\documentstyle[11pt,amstex]{amsart}
%\def \L{\char 147}\def \l{\char 149}
\title
{Remarks on meager $\Sigma^0_2$-supported $\sigma$-ideals on the real line}
\author{Marek Balcerzak}
\address{Institite of Mathematics, {\L}\'od\'z Technical University,
al. Politechniki 11, I-2, 90-924 {\L}\'od\'z, Poland}
\email{mbalce@@krysia.uni.lodz.pl}
\author{Krzysztof Ciesielski}
\address{Department of Mathematics, West Virginia University, Morgantown,
WV 26505-6310, USA}
\email{kcies@@wvnvms.wvnet.edu}
\author{Dorota Rogowska}
\address{Institite of Mathematics, {\L}\'od\'z Technical University,
al. Politechniki 11, I-2, 90-924 {\L}\'od\'z, Poland}
\thanks{This work was partially supported by NSF Cooperative Research
Grant INT-9600548 and its Polish part financed by KBN}
\email{dorotaro@@ck-sg.p.lodz.pl}
\subjclass{54E52, 54A35}
\keywords{meager $\sigma$-ideal, $F_\sigma$ sets, Polish topology}
\newcommand{\I}{{\cal I}}
\newcommand{\J}{{\cal J}}
\newcommand{\M}{{\cal M}}
\newcommand{\E}{{\cal E}}
\newcommand{\A}{{\cal A}}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma_2^0}
\newcommand{\R}{{\Bbb R}}
\newcommand{\con}{{\frak c}}
\newcommand{\add}{\mbox{add}}
\newcommand{\cof}{\mbox{cof}}
\newcommand{\cov}{\mbox{cov}}
\newcommand{\non}{\mbox{non}}
\newtheorem{tw}{Theorem}      %TWIERDZENIA
\newtheorem{wn}{Corollary}    %WNIOSKI
\newtheorem{uw}{Remark}       %UWAGI
\newtheorem{df}{Definition}   %DEFINICJE
\newtheorem{prop}{Proposition}%PROPOZYCJE
\newtheorem{prob}{Problem}    %PROBLEM
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\begin{document}
\maketitle

%ABSTRACT
\begin{abstract}
We consider a condition stating that, for each member $\I$ of some wide
class of $\Sigma^0_2$-supported $\sigma$-ideals on $\Bbb  R$, there exists
a Polish topology on $\Bbb R$ making $\I$ meager.
We show its independence of ZFC.
\end{abstract}

\section*{Introduction}
In \cite{CJ} the authors studied whether,
for a given (proper) $\sigma$-ideal
$\I$ of subsets of $X$, there exists a
possibly good topology $\tau$ on $X$
such that $\I$ equals the $\sigma$-ideal of meager
(i.e., the first category) sets with respect to $\tau$.
Such an $\I$ is called
briefly a $\tau$-{\em meager $\sigma$-ideal}.
In \cite{BR} the above problem was investigated
for the case when $\I$ is
a $\Sigma^0_2$-supported $\sigma$-ideal of
subsets of a given uncountable
Polish space $X$.
Recall that $\I$ is $\Sigma^0_2$-{\em supported\/}
if each set $A \in \I$ is contained
in a set $B \in \I$ of type $F_{\sigma}$.
(In another notation the class of
$F_{\sigma}$ sets is written as $\Sigma^0_2$, see e.g.~\cite{Ke}.)
In particular, it is interesting to know whether there exists a Polish
topology $\tau$ on $X$ such that $\I$ is $\tau$-meager.
These studies are connected with a recent theorem of Kechris and
Solecki from \cite{KS}.
By that theorem, if a $\Sigma^0_2$-supported
$\sigma$-ideal $\I$ on a Polish
space is $ccc$ (i.e., each family of disjoint
Borel sets which are not in
$\I$ is countable), then there exists
a countable family $ \cal F$ of
closed subsets of $X$ such that $\I$ consists of the sets $A \subseteq X$
for which
$A \cap F$ meager in $F$ for every $F \in \cal F$.
In that case, it is possible to find a Polish topology $\tau$ on $X$ such
that $\I$ is $\tau$-meager \cite{BR}.
(See also \cite{R}.)
In the present paper we discuss some further conditions under which there
exists (or does not) a Polish topology making a given
(not necessarily $ccc$)
$\Sigma^0_2$-supported $\sigma$-ideal meager.

We use standard set-theoretical notation as in~\cite{Ci}.
We consider $\sigma$-ideals $\I$ of subsets of the real line $\Bbb R$.
Let $\con=| \R |$. We say that $\A \subseteq \I$
is a {\em base\/} of $\I$ if
$$( \forall B \in \I ) ( \exists A \in {\cal A} ) ( B \subseteq A).$$
We denote:

\begin{eqnarray*}
\add (\I ) & = & \min \left\{ |\A |\colon \A \subseteq\I \mbox{ and } \bigcup
\A \notin \I \right\},\\
\cov (\I ) & = & \min \left\{  | \A |\colon \A \subseteq \I \mbox{ and }
\bigcup \A =\R  \right\},\\
\non (\I )& = &\min \left\{  | E |\colon E \subseteq \R
\mbox{ and } E \notin \I \right\},\\
\cof (\I )& = &\min \left\{  |\A |\colon \A \mbox{ is a base of } \I  \right\}.
\end{eqnarray*}
Recall that
$\add(\I)\leq\min\{\cov(\I),\non(\I)\}\leq\max\{\cov(\I),\non(\I)\}\leq\cof(
\I)$.

Let $ \M \subseteq {\cal P}( \R )$ stand for the $\sigma$-ideal
of meager (in the natural topology) subsets of $\R$, and let $\E
\subseteq {\cal P}( \R )$ denote the $\sigma$-ideal generated by
closed sets of Lebesgue measure zero.
Clearly, $\M$ and $\E $ are $\Sigma^0_2$-supported $\sigma$-ideals.
\bigskip

\section*{Results}

In the reminder of the paper the following statement will be denoted as~($*$).

\medskip

%\begin{quote}

For each $\sigma$-ideal $\I \subseteq {\cal P}( \Bbb R )$ such that
\begin{itemize}%{description}
\item[{\rm (a)}] $[ \Bbb R ]^{\leq \omega}  \subseteq \I , $
\item[{\rm (b)}] $\I$ is $\Sigma^0_2$-supported,
\item[{\rm (c)}] $ ( \forall A \in \I ) ( \exists B \in [ {\R} \setminus A
]^\con )
 ( B \in\I )$,
\end{itemize}%{description}
we have
\begin{itemize}%{description}
\item[{\rm (\dag)}] there exists a Polish topology $\tau$ on $\R$ such that
$\I$ is
$\tau$-meager.
\end{itemize}%{description}
%\end{quote}

\medskip

\noindent{\bf Remarks.} $1^0$ It is evident  that
each $ccc$ $\sigma$-ideal satisfies condition (c).\\
$2^0$ The $\sigma$-ideal $[\R ]^{\le\omega}$ satisfies (a)
and (b) but it
does not satisfy (c), and (\dag) is false for it since $\R$ with any
Polish topology $\tau$ contains a $\tau$-nowhere dense
uncountable ($\tau$-perfect) set.

\medskip

At first let us observe (using methods presented in \cite{BR})
that, in some
models of ZFC, the statement ($*$) is valid.

\begin{tw}
Assume that {\rm $\add( \M )=\cof(\M )= \kappa$.}
If a $\sigma$-ideal $\I \subseteq {\cal P}(\R)$ satisfies the conditions
{\rm (a), (c)}, and {\rm $\add (\I )=\cof (\I )=\kappa$}
then {\rm (\dag )} holds true.
\end{tw}
\begin{pf}
Using $\add (\M )=\cof (\M )=\add (\I)=\cof (\I )=\kappa$,
condition (c), and
Sierpi\'nski-Erd\H{o}s type argument (see \cite[Th.~19.5]{O},
\cite[Th.~2.1]{BR}, or
\cite[Th.~2.1.8]{BJ}) we can find bases
$\{ B_{\alpha}\colon \alpha < \kappa \}$,
$\{D_{\alpha}\colon \alpha < \kappa \}$ of $\M$ and $\I$,
respectively,
such that
\begin{itemize}
\item $B_\alpha \subseteq B_\gamma$ and $D_\alpha \subseteq D_\gamma$
for any $\alpha < \gamma < \kappa$,
\item $|B_0|=|D_0|=|B_{\alpha + 1}\setminus B_\alpha |=
|D_{\alpha +1}\setminus D_\alpha |=\con$
for each $\alpha < \kappa$, and
\item $B_\alpha =\bigcup_{\gamma < \alpha} B_\gamma$ and
$D_\alpha = \bigcup_{\gamma < \alpha}D_\gamma $
for any limit ordinal $\alpha < \kappa$.
\end{itemize}
Then a bijection $f\colon\R \to \R $ such that $f[B_0 ]=D_0$
and $f[B_{\alpha +1} \setminus B_\alpha ]=
D_{\alpha +1} \setminus D_\alpha$
for each $\alpha < \kappa$ is an
isomorphism between $\M$ and $\I$.
This means that for each $E \subseteq \R $ the conditions $E \in \M $
and $f[E] \in\I$ are equivalent. Hence the metric $\rho$ given by
$\rho (x,y)=|f(x)-f(y)|$ for $x,y \in \Bbb R$
generates a Polish topology
on $\Bbb R$ that makes $\I$ meager.
\end{pf}

\begin{wn}\label{cor1}
The continuum hypothesis CH implies {\rm ($*$)}.
In particular {\rm ($*$)} is consistent with ZFC.
\end{wn}

\begin{tw}
If {\rm $\cov ( \M )= \omega_1 < \cof ( \M )$} then {\rm ($*$)} is false.
\end{tw}
\begin{pf}
Since $\cov ( \M )= \omega_1$, we can find a family
$ \{ F_\alpha \colon \alpha < \omega_1 \}$ of $F_\sigma$ sets such that
$\bigcup_{\alpha < \omega_1} F_\alpha = \Bbb R $. Replacing $F_\alpha$ by
$\bigcup_{\gamma < \alpha}F_\gamma$, if necessary,
we can additionaly require that $F_\alpha \subseteq
F_\beta$ if $\alpha < \beta$.
By further modification, we can also
ensure that $|F_\beta \setminus F_\alpha |
=\con $ for every $\alpha < \beta<\omega_1$.

Let $\I$ consist of the sets $A \subseteq \Bbb R$ such
that $A \subseteq F_\alpha$ for some $\alpha < \omega_1$.
Then the $\sigma$-ideal $\I$ satisfies the conditions (a), (b), and (c).
Also observe that $\cof (\I )=\omega_1$.
Suppose that $\I$ satisfies (\dag) and let $\tau$ be a Polish topology
on $\Bbb R$ that makes $\I$ meager.
It is known that there exists a Borel
isomorphism between Polish spaces
$\langle\R , \tau\rangle$ and $\R$
(with natural topology) that preserves
Baire category \cite[Th.~3.15]{CKP}.
We thus have $\cof (\I )=\cof ( \M )$ which contradicts
$\cof (\I )= \omega_1 < \cof ( \M )$.
\end{pf}

Since the condition $\cov ( \M )= \omega_1 < \cof ( \M )$
is consistent with ZFC (see e.g.~\cite{BJ}) we can conclude that

\begin{wn} The condition {\rm ($*$)} is independent of ZFC.
\end{wn}

Another way of obtaining the consistency of $\neg$($*$) is the following
theorem.

\begin{tw}\label{Luz}
The condition {\rm ($*$)} implies the existence of a Luzin set in $\R$.
\end{tw}
\begin{pf}
Let $\J\subseteq\R$ be a $\si$-ideal generated by
$(\M\cap{\cal P}([0,1]))\cup[\R\setminus [0,1]]^{\le\omega}$.
Then $\J$ satisfies conditions (a), (b),
and (c). Hence, by ($*$), there exists a Polish topology $\tau$ on $\R$
making $\J$ meager. It is clear that $X=\R\setminus [0,1]$
is a $\tau$-Luzin set, which means that $|A\cap X|\le\omega$ for each
$\tau$-meager set $A$. Consider a Borel isomorphism $\varphi$ between
$\langle \R ,\tau\rangle$ and $\R$ with the natural topology, such that
$\varphi$ preserves Baire category.
Then $\varphi [X]$  is a Luzin set in~$\R$.
\end{pf}

By Theorem~\ref{Luz} the condition ($*$) is false in any model
of ZFC in which there is no Luzin set. In particular, since
there is no Luzin set under Martin's axiom
MA and the negation of continuum hypothesis (see \cite{Ku})
we can conclude the following corollary.

\begin{wn}
$MA+\neg CH$ implies the negation of {\rm ($*$)}.
\end{wn}

The next corollary gives a partial answer to a question
of Balcerzak and Rogowska~\cite{BR}.

\begin{wn}
The existence of a Polish topology making the ideal $\E$ meager is
independent of ZFC.
\end{wn}
\begin{pf}
Since $\E$ satisfies conditions (a), (b), and (c), it follows from
Corollary~\ref{cor1} that, under CH, there exists a Polish topology which makes
$\E$ meager.
On the other hand, it was proved in \cite{BS} that there are models of ZFC
in which $\non(\E)<\non(\M)$ and such
in which $\cov (\E )>
\cov (\M )$. In those models the existence of a Polish topology $\tau$
that makes $\E$ meager is impossible
since the respective Borel isomorphism
preserving Baire category would witness that $\non (\E )=\non (\M )$
and $\cov (\E )=\cov (\M )$.
\end{pf}

The proof of the above corollary is based on the fact that
the cardinal numbers $\non$ or $\cov$ distinguish between
$\E$ and $\M$. But what if the numbers
$\add$, $\cov$, $\non$, and $\cof$ are the same for $\M$ and some
$\sigma$-ideal $\I$ on $\R$?
Does it imply, in ZFC, that $\I$ can be made meager by a Polish topology?
This leads to the following open problem.

\begin{prob}
Is it provable in ZFC that the property {\rm (\dag )} holds true for every
$\sigma$-ideal $\I\subset{\cal P}(\R )$ fulfilling conditions
{\rm (a), (b), (c),} and the equalities
{\rm $\add (\I )=\add (\M )$, $\cov (\I)
=\cov (\M )$, $\non (\I )=\non (\M )$, $\cof (\I )=\cof (\M )$}?
\end{prob}

It is shown in \cite[Th.3.11]{CJ} that, under CH,
for any $\sigma$-ideal $\I$ on a set $X$
of cardinality $\con$ there exists a Hausdorff topology on $X$ making $\I$
meager.
For the ideal $\E$ we have even better situation --
it follows from Corollary~\ref{cor1} that, under CH,
$\E$ can be made meager by a Polish topology.
However the following general question of Balcerzak and Rogowska~\cite{BR}
remains open.

\begin{prob}
Is it provable in ZFC that there is
a Hausdorff topology on $\R$ making $\E$ meager?
\end{prob}

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\end{document}
