $\sone(\BO,\BG)$?
\end{prob}

We have some approximate results.
For a sequence $\seq{X_n}$ of subsets of $X$, define
$\liminf X_n = \Union_m\bigcap_{n\ge m} X_n$.
For a family $\cF$ of subsets of $X$, $L(\cF)$ denotes
its closure under the operation $\liminf$.
According to \cite{GN},
$X$ is a \emph{$\delta$-set} if
for each $\w$-cover $\cU$ of $X$, $X\in L(\cU)$.
It is easy to see that the $\delta$-property is preserved under taking countable
\emph{increasing} unions.
Clearly $\sone(\Omega,\Gamma)$ implies the $\delta$-property.
The converse is still an open problem \cite{futurespm}.
If a $\delta$-set is a group, we will call it a $\delta$-group.

Let $\Z_2$ denote the usual group $\{0,1\}$ with modulo $2$ addition.
\begin{thm}
Assume that $\p=\c$. Then there exists a subgroup $G$ of $\Cantor$
such that for each $k$ $G^k$
is a countable increasing union of sets satisfying $\sone(\Omega,\Gamma)$.
In particular, all finite powers of $G$ are $\delta$-groups.
\end{thm}
\begin{proof}
As $\p=\c$, there exists a subset $X$ of $\Cantor$ of size continuum which
satisfies $\sone(\Omega,\Gamma)$ \cite{GM}.
We may assume that $0\in X$. Consequently,
$$G := \<X\> = \Union_{n\in\N}\{x_1+\dots+x_n : x_1,\dots,x_n\in X\}$$
is a countable increasing union of continuous images of powers of
$X$. But the property $\sone(\Omega,\Gamma)$ is closed under taking
finite powers and continuous images \cite{coc2}.
Observe that each finite power of $G$ is the countable increasing union
of the same power of the original sets.
\end{proof}

Assuming \CH{}, there exists a set of reals $X$ of size $\c$
satisfying $\sone(\BO,\BG)$ (e.g., \cite{MilNonGamma}).
It is an open problem whether $\sone(\BO,\BG)$ is provably preserved under
taking finite powers \cite{futurespm}. If it is, then we have a positive
answer to Problem \ref{gammagp}.

\begin{thm}\label{borgamma}
Assume that $X\sbst\Cantor$ satisfies $\sone(\BO,\BG)$ in all finite powers.
Then $G=\<X\>$ satisfies $\sone(\BO,\BG)$ in all finite powers.
\end{thm}
\begin{proof}
By the above arguments, it suffices to prove the following.
\begin{lem}
$\sone(\BO,\BG)$ is preserved under taking countable increasing unions.
\end{lem}
To prove the lemma, assume that $X=\Union_n X_n$ is an increasing union,
and observe that $\sone(\BO,\BG)$ implies
$\sone(\BG,\BG)$, which in turn implies that all Borel images of
each $X_k$ in $\NN$ are bounded \cite{CBC}.

Assume that $\cU_n=\{U^n_m : m\in\N\}$, $n\in\N$, are countable Borel $\w$-covers of $X$.
By $\sone(\BO,\BG)$, we may assume that each $\cU_n$ is a $\gamma$-cover of $X_n$.
For each $k$, define a function $\Psi_k$ from $X_k$ to $\NN$ so that for each $x$ and $n$:
$$\Psi_k(x)(n) = \min\{i : \(\forall m\geq i\)\ x\in U^n_m\}.$$
$\Psi_k$ is a Borel map, thus $\Psi_k[X]$ is bounded.
Consequently, $\Union_k\Psi_k[X]$ is bounded, say by the sequence $m_n$.
Then $\seq{U^n_{m_n}}$ is a $\gamma$-cover of $X$, as required.
\end{proof}

We conclude the paper with a group satisfying $\sone(\BG,\BG)$ in all finite powers.
Let $\cN$ denote the collection of null (Lebesgue measure zero) sets of reals.
$\cov(\cN)$ is the minimal size of a cover of $\R$ by null sets, and
$\cof(\cN)$ is the minimal size of a cofinal family in $\cN$ with respect to inclusion.
Let $\kappa$ be an uncountable cardinal.
$S\sbst\R$ is a $\kappa$-Sierpi\'nski set if $|S|\ge\kappa$ and
for each null set $N$, $|S\cap N|<\kappa$. $\b$-Sierpi\'nski sets satisfy
$\sone(\BG,\BG)$, but powers of $\kappa$-Sierpi\'nski sets are never
$\kappa$-Sierpi\'nski sets.
\begin{thm}
Assume that $\cov(\cN)=\cof(\cN)=\b$.
Then there exists a (non $\sigma$-compact)
group $G\sbst\R$ of size $\b$, such that
all finite powers of $G$ satisfy $\sone(\BG,\BG)$.
Consequently, all finite powers of $G$ also satisfy
$\sfin(\BO,\BO)$.
\end{thm}
\begin{proof}
The proof is similar to that of Theorem 7.2 of \cite{BRR2}.
\begin{lem}\label{sier}
Assume that $\cov(\cN)=\cof(\cN)$.
Then there exists a $\cov(\cN)$-Sierpi\'nski set $S$ such that
for each $k$ and each null set $N$ in $\R^k$,
$S^k \cap N$ is contained in a union of less than $\cov(\cN)$
many continuous images of $S^{k-1}$.
\end{lem}
\begin{proof}
Let $\kappa = \cov(\cN)=\cof(\cN)$.
Let $\{N^{(k)}_\alpha : \alpha<\kappa\}$ be a
cofinal family of null sets in $\R^k$, $k\in\N$.

For $J\sbst\R^k$, $x\in\R$, and $i<k$, define
$J_{(x,i)} = \{v_1\frown v_2 : \ell(v_1)=i,\ v_1\frown x\frown v_2\in J\}$.
By the Fubini Theorem, for each null set $N\sbst\R^k$ and $i<k$,
$$\tilde N = \{x : (\E i<k)\ N_{(x,i)}\mbox{ is not null in }\R^{k-1}\}\in\cN.$$

We make an inductive construction on $\alpha<\kappa$ of elements $x_\alpha\in\R$
with auxiliary collections $\cF_\alpha$ of null sets, as follows.
For $\alpha<\kappa$ let $\cP_\alpha = \{N^{(k)}_\alpha : k\in\N\}$.
At step $\alpha$ do the following:
\be
\i Choose
$x_\alpha\nin\Union_{\beta<\alpha}\(\Union_{N\in(\cP_\beta\cup\cF_\beta)\sm P(\R)}\tilde N\cup
\Union_{N\in(\cP_\beta\cup\cF_\beta)\cap P(\R)}N\).$
\i Set $\cF_\alpha = \{N_{(x_\alpha,i)} : \beta<\alpha,\ N\in(\cP_\beta\cup\cF_\beta)\sm P(\R),\ i\in\N\}$.
\ee
This is possible because $x_\alpha$ is required to avoid membership in
a union of less than $\cov(\cN)$ many null sets.

Take $S = \{x_\alpha : \alpha<\kappa\}$. Then $S$ is a $\kappa$-Sierpi\'nski set.
Fix $k$. For each null $N\sbst\R^k$, there exists
$\beta<\kappa$ with $N\sbst N^{(k)}_\beta$.
Whenever $\beta<\alpha_0<\dots<\alpha_{k-1}$, and $\pi$ is a permutation on $\{0,\dots,k-1\}$,
$(N^{(k)}_\beta)_{(x_{\alpha_0},\pi\inv(0))} \in\cF_{\alpha_0}$,
thus
$(N^{(k)}_\beta)_{(x_{\alpha_0},\pi\inv(0)),(x_{\alpha_1},\pi\inv(1))} \in\cF_{\alpha_1}$,
\dots,
$(N^{(k)}_\beta)_{(x_{\alpha_0},\pi\inv(0)),\dots,(x_{\alpha_{k-2}},\pi\inv(k-2))} \in\cF_{\alpha_{k-2}}$,
thus $x_{\alpha_{k-1}}\nin (N^{(k)}_\beta)_{(x_{\alpha_0},\pi\inv(0)),\dots,(x_{\alpha_{k-2}},\pi\inv(k-2))}$,
that is, $(x_{\alpha_{\pi(0)}},\dots,x_{\alpha_{\pi(k-1)}})\nin N^{(k)}_\beta$.


Consequently, $S^k\cap N$ is contained in the union of all sets of the form
$S^i\x\{x_\xi : \xi\le\beta\}\x S^{k-i-1}$ ($i< k$)---a union of $|\beta|<\kappa$
copies of $S^{k-1}$---and $\{v\in S^k : v_i=v_j\}$ ($i<j<k$),
which are continuous images of $S^{k-1}$.
\end{proof}
Now assume that $\cov(\cN)=\cof(\cN)=\b$, and let $S$ be a
$\b$-Sierpi\'nski set as in Lemma \ref{sier}.
We will show by induction that for each $k$, $S^k$ satisfies $\sone(\BG,\BG)$.
By \cite{CBC} it is enough to show that for each null $N\sbst\R^k$,
$S^k\cap N$ satisfies $\sone(\BG,\BG)$.
By Lemma \ref{sier} and the induction hypothesis, $S^k\cap N$ is contained in a union of less than
$\b$ many sets satisfying $\sone(\BG,\BG)$.
In \cite{AddQuad} it is shown that
$\sone(\BG,\BG)$ is preserved under taking unions of size
less than $\b$, and in \cite{ideals} it is shown that
$\sone(\BG,\BG)$ is preserved under taking subsets.
This proves the assertion.

So all finite powers of $S$ satisfy $\sone(\BG,\BG)$,
and $G=\<S\>$ works, since as we mentioned
before, $\sone(\BG,\BG)$ is preserved under taking countable
unions.
Finally, by \cite{CBC} $\sone(\BG,\B)$ in all finite powers implies
$\sfin(\BO,\BO)$.
\end{proof}

It seems that the following was not known before.

\begin{cor}
Assume that $\cov(\cN)=\cof(\cN)=\b$. Then
there exists a set of reals satisfying
$\sone(\BG,\BG)$ and $\sfin(\BO,\BO)$, but not $\sone(\O,\O)$.
\end{cor}

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Topology and its Applications \textbf{127} (2003),
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325--330.

\bibitem{KMlinear}
A.\ Krawczyk and H.\ Michalewski,
\emph{Linear metric spaces close to being $\sigma$-compact},
Technical Report 46 (2001) of the Institute of Mathematics,
Warsaw University.
\texttt{http://www.minuw.edu.pl/english/research/reports/tr-imat/46/products.ps}

\bibitem{MilNonGamma}
A.\ W.\ Miller,
\emph{A Nonhereditary Borel-cover $\gamma$-set},
Real Analysis Exchange \textbf{29} (2003/4),
601--606.
%\arx{math.LO/0304414}

\bibitem{coc1}
M.\ Scheepers,
\emph{Combinatorics of open covers I: Ramsey theory},
Topology and its Applications \textbf{69} (1996),
%Topology Appl.\ \textbf{69} (1996),
31--62.

\bibitem{LecceSurvey}
M.\ Scheepers,
\emph{Selection principles and covering properties in topology},
Note di Matematica \textbf{22} (2003),
%Note Mat.\ \textbf{22} (2003),
3--41.

\bibitem{CBC}
M.\ Scheepers and B.\ Tsaban,
\emph{The combinatorics of Borel covers},
Topology and its Applications \textbf{121} (2002),
%Topology Appl.\ \textbf{121} (2002),
357--382.
%\arx{math.GN/0302322}

\bibitem{TkaIntro}
M.\ Tka\v{c}enko,
\emph{Introduction to topological groups},
Topology and its Applications \textbf{86} (1998),
%Topology Appl.\ \textbf{86} (1998),
179--231.

\bibitem{futurespm}
B.\ Tsaban,
\emph{Selection principles in Mathematics: A milestone of open problems},
Note di Matematica \textbf{22} (2003),
%Note Mat.\ \textbf{22} (2003),
179--208.
%\\ \arx{math.GN/0312182}

\newcommand{\Bc}[9]{\bibitem{#1} {#2}, \emph{#3}, in: \textbf{#4} (#5), #6 #7, #8--#9.}
\Bc{AddQuad}{B. Tsaban}{Additivity numbers of covering properties}{Selection Principles and Covering Properties in Topology}
{L. Ko\v{c}inac, ed.}{Quaderni di Matematica 18, Seconda Universita di Napoli, Caserta}{2006}{245}{282}

\end{thebibliography}

\end{document}

math0307225.extracted_bib
\begin{thebibliography}{00} %\input{../bib.tex}

\bibitem{coc11}
L.\ Babinkostova, Lj.\ D.R.\  Ko\v{c}inac, and M.\ Scheepers,
\emph{Combinatorics of open covers (XI): Menger- and Rothberger-bounded groups},
Topology and its Applications \textbf{154} (2007), 1269--1280.

\bibitem{BNS}
T.\ Banakh, P.\ Nickolas, and M.\ Sanchis,
\emph{Filter games and pathological subgroups of a countable product of lines},
Journal of the Australian Mathematical Society \textbf{81} (2006), 321--350.

\bibitem{covM}
T.\ Bartoszy\'nski,
\emph{Combinatorial aspects of measure and category},
Fundamenta Mathematicae \textbf{127} (1987),
%Fund.\ Math.\ \textbf{127} (1987),
209--213.

\bibitem{huremen2}
T.\ Bartoszy\'nski, S.\ Shelah, and B.\ Tsaban,
\emph{Additivity properties of topological diagonalizations},
Journal of Symbolic Logic \textbf{68} (2003),
%J.\ Symbolic Logic \textbf{68} (2003),
1254--1260.

\bibitem{ideals}
T.\ Bartoszy\'nski and B.\ Tsaban,
\emph{Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures},
Proceedings of the American Mathematical Society \textbf{134} (2006),
605--615.

\bibitem{BRR2}
L.\ Bukovsky, I.\ Reclaw, M.\ Repicky,
\emph{Spaces not distinguishing convergences of real-valued functions},
Topology and its Applications \textbf{112} (2001),
%Topology Appl.\ \textbf{112} (2001),
13--40.

\bibitem{GM}
F.\ Galvin and A.\ W.\ Miller,
\emph{$\gamma$-sets and other singular sets of real numbers},
Topology and its Applications \textbf{17} (1984),
%Topology Appl.\ \textbf{17} (1984),
145--155.

\bibitem{GN}
J.\ Gerlits and Zs.\ Nagy,
\emph{Some properties of $C(X)$, I},
Topology and its Applications \textbf{14} (1982),
%Topology Appl.\ \textbf{14} (1982),
151--161.

\bibitem{Hernandez}
C.\ Hernandez,
\emph{Topological groups close to being $\sigma$-compact},
Topology and its Applications \textbf{102} (2000),
%Topology Appl.\ \textbf{102} (2000),
101--111.

\bibitem{HRT}
C.\ Hernandez, D.\ Robbie and M.\ Tkachenko,
Some properties of $o$-bounded and strictly $o$-bounded groups,
Applied General Topology \textbf{1} (2000),
%Appl.\ Gen.\ Topol.\ \textbf{1} (2000),
29--43.

\bibitem{coc2}
W.\ Just, A.\ W.\ Miller, M.\ Scheepers, and P.\ J.\ Szeptycki,
\emph{The combinatorics of open covers II},
Topology and its Applications \textbf{73} (1996),
%Topology Appl.\ \textbf{73} (1996),
241--266.

\bibitem{KMexample}
A.\ Krawczyk and H.\ Michalewski,
\emph{An example of a topological group},
Topology and its Applications \textbf{127} (2003),
%Topology Appl.\ \textbf{127} (2003),
325--330.

\bibitem{KMlinear}
A.\ Krawczyk and H.\ Michalewski,
\emph{Linear metric spaces close to being $\sigma$-compact},
Technical Report 46 (2001) of the Institute of Mathematics,
Warsaw University.
\texttt{http://www.minuw.edu.pl/english/research/reports/tr-imat/46/products.ps}

\bibitem{MilNonGamma}
A.\ W.\ Miller,
\emph{A Nonhereditary Borel-cover $\gamma$-set},
Real Analysis Exchange \textbf{29} (2003/4),
601--606.
%\arx{math.LO/0304414}

\bibitem{coc1}
M.\ Scheepers,
\emph{Combinatorics of open covers I: Ramsey theory},
Topology and its Applications \textbf{69} (1996),
%Topology Appl.\ \textbf{69} (1996),
31--62.

\bibitem{LecceSurvey}
M.\ Scheepers,
\emph{Selection principles and covering properties in topology},
Note di Matematica \textbf{22} (2003),
%Note Mat.\ \textbf{22} (2003),
3--41.

\bibitem{CBC}
M.\ Scheepers and B.\ Tsaban,
\emph{The combinatorics of Borel covers},
Topology and its Applications \textbf{121} (2002),
%Topology Appl.\ \textbf{121} (2002),
357--382.
%\arx{math.GN/0302322}

\bibitem{TkaIntro}
M.\ Tka\v{c}enko,
\emph{Introduction to topological groups},
Topology and its Applications \textbf{86} (1998),
%Topology Appl.\ \textbf{86} (1998),
179--231.

\bibitem{futurespm}
B.\ Tsaban,
\emph{Selection principles in Mathematics: A milestone of open problems},
Note di Matematica \textbf{22} (2003),
%Note Mat.\ \textbf{22} (2003),
179--208.
%\\ \arx{math.GN/0312182}

\newcommand{\Bc}[9]{\bibitem{#1} {#2}, \emph{#3}, in: \textbf{#4} (#5), #6 #7, #8--#9.}
\Bc{AddQuad}{B. Tsaban}{Additivity numbers of covering properties}{Selection Principles and Covering Properties in Topology}
{L. Ko\v{c}inac, ed.}{Quaderni di Matematica 18, Seconda Universita di Napoli, Caserta}{2006}{245}{282}

\end{thebibliography}