\documentstyle[11pt,amstex,amsfonts,amssymb]{amsart} \textheight 19.4cm \textwidth 12.6cm \voffset -1.7cm \hoffset 1.8cm %\hoffset 2.4cm \setcounter{page}{159} \setcounter{section}{0} %\def \L{\char147} %\def \l{\char149} \renewcommand{\dot}{\overset{\boldsymbol .}} \renewcommand{\refname}{\bf References} \renewcommand{\proofname}{\em\bf Proof} \renewcommand{\abstractname}{\bf Abstract} \def \ss{\smallskip} \def \ms{\medskip} \def \bs{\bigskip} %\newcommand{\mathcal}{\cal} %\newcommand{\forces}{\mathrel{\|}\joinrel\mathrel{-}} \newcommand{\forces}{\Vdash} \newcommand{\fs}{2^{<\omega}} \newcommand{\ch}{} \newcommand{\cont}{{\frak c}}%{{\mathfrak c}} \newcommand{\suc}{\operatorname {succ}} \newcommand{\cf}{\operatorname {cf}} %\newcommand{\cc}{\operatorname{con}} \newcommand{\ccc}{\operatorname {con}} \newcommand{\supp}{\operatorname {supp}} \newcommand{\nor}{\operatorname{norm}} \newcommand{\norsup}{\operatorname{\underline{norm}}} \newcommand{\sq}{\subseteq} \newcommand{\real}{{\Bbb R}} \newcommand{\R}{{\real}} \newcommand{\rational}{{\Bbb Q}} \newcommand{\Q}{{\rational}} \newcommand{\integer}{{\Bbb Z}} \newcommand{\N}{{\Bbb N}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\NN}{{\cal N}} \newcommand{\nnn}{\operatorname {N}} \newcommand{\Sg}{{\Sigma}} \newcommand{\s}{{\sigma}} \newcommand{\h}{{\aleph}} \newcommand{\la}{{\langle}} \newcommand{\ra}{{\rangle}} \newcommand{\restr}{{\hbox{$\,|\grave{}\,$}}} \newcommand{\srestr}{{\hbox{${\scriptstyle\,|\grave{}\,}$}}} \newcommand{\A}{{\cal A}} \newcommand{\M}{{\cal M}} \newcommand{\F}{{\cal F}} \newcommand{\G}{{\cal G}} \newcommand{\K}{{\cal K}} \newcommand{\C}{\operatorname {C}} \newcommand{\B}{{\cal B}} \newcommand{\T}{{\cal T}} \newcommand{\D}{{\cal D}} \renewcommand{\H}{{\cal H}} \newcommand{\E}{{\cal E}} \newcommand{\e}{{\emptyset}} \newcommand{\ep}{{\varepsilon}} \newcommand{\g}{{\gamma}} \renewcommand{\d}{{\delta}} \newcommand{\const}{\operatorname {Const}} \newcommand{\bor}{\mbox{${\cal B}or$}} \newcommand{\proj}{\operatorname {pr}} %double equivalence \newcommand{\Equi}{\mbox{$\Leftrightarrow$}} \newcommand{\cm}{{\operatorname {C}_n}} \newcommand{\cc}{{\operatorname {C}_c}} \newcommand{\nd}{{\cal ND}} \newcommand{\continuum}{{\cont}} \def\poset{{\Bbb P}} \def\range{\mathop{\operatorname {range}}} \def\card{\mathop{\operatorname {Card}}} \def\cl{\mathop{\operatorname{cl}}} \def\inter{\mathop{\operatorname {int}}} \def\cof{\mathop{\operatorname {cof}}} \def\k{{\kappa}} \def\dom{\operatorname {dom}} %\def\proof{\noindent {\sc Proof. }} %\def\qed{\hfill\vrule height6pt width6pt depth1pt\medskip} %\def\endproof{{\qed}} \def\AA{{\cal A}} \newcommand{\tr}{\operatorname {tr}} \newcommand{\sru}{\operatorname {sru}} \renewcommand{\P}{{\cal P}} \def\implies{\longrightarrow} %% Theorems, etc. \newtheorem{theorem}{Theorem}%[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{problem}[theorem]{Problem} \newtheorem{Fact}[theorem]{Fact} \theoremstyle{definition} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newcommand{\thm}[2]{\begin{theorem}\label{#1}{#2}\end{theorem}} \newcommand{\cor}[2]{\begin{corollary}\label{#1}{#2}\end{corollary}} \newcommand{\prop}[2]{\begin{proposition}\label{#1}{#2}\end{proposition}} \newcommand{\lem}[2]{\begin{lemma}\label{#1}{#2}\end{lemma}} \newcommand{\pr}[2]{\begin{problem}\label{#1}{#2}\end{problem}} \newcommand{\ex}[2]{\begin{example}\label{#1}{#2}\end{example}} \newcommand{\defi}[2]{\begin{definition}\label{#1}{#2}\end{definition}} \newcommand{\rem}[2]{\begin{remark}\label{#1}{#2}\end{remark}} \newcommand{\fact}[2]{\begin{Fact}\label{#1}{#2}\end{Fact}} \title[C{ategory analogue of sup-measurability problem}] {\vspace {3.0cm}{\Large \bf Category analogue of sup-measurability problem}} \author [K. Ciesielski and S. S{helah}] {\bf K. Ciesielski and S. Shelah} \subjclass{Primary: 03E35, 26A15; Secondary: 26A30, 26B40} \keywords{Baire property, composition of functions, sup-measurable functions} \thanks{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. \\ The work of the second author was supported in part by a grant from ``Basic Research Foundation'' founded by the Israel Academy of Sciences and Humanities. Publication 695.\\ \ \\ ISSN 1425-6908 \quad \copyright \ Heldermann Verlag} \dedicatory{\footnotesize {\it Received September 3, 1999 and, in revised form, April 11, 2000}} %\date{} \begin{document} \leftline{\footnotesize \sc Journal of Applied Analysis} \leftline{\footnotesize Vol. 6, No. 2 (2000), pp. 159--172} \maketitle \thispagestyle{empty} \bigskip \begin{abstract} \ \ \ A function $F\colon\R^2\to\R$ is called {\em sup-measurable} if \break $F_f\colon \R\to\R$ 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} \ms \stepcounter{section} \section*{\bf 1. Introduction} \ms 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 \[ [s]\stackrel{\text {def}}{=}\{x\in 2^\omega\colon s\subset f\}, \] 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 $2^\omega$. \ms 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[Corollary~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[Proposition 1.5]{B}.) \ms \prop{pr1}{%\em The following conditions are equivalent. \begin{itemize} \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\linebreak $f\colon\R\to\R$. \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{itemize} } \ms 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. \ms \thm{thMain}{%\em 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} } \ms Before proving this theorem let us notice that it implies easily the following corollary. \ms \cor{cor1}{%\em \quad The existence of Baire sup-measurable function \linebreak $F\colon\R^2\to\R$ without the Baire property is independent from the set theory ZFC. } \ss \begin{pf} 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. \end{pf} \bs \stepcounter{section} \section*{\bf 2. Reduction of the proof of Theorem~\ref{thMain} to the main lemma} \ms 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{itemize} \item[(m)] every non-Baire subset $A^*$ of $2^\omega$ contains a non-Baire subset $A$ of cardinality $\omega_1$. \end{itemize} 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 \[ V[G_\alpha]\models `` \proj(A\cap f) \text{ is not Baire in $2^\omega$,''} \] 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[Chapter~IV]{Sh:f} (compare also \cite[Chapter~IV]{ShPF}) and here we will recall only the fragments that are relevant to our specific situation. In particular if \[ \Gamma\stackrel{\text {def}}{=}\{\lambda<\omega_1\colon \lambda \mbox{ is a limit ordinal}\} \] 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 $\delta+1\subset M_\delta$, $M_\delta\models ``\delta$\text{ 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. \ms \fact{f1}{%\em 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. } \ms 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. \ms \lem{lemMain}{%\em 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} } \ms 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\right. $ $\left. g^{-1}(\omega_1 \setminus \{0\}) \in [\alpha]^{<\omega}\right\}.$ This will allow us to treat the $\diamondsuit_{\omega_2}$-sequence \linebreak $\la X_\alpha\colon \alpha<\omega_2\ra$ 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}. \ms \lem{lemEx}{%\em 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} } \begin{pf} 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. \end{pf} \ms Now, the iteration $P_{\omega_2}$ is defined by choosing by induction the sequence \linebreak $\la \la P_\alpha,\dot A_\alpha,\dot\M_\alpha, \dot Q_\alpha, \dot f_\alpha \ra\colon\alpha<\omega_2\ra$ 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{\bf 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{\bf 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 $\dot A_\alpha$. \bs \stepcounter{section} \section*{\bf 3. 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 $r_0,r_1\in 2^\omega$. \ms \fact{fact2}{%\em 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 $2^\omega$. } \ss \begin{pf} 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 $\cl(h[D])=2^\omega$. \end{pf} \ms 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{itemize} \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{itemize} 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