%version of April 10, 2000

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\author{
Krzysztof Ciesielski%
\thanks{%\endgraf 
The work of the first author was partially supported by 
NSF Cooperative
Research Grant INT-9600548, with its Polish part %being 
financed by Polish Academy of Science PAN.}
\\
{\footnotesize Department of Mathematics,}
{\footnotesize West Virginia University,}\\
{\footnotesize Morgantown, WV 26506-6310, USA}\\
{\footnotesize e-mail: K\_Cies@math.wvu.edu}\\
{\footnotesize web page: {\tt http://www.math.wvu.edu/homepages/kcies}}
\and
Saharon Shelah%
\thanks{This work was supported in part by a grant from
``Basic Research Foundation'' 
founded by the Israel Academy of Sciences and Humanities.
Publication 695. 
\endgraf 
The authors wish to thank Professor
Andrzej Ros{\l}anowski
for reading earlier versions of this paper
and helping in improving its final form.
}\\
{\footnotesize Institute of Mathematics,}
{\footnotesize the Hebrew University of Jerusalem}\\
{\footnotesize 91904 Jerusalem, Israel}\\
{\footnotesize and}\\
{\footnotesize Department of Mathematics,}
{\footnotesize Rutgers University}\\
{\footnotesize New Brunswick, NJ 08854, USA}}


 
 
\title{Category analogue of sup-measurability problem}

\date{}

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\newcommand{\fact}[2]{\begin{Fact}\label{#1}{\sl #2}\end{Fact}} 
 
 
\begin{document}
 
\maketitle

\begin{abstract} 
A function $F\colon\R^2\to\R$ is {\em sup-measurable\/}
if $F_f\colon\R\to\R$ given by $F_f(x)=F(x,f(x))$, $x\in\R$, 
is measurable for each measurable function $f\colon\R\to\R$. 
It is known that under different set theoretical assumptions, 
including CH, there are sup-measurable non-measurable 
functions, as well as their category analogues. 
In this paper we will show
that the existence of the category analogues of 
sup-measurable non-measurable functions is independent of ZFC.
A similar result for the original measurable case 
is the subject of a work in prepartion by
Ros{\l}anowski and Shelah. 
\end{abstract} 

\section{Introduction} 

Our terminology is standard and follows that from~\cite{BJ},
\cite{CiBook}, \cite{Ku}, or~\cite{Sh:f}.
In particular, $\proj\colon X\times Y\to X$ will stand 
for the projection onto the first coordinate. 
A subset $A$ of a Polish space $X$ is {\em nowhere meager\/}
provided $A\cap U$ is not meager for every non-empty
open subset of $X$. 

The ternary Cantor subset of $\real$ will be identified 
with with its homeomorphic copy, $2^\omega$, 
which stands for the set of all function $x\colon\omega\to\{0,1\}$
considered with the product topology. 
In particular, the basic open subsets of $2^\omega$
are in the form
\[
[s]\stackrel{\rm def}{=}\{x\in 2^\omega\colon s\subset f\},
\]
where $s\in 2^{<\omega}$. 
Also, since $\real\setminus\rational$ is homeomorphic to
$2^\omega\setminus E$ for some countable set $E$
(the set of all eventually constant functions in $2^\omega$)
in our more technical part of the paper we will be able
replace $\real$ with $2^\omega$. 

\bigskip

The study of sup-measurable 
functions%
\footnote{This is abbreviation from {\em  
              superposition-measurable function}.}
comes from the theory of 
differential equations.
More precisely it comes from 
the question: For which functions
$F\colon\R^2\to\R$
does the Cauchy problem
\begin{equation} \label{Cy}
y'=F(x,y),\;\;\;y(x_0)=y_0
\end{equation}
have a (unique) {\em a.e.-solution\/} in the class
of locally absolutely continuous functions on $\real$
in the  sense that 
$y(x_0)=y_0$ and $y'(x)=F(x,y(x))$ for 
almost all $x\in\real$?
(For more on this motivation see \cite{Kh2} or \cite{BC}.
Compare also \cite{Kh3}.)
It is not hard to find measurable functions which are not sup-measurable. 
(See \cite{Sr} or \cite[Cor.~1.4]{B}.)
Under the continuum hypothesis
CH or some weaker set-theoretical assumptions 
nonmeasurable sup-measurable functions were constructed 
in \cite{GL}, \cite{Kh1}, \cite{B}, and \cite{Kh2}. 
The  independence from ZFC of the existence of such an example 
is the  subject of a work in prepartion by
Ros{\l}anowski and Shelah. 

A function $F\colon\R^2\to\R$ 
is a {\em category analogue of 
sup-measurable 
function\/} (or {\em Baire sup-measurable}) 
provided 
$F_f\colon\R\to\R$ given by $F_f(x)=F(x,f(x))$, $x\in\R$, 
has the Baire property 
for each function $f\colon\R\to\R$ with the Baire property. 
A Baire sup-measurable function without the Baire property
has been constructed under CH in~\cite{GG}.
(See also \cite{B} and \cite{BC}.) 
The main goal of this paper is to show that
the existence of such functions cannot be proved in ZFC.
For this we need the following easy fact.
(See \cite[Prop. 1.5]{B}.)

\prop{pr1}{
The following conditions are equivalent. 
\begin{description}
\item[(i)] There is a Baire sup-measurable function
$F\colon\R^2\to\R$ without the Baire property. 

\item[(ii)] There is a function
$F\colon\R^2\to\R$ without the Baire property
such that $F_f$ has the Baire property for every 
Borel function $f\colon\R\to\R$.


\item[(iii)] There is a set 
$A\subset\R^2$ without the Baire property 
such that the projection 
$\proj(A\cap f)=\{x\in\real\colon \la x,f(x)\ra\in A\}$ 
has the Baire property for each 
Borel function $f\colon\R\to\R$. 

\item[(iv)] There is a Baire sup-measurable function
$F\colon\R^2\to\{ 0,1\}$ without the Baire property.
\end{description}
}

The equivalence of (i) and (ii) follows from the fact that
the function $F\colon\R^2\to\R$ is Baire sup-measurable
if and only if $F_f$ has a Baire property for every 
Borel function $f\colon\R\to\R$.%
\footnote{It is also true that 
$F\colon\R^2\to\R$ is Baire sup-measurable
provided $F_f$ has the Baire property for every 
Baire class one function $f\colon\R\to\R$, and that 
$F\colon\R^2\to\R$ is sup-measurable
provided $F_f$ is measurable for every 
continuous function $f\colon\R\to\R$.
See for example \cite[lemma~1 and remark~1]{BC}.}



The main theorem of the paper is the following. 

 
\thm{thMain}{It is consistent with the set theory ZFC that 
\begin{itemize}
\item[$\varphi$:] 
for every $A\subset 2^\omega\times 2^\omega$ 
for which the sets $A$ and $A^c=(2^\omega\times 2^\omega)\setminus A$
are nowhere meager in 
$2^\omega\times 2^\omega$ 
there exists  a homeomorphism 
$f$ from $2^\omega$ onto $2^\omega$ such that
the set 
$\proj(A\cap f)$ does not have the  
Baire property in $2^\omega$.
\end{itemize}
}

Before proving this theorem let us notice that
it implies easily the following corollary. 

\cor{cor1}{
The existence of Baire sup-measurable function
$F\colon\R^2\to\R$
without the Baire property is independent from 
the set theory ZFC. 
}

\proof As mentioned above under CH there exist 
Baire sup-measurable functions
without the Baire property.
So, it is enough to show that the property $\varphi$ 
from Theorem~\ref{thMain} implies that there 
are no such functions.

So, take an arbitrary $A\subset\R^2$ without the Baire property.
By (iii) of Proposition~\ref{pr1} it
is enough to show there exists 
a Borel 
function $f\colon\R\to\R$ for which the set 
$\proj(A\cap f)$ does not have the Baire property. 

We will first show this under the additional assumption
that the sets $A$ and $\real^2\setminus A$
are nowhere meager in $\real^2$. 
But then the set $A_0=A\cap(\real\setminus\rational)^2$
and its complement are 
nowhere meager in $(\real\setminus\rational)^2$.
Moreover, since
$\real\setminus\rational$ is homeomorphic to
$2^\omega\setminus E$ for some countable set $E$
%(the set of all eventually constant functions in $2^\omega$)
we can consider $A_0$ as a subset of 
$(2^\omega\setminus E)^2\subset 2^\omega\times 2^\omega$.
Then $A_0$ and its complement are 
still nowhere meager in $2^\omega\times 2^\omega$.
Therefore, by $\varphi$, 
there exists an autohomeomorphism 
$f$ of $2^\omega$
such that the set 
$\proj(A_0\cap f)=
\{x\in 2^\omega\setminus E\colon\la x,f(x)\ra\in A_0\}$  
does not have the  
Baire property in $2^\omega$.
Now, as before, $f\restriction(2^\omega\setminus E)$
can be considered as defined on 
$\real\setminus\rational$. 
So if $\bar f\colon\real\to\real$ is an extension
of $f\restriction(2^\omega\setminus E)$
(under such identification) to $\real$ 
which is  constant on $\rational$,  then $\bar f$ 
is Borel and the set 
$\proj(A_0\cap \bar f)$  
does not have the Baire property in $\real$.

Now, if $A$ is an arbitrary subset of 
$\R^2$ without the Baire property we can find 
non-empty open intervals 
$U$ and $W$ in $\real$ such that $A$ and $(U\times W)\setminus A$
are nowhere meager 
in $U\times W$. 
Since $U$ and $W$ are homeomorphic with $\real$ 
the above case implies the existence of
Borel function $f_0\colon U\to W$ such that
$\proj(A\cap f_0)$  
does not have the Baire property in $U$.
So any Borel extension $f\colon\real\to\real$ of $f_0$ 
works. 
\qed


\section{Reduction of the proof of Theorem~\ref{thMain}
to the main lemma}

The theorem will be proved by the method of iterated forcing,
a knowledge of which is needed from this point on. 

The idea of the proof is quite simple. 
For every nowhere meager subset $A$ of $2^\omega\times 2^\omega$ 
for which $A^c=(2^\omega\times 2^\omega)\setminus A$
is also nowhere meager
we will find 
a natural ccc forcing notion $Q_A$ 
which adds the required homeomorphism~$f$. 
%continuous function
Then we will start with the constructible universe 
$V=L$ and 
iterate with finite support these 
notions of forcing in such a way that every
nowhere meager  
set $A^*\subset 2^\omega\times 2^\omega$,
with $(2^\omega\times 2^\omega)\setminus A^*$ nowhere meager, 
will be taken care of by some $Q_A$ 
at an appropriate step of iteration. 

There are two technical problems with carrying through this idea. 
First is that we cannot possibly list in our iteration 
all nowhere meager  
subsets of 
$2^\omega\times 2^\omega$ with nowhere meager complements 
since the iteration can be of length at most 
continuum $\cont$ and there are $2^\cont$ such 
sets. This problem will be solved by defining our
iteration as 
$P_{\omega_2}=\la\la P_\alpha,\dot Q_\alpha\ra\colon\alpha<\omega_2\ra$ 
such that the generic extension $V[G]$ of $V$ with respect to 
$P_{\omega_2}$
will satisfy $2^\omega=2^{\omega_1}=\omega_2$ and
have the property that 
\begin{description}
\item{(m)} every non-Baire subset $A^*$ of $2^\omega$ contains 
           a non-Baire subset $A$ of cardinality $\omega_1$.
\end{description}
Thus in the iteration we will use only the forcing notions 
$Q_\alpha=Q_A$ 
for the sets $A$ of cardinality $\omega_1$, 
whose number is equal to $\omega_2$, the length of iteration.
Condition (m) will guarantee that this will give us enough
control of all nowhere meager  %non-Baire 
subsets $A^*$ of $2^\omega\times 2^\omega$.

The second problem is that 
even if at some stage $\alpha<\omega_2$ of our iteration
we will add a homeomorphism %continuous function 
$f$ appropriate
for a  given %nowhere meager  %non-Baire 
set $A\subset 2^\omega\times 2^\omega$,
that is such that
\[
V[G_\alpha]\models ``
\proj(A\cap f) \textrm{ is not Baire in $2^\omega$,''}  
\]
where $G_\alpha=G\cap P_\alpha$, 
then in general there is no guarantee that the set 
$\proj(A\cap f)$ will remain non-Baire in the final model $V[G]$.
The preservation of non-Baireness of each appropriate 
set $\proj(A\cap f)$ 
will be achieved by carefully crafting our iteration following
a method known as 
the {\em oracle-cc\/} forcing iteration.



The theory of the oracle-cc forcings
is described in details in \cite[Ch.~IV]{Sh:f}
(compare also \cite[Ch.~IV]{ShPF})
and here we will recall only the fragments 
that are relevant to our specific situation.
In particular if 
\[
\Gamma\stackrel{\rm def}{=}\{\lambda<\omega_1\colon \lambda \mbox{ is a limit
ordinal}\}
\] 
then 
\begin{itemize}
\item an {\em $\omega_1$-oracle\/} is any sequence 
$\M=\la M_\delta\colon\delta\in\Gamma\ra$
where $M_\delta$ is a countable transitive model of ZFC$^-$
that is, 
ZFC without the power set axiom) with a property %such 
that 
$\delta+1\subset M_\delta$, 
$M_\delta\models ``\delta$\textrm{ is countable},''
and the set $\{\delta\in\Gamma\colon A\cap\delta\in M_\delta\}$
is stationary in $\omega_1$ for every $A\subset\omega_1$. 
\end{itemize}
The existence of an $\omega_1$-oracle is equivalent to 
the diamond principle $\diamondsuit$. 

With each $\omega_1$-oracle $\M=\la M_\delta\colon\delta\in\Gamma\ra$
there is associated a filter $D_{\M}$ 
generated by the sets 
$I_\M(A)=\{\delta\in\Gamma\colon A\cap\delta\in M_\delta\}$
for $A\subset\omega_1$. 
It is proved in~\cite[Claim~1.4]{Sh:f} that $D_{\M}$
is a proper normal filter containing every closed 
unbounded subset of $\Gamma$. 


We will also need the following fact which, for our purposes, can be
viewed as a definition of $\M$-cc property.  

\fact{f1}{Let $P$ be a forcing notion of cardinality 
$\leq\omega_1$, 
$e\colon P\to\omega_1$ be one-to-one, and 
$\M=\la M_\delta\colon\delta\in\Gamma\ra$
be an $\omega_1$-oracle.
If there exists a $C\in D_\M$ 
such that for every $\delta\in\Gamma\cap C$
\begin{quote}
$e^{-1}(E)$ is predense in $P$ for every set 
$E\in M_\delta\cap\P(\delta)$, for which 
$e^{-1}(E)$ is predense in $e^{-1}(\{\gamma\colon \gamma<\delta\})$,
\end{quote}
then $P$ has the $\M$-cc property.
}
This follows immediately from the definition
of $\M$-cc property \cite[Definition~1.5, p. 150]{Sh:f}.

%the fact that the set  
%$S=\{\delta\in\Gamma\colon e^{-1}(\delta)\in M_\delta\}$
%is stationary in $\omega_1$.



Our proof will rely on the following main lemma.

\lem{lemMain}{For every 
$A\subset 2^\omega\times 2^\omega$
for which $A$ and $A^c=(2^\omega\times 2^\omega)\setminus A$
are nowhere meager in $2^\omega\times 2^\omega$
and for every $\omega_1$-oracle $\M$
there exists an $\M$-cc forcing notion $Q_A$
of cardinality $\omega_1$ such that $Q_A$ forces
\begin{quote}
there exists 
%a continuous function $f$ from $D\subset 2^\omega$ into 
an autohomeomorphism $f$ of 
$2^\omega$ such that the sets 
$\proj(f\cap A)$ and $\proj(f\setminus A)$ are nowhere 
meager in $2^\omega$.
\end{quote}
}

The proof of Lemma~\ref{lemMain} represents the core of our
argument and will be presented in the next section.
In the remainder of this section
we will sketch how Lemma~\ref{lemMain} 
implies Theorem~\ref{thMain}.
Since this follows the standard path,
as described in~\cite[Chapter~IV]{Sh:f},
the readers familiar with this treatment may 
proceed directly to the next section. 

First of all, to define an appropriate iteration
we will treat forcings $Q_A$ from
Lemma~\ref{lemMain} as defined on $\omega_1$.
More precisely, in the iteration we will always replace
$Q_A$ with its order isomorphic copy
$\la \omega_1,\leq_A\ra$. 
%Thus, 
So, we can treat any finite support iteration 
$P_\alpha=\la\la P_\beta,\dot Q_\beta\ra\colon\beta<\alpha\ra$ 
of $Q_A$ forcing notions as having an absolute and fixed
universe, say 
$U_\alpha=\left\{g\in(\omega_1)^{\omega_2}\colon
g^{-1}(\omega_1\setminus\{0\})\in[\alpha]^{<\omega}\right\}$.
This will allow us to treat 
the $\diamondsuit_{\omega_2}$-sequence
$\la X_\alpha\colon \alpha<\omega_2\ra$ 
as a sequence of $P_\alpha$-names
of subsets of $2^\omega\times 2^\omega$. 
(After appropriate coding.)

We will also need the following variant of~\cite[Example~2.2]{Sh:f}.

\lem{lemEx}{Assume that $\diamondsuit_{\omega_1}$ holds and that
$S\subset 2^\omega$ is such that 
$S$ and $S^c$ are 
nowhere meager in $2^\omega$. 
Then there exists an $\omega_1$-oracle $\M$ such that 
if $P$ is an arbitrary $\M$-cc forcing then $P$ forces that
\begin{itemize}
\item[]  
$S$ and $S^c$ are nowhere meager in $2^\omega$.
\end{itemize}
}

\proof By \cite[Example~2.2]{Sh:f} for any non-empty basic open set
$W$ of $2^\omega$ there are oracles
$\M^0_W$ and $\M^1_W$ such any $\M^0_W$-cc forcing
forces that $S\cap W$ is not meager, and any 
$\M^1_W$-cc forcing
forces that $S^c\cap W$ is not meager. 
So, by \cite[Claim~3.1]{Sh:f}, there is a single 
$\omega_1$-oracle $\M$ which 
``extends'' all oracles $\M^i_W$, and it clearly 
does the job. 
\qed


Now, the iteration $P_{\omega_2}$ is defined 
by choosing by induction the sequence
$\la \la P_\alpha,\dot A_\alpha,\dot\M_\alpha,
\dot Q_\alpha, 
\dot f_\alpha
\ra\colon\alpha<\omega_2\ra$ 
such that for every $\alpha<\omega_2$
\begin{itemize}
\item[(a)] $P_\alpha=\la \la P_\beta,\dot
      Q_\beta\ra\colon\beta<\alpha\ra$ is a finite support iteration,

\item[(b)] $\dot A_\alpha$ is a $P_\alpha$-name 
for which $P_\alpha$ forces that
\begin{itemize}
\item[]  
\mbox{$\dot A_\alpha$ and $(\dot A_\alpha)^c$
are 
nowhere meager subsets of $2^\omega\times 2^\omega$,}
\end{itemize}

\item[(c)] $\dot\M_\alpha$ is a $P_\alpha$-name for which 
$P_\alpha$ forces that \medskip

$\dot\M_\alpha$ is an $\omega_1$-oracle and 
        for every $\dot Q$ satisfying $\dot\M_\alpha$-cc we have

\begin{itemize}
\item[(i)]
for every  $\beta<\alpha$ 
if $P_\alpha=P_\beta * \dot P_{\beta,\alpha}$ then 
\[
P_\beta\forces ``\dot P_{\beta,\alpha} * \dot Q\ \mbox{ is
$\dot\M_\beta$-cc,''}
\]

\item[(ii)]
if $\alpha=\gamma+1$ then
\[
\mbox{$\dot Q\forces 
``\proj(\dot f_\gamma \cap\dot A_\gamma),
\proj(\dot f_\gamma \setminus\dot A_\gamma)\subset 2^\omega$ 
are nowhere meager in $2^\omega$,''}
\]
\end{itemize}

\item[(d)] $\dot Q_\alpha$ is a $P_\alpha$-name 
for a forcing such that $P_\alpha$ forces
\begin{itemize}
\item[] $\dot Q_\alpha$ is an $\dot\M_\alpha$-cc forcing 
$Q_{\dot A_\alpha}$
from Lemma~\ref{lemMain},
\end{itemize}

\item[(e)] $\dot f_\alpha$ is a $P_{\alpha+1}$-name 
for which $P_{\alpha+1}$ forces that 
\begin{itemize}
\item[] $\dot f_\alpha$ is a 
$\dot Q_\alpha$-name for the function $f$ from Lemma~\ref{lemMain}.
\end{itemize}
\end{itemize}

If for some $\alpha<\omega_2$ the sequence 
$\la \la P_\beta,\dot A_\beta,\dot\M_\beta,
\dot Q_\beta,\dot f_\beta\ra\colon\beta<\alpha\ra$ 
has been defined
then we proceed as follows.
Forcing $P_\alpha$ is already determined by (a). 
We choose $\dot A_\alpha$ as $X_\alpha$ from 
the $\diamondsuit_{\omega_2}$-sequence if 
it satisfies (b)
and arbitrarily, still maintaining (b), otherwise.
Since steps (d) and (e) are facilitated by 
Lemma~\ref{lemMain}, it is enough to construct $\dot\M_\alpha$
satisfying (c). For this we will consider two cases.

\medskip

\noindent{\sc Case 1}: $\alpha$ is a limit ordinal.

For a moment fix a $\beta<\alpha$ and work in $V^{P_\beta}$.
Let $\M_\beta$ and $P_{\beta,\alpha}$
be the interpretations of $\dot\M_\beta$
and $\dot P_{\beta,\alpha}$, respectively. 
By the inductive assumption for every $\beta<\gamma<\alpha$ 
forcing $P_{\beta,\gamma}$ is $\M_\beta$-cc.
So, by \cite[Claim~3.2]{Sh:f},
$P_{\beta,\alpha}$ is $\M_\beta$-cc.
Thus, by \cite[Claim~3.3]{Sh:f},
in $(V^{P_\beta})^{P_{\beta,\alpha}}=V^{P_\alpha}$
there is an $\omega_1$-oracle $\M_\beta^*$
such that if $Q$ is $\M_\beta^*$-cc then
$P_{\beta,\alpha} * Q$ is $\M_\beta$-cc.

So, in $V^{P_\alpha}$, we have 
$\omega_1$-oracles $\M_\beta^*$ for every $\beta<\alpha$. 
Thus, by \cite[Claim~3.1]{Sh:f}, in $V^{P_\alpha}$
there exists an $\omega_1$-oracle $\M_\alpha$
which is stronger than all $\M_\beta^*$'s in a sense that
if $Q$ is $\M_\alpha$-cc then $Q$ is also $\M_\beta^*$-cc. 
So, there is a $P_\alpha$-name
$\dot\M_\alpha$ for $\M_\alpha$ for which (c) holds. 

\medskip

\noindent{\sc Case 2}: $\alpha$ is a successor ordinal, 
$\alpha=\gamma+1$. Then $P_\alpha=P_\gamma * \dot Q_\gamma$.

Since, by (d), $P_\gamma$ forces that $\dot Q_\gamma$ is
$\dot\M_\gamma$-cc, using (c) for $\alpha=\gamma$ we conclude that
\[
P_\beta\forces ``\dot P_{\beta,\alpha}\ \mbox{ is $\dot\M_\beta$-cc''}
\]
for every $\beta<\gamma$. So, proceeding as in Case~1, in $V^{P_\alpha}$
we can find $\omega_1$-oracles $\dot\M_\beta^*$
such that
\[
P_\beta\forces ``\dot P_{\beta,\alpha} * \dot Q\ \mbox{ is
$\dot\M_\beta$-cc''}
\]
for every $Q$ which is $\dot\M_\beta^*$-cc.
Let $\M$ be an $\omega_1$-oracle from 
Lemma~\ref{lemEx} used with
$S=\proj(\dot f_\gamma \cap\dot A_\gamma)$.
As above we can find, in $V^{P_\alpha}$, an $\omega_1$-oracle 
$\M_\alpha$ which is 
stronger than all $\M_\beta^*$'s and $\M$.
Then, there is a $P_\alpha$-name
$\dot\M_\alpha$ for $\M_\alpha$ for which (c) holds. 
This finishes the construction of the iteration. 

\bigskip

To finish the argument first note that 
the interpretations of 
$\proj(\dot f_\alpha\cap\dot A_\alpha)$ and
$\proj(\dot f_\alpha\setminus\dot A_\alpha)$
in the final model $V[G]$ remain nowhere meager in $2^\omega$.
This is the case since, by (e), 
$P_{\alpha+1}$ forces that 
\begin{center}
$\proj(\dot f_\alpha\cap\dot A_\alpha)$ and
$\proj(\dot f_\alpha\setminus\dot A_\alpha)$
are nowhere meager in $2^\omega$,
\end{center}
and, by (c)(i), that
\begin{center}
every $\dot P_{\alpha+1,\gamma}$ is $\dot\M_{\alpha+1}$-cc
\end{center}
while, by condition (c)(ii), every $\dot\M_{\alpha+1}$-cc forcing
preserves nowhere meagerness of
$\proj(\dot f_\alpha\cap\dot A_\alpha)$ and
$\proj(\dot f_\alpha\setminus\dot A_\alpha)$.
To finish this part of the argument it is enough to note that
$P_{\alpha+1}$ forces that
``$\dot P_{\alpha+1,\omega_2}$ is $\dot\M_{\alpha+1}$-cc''
which follows from \cite[Claim~3.2]{Sh:f}.

To complete the argument it is enough to show that 
each nowhere meager subset
$A^*$ of $2^\omega\times 2^\omega$
from $V[G]$ with nowhere meager complement 
contains an interpretation of 
some $\dot A_\alpha$. However, $P_{\omega_2}$ is ccc. So, 
if $\dot A$ is a $P_{\omega_2}$-name for $A^*$ then 
the set
\[
\left\{\alpha\in\Gamma\colon 
P_\alpha\forces \dot A\cap V^{P_\alpha}\mbox{ is nowhere meager in }
2^\omega\times 2^\omega\right\}
\]
contains a closed unbounded subset of $\Gamma$. 
Thus $\diamondsuit_{\omega_2}$ guarantees that
$A^*$ contains an interpretation of 
some $\dot A_\alpha$. 

\section{Proof of Lemma~\ref{lemMain}}

Let $\K$ be the family of all sequences
$\bar h=\la h_\xi\colon\xi\in\Gamma\ra$ such that 
each $h_\xi$ is a function from
a countable set $D_\xi\subset 2^\omega$ onto
$R_\xi\subset 2^\omega$ and that
\[
D_\xi\cap D_\eta=R_\xi\cap R_\eta=\emptyset
\ \mbox{ for every distinct }\ \xi,\eta\in\Gamma.
\]
For each $\bar h\in\K$ we will define a forcing notion
$Q_{\bar h}$. 
Forcing $Q_A$ satisfying Lemma~\ref{lemMain}
will be chosen as $Q_{\bar h}$ for some $\bar h\in\K$.

So fix an $\bar h\in\K$. Then $Q_{\bar h}$ is defined
as the set of all triples $p=\la n,\pi,h\ra$ for which 

\begin{itemize}
\item[(A)] $h$ is a function from a finite subset $D$ of
$\bigcup_{\xi\in\Gamma}D_\xi$ into $2^\omega$;

\item[(B)] $n<\omega$ and $\pi$ is a permutation of $2^n$;

\item[(C)] $|D\cap D_\xi|\leq 1$ for every $\xi\in\Gamma$;

\item[(D)] if $x\in D\cap D_\xi$ then $h(x)=h_\xi(x)$ 
      and $h(x)\restriction n=\pi(x\restriction n)$.
\end{itemize}
Forcing $Q_{\bar h}$ is ordered as follows. 
Condition $p'=\la n',\pi',h'\ra$ is stronger than 
$p=\la n,\pi,h\ra$, $p'\leq p$, 
provided 
\begin{equation}\label{ord}
\mbox{$n\leq n'$, \ $h\subset h'$,\ \  and\ \ 
$\pi'(s)\restriction n=\pi(s\restriction n)$\ \ 
for every $s\in 2^{n'}$.}
\end{equation}
Note that 
the second part of (D) says that for every 
$x\in D$ and $s\in 2^n$
\begin{equation}\label{conD}
x\in [s]\ \  \mbox{ if and only if  }\ \  h(x)\in [\pi(s)].
\end{equation}
Also, if $n<\omega$ we will write $[s]\restriction 2^n$
for $\{x\restriction 2^n\colon x\in[s]\}$. 
Note that in this notation the part of (\ref{ord}) concerning 
permutations says that
$\pi'$ extends $\pi$ in a sense that $\pi'$ 
maps $[t]\restriction 2^{n'}
$ onto 
$[\pi(t)]\restriction 2^{n'}$
for every $t\in 2^{n}$.


In what follows we will use the following basic property of $Q_{\bar h}$.

\begin{itemize}
\item[($*$)] For every 
$q=\la n,\pi,h\ra\in Q_{\bar h}$ and $m<\omega$
there exist an $n'\geq m$ and a permutation $\pi'$ of $2^{n'}$
%with the property 
such that 
$q'=\la n',\pi',h\ra
\in Q_{\bar h}$ and $q'$ extends $q$. 
\end{itemize}

The choice of such $n'$ and $\pi'$ is easy. 
First pick $n'\geq \max\{m,n\}$ such that
$x\restriction n'\neq y\restriction n'$
for every different $x$ and $y$ from either domain $D$ 
or range $R=h[D]$ of $h$.
This implies that for every $t\in 2^{n}$
the set 
$D_t=
\{x\restriction n'\colon x\in D\cap [t]\}\subset [t]\restriction
2^{n'}$  has the same cardinality as $D\cap [t]$
and $H_t=
\{x\restriction n'\colon x\in h[D]\cap [\pi(t)]\}
\subset [\pi(t)]\restriction 2^{n'}$ 
has the same cardinality as $h[D]\cap [\pi(t)]$.
Since, by (\ref{conD}), we have also
$|D\cap [t]|=|h[D]\cap [\pi(t)]|$
we see that $|D_t|=|H_t|$. 
Define $\pi'$ on $D_t$ by 
$\pi'(x\restriction n')=h(x)\restriction n'$
for every $x\in D_t$.  Then $\pi'$ is a bijection 
from $D_t$ onto $H_t$ 
and this definition ensures that an appropriate part of
the condition (D) for $h$ and $\pi'$ is
satisfied.  Also, if for each $t\in 2^{n}$
we extend $\pi'$ onto $[t]\restriction 2^{n'}$
as a bijection from $([t]\restriction 2^{n'})\setminus D_t$
onto $([\pi(t)]\restriction 2^{n'})\setminus H_t$,
then the condition (\ref{ord}) will be satisfied. 
Thus such defined $q'=\la n',\pi',h\ra$ belongs to 
$Q_{\bar h}$ and extends $q$.


Next note that forcing $Q_{\bar h}$ 
has the following property, described in Fact~\ref{fact2}, 
needed to prove Lemma~\ref{lemMain}. 
In what follows we will 
consider $2^\omega$ with the standard
distance: 
$$
d(r_0,r_1)=2^{-\min\{n<\omega\colon r_0(n)\neq r_1(n)\}}
$$
for different $r_0,r_1\in 2^\omega$.



\fact{fact2}{Let $\bar h=\la h_\xi\colon\xi\in\Gamma\ra\in\K$ and 
$f=\bigcup\{h\colon \la n,\pi,h\ra\in H\}$, where 
$H$ is a $V$-generic filter over $Q_{\bar h}$.
Then $f$ is a uniformly continuous one-to-one function from a subset
$D$ of $2^\omega$ into~$2^\omega$.
Moreover, if for every $\xi\in\Gamma$ the graph of $h_\xi$
is dense in $2^\omega\times 2^\omega$, then $D$ and $f[D]$ are dense in
$2^\omega$ and $f$ can be uniquely extended to an autohomeomorphism 
$\tilde f$ of $2^\omega$. 
}

\proof Clearly $f$ is a one-to-one function from a subset
$D$ of $2^\omega$ into~$2^\omega$.
To see that it is uniformly continuous 
choose an $\ep>0$.
We will find $\delta>0$ such that 
$r_0,r_1\in D$ and $d(r_0,r_1)<\delta$ imply $d(f(r_0),f(r_1))<\ep$. 
For this note that, by ($*$), the set
\[
S=\{q=\la n,\pi,h\ra\in Q_{\bar h}\colon 2^{-n}<\ep\}
\]
is dense in $Q_{\bar h}$. So take a $q=\la n,\pi,h\ra\in H\cap S$
and put $\delta=2^{-n}$.
We claim that this $\delta$ works.

Indeed, take $r_0,r_1\in D$ such that %with 
$d(r_0,r_1)<\delta$.
Then there exists a %is 
$q'=\la n',\pi',h'\ra\in H$ 
stronger than $q$ 
such that $r_0$ and $r_1$ are in the domain of~$h'$.
Therefore, $n\leq n'$ and for $j<2$
\[
f(r_j)\restriction n
=h'(r_j)\restriction n
=(h'(r_j)\restriction n')\restriction n
=\pi'(r_j\restriction n')\restriction n
%=\pi((r_j\restriction n')\restriction n)
=\pi(r_j\restriction n)
\]
by the conditions (D) and (\ref{ord}).
Since $d(r_0,r_1)<\delta=2^{-n}$ 
implies that \mbox{$r_0\restriction n=r_1\restriction n$}
we obtain
\[
f(r_0)\restriction n
=\pi(r_0\restriction n)
=\pi(r_1\restriction n)
=f(r_1)\restriction n
\]
that is, 
$d(f(r_0),f(r_1))\leq 2^{-n}<\ep$. So $f$ is uniformly continuous. 

Essentially the same argument (with the same values of $\ep$ and $\delta$)
shows that $f^{-1}\colon f[D]\to D$ is uniformly continuous. 
Thus, if $\tilde f$ is the unique continuous extension of 
$f$ into $\cl(D)$, then $\tilde f$ is a homeomorphism from
$\cl(D)$ onto $\cl(f[D])$.

To finish the argument assume that all functions $h_\xi$ have dense
graphs, take a $t\in 2^{m}$ for some $m<\omega$,
and notice that the set
\[
S_t=\{q=\la n,\pi,h\ra\in Q_{\bar h}\colon 
\mbox{ the domain $D'$ of $h$ intersects }[t]\}
\]
is dense in $Q_{\bar h}$. Indeed, if 
$q=\la n,\pi,h\ra\in Q_{\bar h}$
then, by ($*$), strengthening $q$ if necessary, 
we can assume that $m\leq n$.
Then, refining $t$ if necessary, we can also assume that $m=n$,
that is, that $t$ is in the domain of $\pi$.
Now, if $[t]$ intersects the domain of 
$h$, then already $q$ belongs to $S_t$.
Otherwise take $\xi\in\Gamma$ with $D'\cap D_\xi=\emptyset$
and pick $\la x,h_\xi(x)\ra\in [t]\times[\pi(t)]$,
which exists by the density of the graph of $h_\xi$. 
Then $\la n,\pi,h\cup\{\la x,h_\xi(x)\ra\}\ra$ belongs to $S_t$ 
and extends $q$. 

This shows that $D\cap [t]\neq\emptyset$ for every 
$t\in 2^{<\omega}$, that is, $D$ is dense in $2^\omega$. 

A similar argument shows that for every 
$t\in 2^{<\omega}$ the set
\[
S^t=\{q=\la n,\pi,h\ra\in Q_{\bar h}\colon 
\mbox{ the range of $h$ intersects }[t]\}
\]
is dense in $Q_{\bar h}$, which implies that  
$h[D]$ is dense in $2^\omega$. 
Thus $\tilde f$ is a homeomorphism from $\cl(D)=2^\omega$
onto $\cl(h[D])=2^\omega$.
\qed

Now take $A\subset 2^\omega\times 2^\omega$
for which $A$ and $A^c=(2^\omega\times 2^\omega)\setminus A$
are nowhere meager in $2^\omega\times 2^\omega$
and fix an $\omega_1$-oracle 
$\M=\la M_\delta\colon\delta\in\Gamma\ra$. 
By Fact~\ref{fact2} in order to prove Lemma~\ref{lemMain}
it is enough to find an 
$\bar h=\la h_\xi\colon\xi\in\Gamma\ra\in\K$ such that
\begin{equation}\label{eqN2}
\mbox{$Q_A=Q_{\bar h}$ is $\M$-cc} 
\end{equation}
and $Q_{\bar h}$ forces that, in $V[H]$, 
\begin{equation}\label{eqN3}
\mbox{the sets 
$\proj(f\cap A)$ and $\proj(f\setminus A)$ are nowhere 
meager in $2^\omega$.} 
\end{equation}
(In (\ref{eqN3}) function $f$ is defined as in Fact~\ref{fact2}.)

To define $\bar h$ we will construct 
a sequence
$\la \la x_\alpha,y_\alpha\ra\in 2^\omega\times 2^\omega\colon
\alpha<\omega_1\ra$ 
aiming at 
$h_\xi=\{\la x_{\xi+n},y_{\xi+n}\ra\colon n<\omega\}$,
where $\xi\in\Gamma$.

Let $\{\la s_n,t_n\ra\colon n<\omega\}$
be an enumeration of
$2^{<\omega}\times 2^{<\omega}$ with each pair $\la s,t\ra$ 
appearing for an odd $n$ and for an even $n$.  
Points $\la x_{\xi+n},y_{\xi+n}\ra$
are chosen inductively in such a way that
\begin{description}
\item{(i)} $\la x_{\xi+n},y_{\xi+n}\ra$ 
      is a Cohen real over 
      $M_\delta[\la \la x_\alpha,y_\alpha\ra\colon \alpha<\xi+n\ra]$
      for every $\delta\leq\xi$, $\delta\in\Gamma$, 
      that is, $\la x_{\xi+n},y_{\xi+n}\ra$ is outside all 
      meager subsets of $2^\omega\times 2^\omega$ which are
      coded in 
      $M_\delta[\la \la x_\alpha,y_\alpha\ra\colon \alpha<\xi+n\ra]$;
%      (or any other countable model of ZFC$^-$ containing 
%      $M_\delta$ and the sequence 
%      $\la \la x_\alpha,y_\alpha\ra\colon \alpha<\delta+n\ra$);
\item{(ii)} $\la x_{\xi+n},y_{\xi+n}\ra\in A$ if $n$ is even, and
      $\la x_{\xi+n},y_{\xi+n}\ra\in A^c$ otherwise. 
\item{(iii)} $\la x_{\xi+n},y_{\xi+n}\ra\in [s_n]\times[t_n]$. 
\end{description}
The choice of $\la x_{\xi+n},y_{\xi+n}\ra$
is possible since both sets $A$ and $A^c$
are nowhere meager, and we consider
each time only countably many meager sets. 
Condition (iii) guarantees that the graph of each of $h_\xi$
will be dense in $2^\omega\times 2^\omega$. 
Note that if $\Gamma\ni\delta\leq \alpha_0<\cdots<\alpha_{k-1}$,
where $k<\omega$, then (by the product lemma in $M_\delta$)
\begin{equation}\label{PLem}
\mbox{$\la\la x_{\alpha_i},y_{\alpha_i}\ra\colon i<k\ra$
is an $M_\delta$-generic Cohen real in 
$\left(2^\omega\times 2^\omega\right)^k$.}
\end{equation}

For $q=\la n,\pi,h\ra\in Q_{\bar h}$
define
$$
\hat q=\bigcup_{\la s,t\ra\in\pi}[s]\times[t].
$$
Clearly $\hat q$ is an open subset of $2^\omega\times 2^\omega$
and condition (\ref{ord}) implies that 
for every $q,r\in Q_{\bar h}$ with $r=\la n',\pi',h'\ra$
\begin{equation}\label{BBBB}
\mbox{if $q\leq r$ then $\hat q\subset\hat r$
and $\hat q\cap([s]\times[t])\neq\emptyset$ 
for every $\la s,t\ra\in\pi'$.}
\end{equation}
Also for $\delta\in\Gamma$ let
$(Q_{\bar h})^\delta=\left\{\la n,\pi,h\ra\in Q_{\bar h}
\colon h\subset\bigcup_{\zeta<\delta}h_\zeta\right\}$.
To prove (\ref{eqN2}) and (\ref{eqN3}) we will use also the following fact.













\fact{FACT}{Let $\delta\in\Gamma$ 
and let $E\in M_\delta$ be 
a predense subset of $(Q_{\bar h})^\delta$. Then
for every $k<\omega$ and
$p=\la n,\pi,h\ra\in (Q_{\bar h})^\delta$ 
the set 
\begin{equation}\label{CCC}
B_{p}^k=\bigcup\left\{
(\hat q)^k\colon 
\mbox{ $q$ extends $p$ and some $q_0\in E$}\right\}
\end{equation}
is dense in $(\hat p)^k\subset\left(2^\omega\times 2^\omega\right)^k$.
}

\proof 
By way of contradiction assume that $B_{p}^k$ is not dense in $(\hat p)^k$.
Then there exist %are 
$m<\omega$ and 
$s_0,t_0,\ldots,s_{k-1},t_{k-1}\in 2^m$
with the property %such 
that $P=\prod_{i<k}([s_i]\times[t_i])\subset (\hat p)^k$
is disjoint from $B_{p}^k$. 
Increasing $m$ and refining the 
$s_i$'s and $t_j$'s, if necessary, we may assume that $m\geq n$, 
all $s_i$'s and $t_j$'s are different, 
$\bigcup_{i<k}[s_i]$
is disjoint from the domain $D$ of $h$,
and $h[D]\cap\bigcup_{i<k}[t_i]=\emptyset$. 
We can also assume that $x\restriction m\neq y\restriction m$ for every different
$x$ and $y$ from $D$ and from $h[D]$. Now, refining slightly the argument for  
($*$) we can find 
$r=\la m,\pi',h\ra\in (Q_{\bar h})^\delta$ extending $p$ such that
$\pi'(s_i)=t_i$ for every $i<k$.
(Note that $P\subset (\hat p)^k$.)
We will obtain a contradiction with the predensity of 
$E$ in $(Q_{\bar h})^\delta$ by showing that 
$r$ is incompatible with every element of $E$.

Indeed if $q$ were an extension of $r\leq p$ and an element $q_0$ of
$E$, then we would have 
$(\hat q)^k\subset B^k_p$.
But then, by (\ref{BBBB}) and the fact that $\la s_i,t_i\ra\in\pi'$
for $i<k$,
we would also have $(\hat q)^k\cap P\neq\e$, 
%since $\e\neq(\hat q)^k\subset (\hat r)^k=P$, 
contradicting $P\cap B^k_p=\e$. 
This finishes the proof of Fact~\ref{FACT}. \qed



Now we are ready to prove (\ref{eqN2}), that is, that 
$Q_{\bar h}$ is $\M$-cc. So, fix a bijection
$e\colon Q_{\bar h}\to\omega_1$ and let
\[
C=\left\{\delta\in\Gamma\colon 
(Q_{\bar h})^\delta=e^{-1}(\delta)\in M_\delta\right\}.
\]
Then $C\in D_{\mathcal M}$.
(Just use a suitable nice codding or \cite[Claim~1.4(4)]{Sh:f}.) 
Take a $\delta\in C$ and fix an $E\subset \delta$,
$E\in M_\delta$, for which 
$e^{-1}(E)$ is predense in 
$(Q_{\bar h})^\delta$. 
By Fact~\ref{f1} it is enough to show that
\begin{equation*}\label{AAA}
\mbox{$e^{-1}(E)$ is predense in $Q_{\bar h}$.}
\end{equation*}

 


Take
$p_0=\la n,\pi,h_0\ra$ from $Q_{\bar h}$,
let $h=h_0\restriction\bigcup_{\eta<\delta}D_\eta$ and
$h_1=h_0\setminus h$, 
and notice that the condition 
$p=\la n,\pi,h\ra$ 
belongs to $(Q_{\bar h})^\delta$.
Assume that $h_1=\{\la x_i,y_i\ra\colon i<k\}$.
Since $s(h_1)=\la\la x_i,y_i\ra\colon i<k\ra\in (\hat p)^k$,
$B_{p}^k\in M_\delta$ (as defined from 
$(Q_{\bar h})^\delta\in M_\delta$)
and, by
Fact~\ref{FACT},  
$B_{p}^k$ is dense in $(\hat p)^k$
condition (\ref{PLem}) implies that 
$s(h_1)\in B_{p}^k$.
So there are 
$q=\la n_0,\pi_0,g\ra\in(Q_{\bar h})^\delta$ 
extending $p$ and some $q_0\in e^{-1}(E)$ for which
$s(h_1)\in \hat q^k$.
But then $p'=\la n_0,\pi_0,g\cup h_1\ra$
belongs to $Q_{\bar h}$ and extends 
$q$.
This finishes the proof of (\ref{eqN2}).

The proof of (\ref{eqN3}) is similar. We will prove only that
$\proj(f\setminus A)=\proj(f\cap A^c)$ is nowhere meager in
$2^\omega$, the argument for $\proj(f\cap A)$ being essentially the
same. 



By way of contradiction assume that $\proj(f\setminus A)$ is not
nowhere meager in~$2^\omega$. So there exists
an $s^*\in 2^{<\omega}$ such  
that $\proj(f\setminus A)$ is meager in $[s^*]$. Let a condition  
$p^*\in
Q_{\bar{h}}$ and $Q_{\bar h}$-names $\dot{U}_m$, for $m<\omega$, be such
\[
p^*\forces_{Q_{\bar{h}}}\mbox{each $\dot U_m$ is an open dense subset of
$[s^*]$ and }\proj(f\setminus A)\cap\bigcap_{m<\omega}
\dot{U}_m=\emptyset.
\]
For each $m<\omega$, since $p^*$ forces that 
$\dot U_m$ is an open dense subset of $[s^*]$, 
for every 
$t\in\fs$ extending $s^*$ there is a maximal antichain 
$\la p^m_{s,k}\colon k<\kappa_s^m\rangle$
in $Q_{\bar{h}}$ 
forcing 
that $\dot U_m\cap[t]$ contains some basic open subset $[s]$. 


Note that each of these antichains must be  countable, since 
the forcing notion $Q_{\bar h}$ is ${\mathcal M}$-cc and %thus 
therefore ccc.
Combining all these antichains we get a sequence 
$\la p^m_{s,k}\in Q_{\bar{h}}\colon m<\omega, s\in\fs, k<\kappa_s^m\ra$ 
such that 
\begin{itemize}
\item $\kappa_s^m\leq\omega$,
\item $p^m_{s,k}\forces_{Q_{\bar{h}}} [s]\subseteq \dot{U}_m$,
\item for every $q\in Q_{\bar{h}}$ extending $p^*$ 
and $t\in\fs$ extending $s^*$ there are  
$s\in\fs$
and $k<\kappa_s^m$ such that the conditions $q$ and $p^m_{s,k}$ are
compatible and $t\subset s$. 
\end{itemize}
Note that for sufficiently large $\delta\in \Gamma$ we have
$p^m_{s,k}\in (Q_{\bar{h}})^\delta$ for all $m<\omega$, $s\in \fs$, 
and $k<\kappa_s^m$. 



Now, by the definition of $\omega_1$-oracle, the set 
$B_0$ of all $\delta\in\Gamma$ for which
\[
\la p^m_{s,k}\in Q_{\bar{h}}\colon m<\omega, s\in\fs, k<\kappa_s^m\ra
\in M_\delta\ \ 
\mbox{ and } \ \ 
(Q_{\bar h})^{\delta}\in M_\delta
\]
is stationary in $\omega_1$. (Just use a suitable nice coding, or see
\cite[Ch.IV, Claim~1.4(4)]{Sh:f}). Thus, using clause (iii) of the  
choice of
$x_\xi$'s, we may find a $\delta\in B_0$, an odd $j<\omega$, and a
condition $p_0=\langle n_0,\pi_0,h_0\rangle\in Q_{\bar{h}}$ such that 
\begin{itemize}
\item $p_0\leq p^*$, $s^*\subset x_{\delta+j}$, and
\item $x_{\delta+j}$ belongs to the domain of $h_0$.
\end{itemize}
Then $p_0\forces``x_{\delta+j}\in [s^*]\cap {\rm proj}(f\setminus A)$''
(remember $j$ is odd so $\langle x_{\delta+j},y_{\delta+j}\rangle\in
A^c$). We will show that  
\[
p_0\forces x_{\delta+j}\in \bigcap_{m<\omega}\dot{U}_m,
\]
which will finish the proof. 




So, assume that this is not the case. Then there exist an $i<\omega$ and  
a $p_1=\langle n,\pi,h_1\rangle\in Q_{\bar h}$ stronger than $p_0$ such  
that
$p_1\forces``x_{\delta+j}\notin\dot{U}_i$.'' Let us define $h=h_1\restriction
\{x_\alpha:\alpha<\delta\}$ and $h_1\setminus h=\{\langle  
a_l,b_l\rangle
\colon l<m\}$. Notice that the condition $p=\langle n,\pi,h\rangle$  
belongs
to $(Q_{\bar h})^\delta$. We can also assume that $\langle x_{\delta+j},
y_{\delta+j}\rangle=\langle a_0,b_0\rangle$. 

Now consider the set
$Z$ of all 
$\la z_0,z_0',\ldots,z_{m-1},z_{m-1}'\ra\in (2^\omega\times 2^\omega)^m$ 
for which 
\begin{itemize}
\item there exist $s\in \fs$, $k<\kappa_s^i$, and
$q\in (Q_{\bar{h}})^\delta$ such that $s\subset z_0$,
$q$ extends $p$ and $p^i_{s,k}$, and
$\la z_0,z_0',\ldots,z_{m-1},z_{m-1}'\ra\in (\hat{q})^m$.
\end{itemize}



\noindent {\bf Claim} {\em The set $Z$ belongs to the model $M_\delta$  
and
it is an open dense subset of~$(\hat{p})^m$.}
\medskip

\noindent {\sc Proof.} It should be clear that $Z$ is (coded) in  
$M_\delta$.
(Remember the choice of~$\delta$.) To show that it is dense in  
$(\hat{p})^m$
we proceed like in the proof of Fact~\ref{FACT}. 
We choose $s_0,t_0,\ldots,s_{m-1},t_{m-1}$ 
and $r$ exactly as there. Next pick a condition
$q\in Q_{\bar{h}}$, a sequence $s\in\fs$, and $k<\kappa_s^m$ such that
\[
s_0\subset s \ \mbox{ and  \ \ $q$ \ extends \ $p^i_{s,k}$ \ and \ $r$.}
\]
(Remember the choice of the $p^i_{s,k}$'s.) Clearly we can demand that
$q\in (Q_{\bar{h}})^\delta$. Now note that it is possible to choose a 
$\bar{z}=\la z_0,z_0',\ldots,z_{m-1},z_{m-1}'\ra\in (\hat{q})^m$ such that 
$s\subset z_0$, $s_i\subset z_i$, $t_i\subset z_i'$. 
Then $\bar{z}\in Z\cap  
\prod_{i<k}([s_i]\times[t_i])$.  

Since $Z$ is clearly open, this completes the proof of Claim. 

\medskip



Now, by (\ref{PLem}) 
and the Claim above, $\la\la a_l,b_l\ra\colon l<m\ra$
belongs to $Z$ 
since $\la\la a_l,b_l\ra\colon l<m\ra$
belongs to $(\hat{p}_1)^m=(\hat{p})^m$. But this means that there  
exist 
$q=\langle n^q,\pi^q,h^q\rangle\in (Q_{\bar{h}})^\delta$ and
$s\in\fs$ such that:
\begin{itemize}
\item $q\leq p$, $q\forces$``$[s]\subseteq \dot{U_i}$'', and 
\item $\langle\langle a_l,b_l\rangle\colon l<m\rangle\in (\hat q)^m$,  
and
$x_{\delta+j}=a_0\in [s]$.  
\end{itemize}
But then $p_2=\langle n^q,\pi^q,h^q\cup\{\langle a_l,b_l\rangle\colon  
l<m\}\rangle$ 
belongs to $Q_{\bar h}$ and extends both $q$ and $p_1$. So,  
$p_2$
forces that $x_{\delta+j}=a_0\in[s]\subseteq\dot{U}_i$,  
contradicting our
assumption that $p_1\forces``x_{\delta+j}\notin\dot U_i$.''  





This finishes the proof of (\ref{eqN3}) 
and of Lemma~\ref{lemMain}. 



\begin{thebibliography}{Ro}

\bibitem%[B]
{B} M. Balcerzak, {\it Some remarks on sup-measurablilty}, 
Real Anal. Exchange
{\bf 17} (1991--92), 597--607.


\bibitem{BC}
M.~Balcerzak, K.~Ciesielski,
{\sl On sup-measurable functions problem},
Real Anal. Exchange {\bf 23} (1997--98), 787--797.

\bibitem{BJ} T.~Bartoszy\'{n}ski, H.~Judah, 
{\em Set Theory}, A~K~Peters, 1995.

\bibitem{CiBook} K.~Ciesielski,
{\sl Set Theory for the Working Mathematician}, 
London Math. Soc. Student Texts {\bf 39}, Cambridge Univ. Press 1997.

\bibitem%[GG]
{GG} E. Grande, Z. Grande,
{\it Quelques remarques sur la superposition $F(x,f(x))$}, 
Fund. Math. {\bf 121} (1984), 199--211. 

\bibitem%[GL]
{GL} Z. Grande, J. Lipi\'nski, 
{\it Un example d'une fonction sup-mesurable qui n'est pas mesurable}, 
Colloq. Math. {\bf 39} (1978), 77--79. 

\bibitem%[Kh1]
{Kh1} A.~B. Kharazishvili,
{\it Some questions from the theory of invariant measures}, 
Bull. Acad. Sci. Georgian SSR {\bf 100} (1980) (in Russian). 

\bibitem%[Kh2]
{Kh2} A.~B. Kharazishvili, {\it Sup-measurable and 
weakly sup-measurable
mappings in the theory of ordinary differential equations}, 
J. Appl. Anal. {\bf 3}(2) (1997), 211--223. 

\bibitem%[Kh3]
{Kh3} A.~B. Kharazishvili, {\sl 
Strange Functions in Real Analysis}, 
Pure and Applied Mathematics
{\bf 229}, Marcel Dekker, 2000. 

   
\bibitem{Ku} K.~Kunen, {\em Set Theory}, North-Holland, 1983.

\bibitem{ShPF} S.~Shelah, 
{\em Proper Forcing}, Lectures Notes in Math. 940, Springer-Verlag 1982. 

\bibitem{Sh:f} S.~Shelah,
{\em {Proper and improper forcing}},
{Perspectives in Mathematical Logic}, Springer-Verlag 1998.


\bibitem{Sr} J.~W. \u{S}ragin, {\it Conditions for measurability 
of superpositions},
Dokl. Akad. Nauk SSSR {\bf 197} (1971), 295--298 (in Russian).   

\end{thebibliography}

\end{document}







