% Corrected by Krzysztof Chris Ciesielski according to Jas suggestions. 
% Sent 10/15/2002

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\title{Uncountable $\gamma$-sets under
axiom CPA$_{\rm cube}^{\rm game}$}

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\markboth{K.~Ciesielski, A.~Mill\'an, and J.~Pawlikowski
\ \ \ \ \ \UpdateDate
}{Uncountable $\gamma$-sets under CPA$_{\rm cube}^{\rm game}$
\ \ \ \ \ \ \ \ \ \UpdateDate
}


\author{
Krzysztof Ciesielski%
\thanks{AMS classification numbers: Primary 03E35; Secondary
03E17, 26A03. \endgraf
\ \ Key words and phrases: $\gamma$-set, strongly meager. \endgraf
\ \ The work of the first author was partially supported by
NATO Grant PST.CLG.977652.
}\\
{\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/\~{}kcies}}
\and
Andr\'es Mill\'an Mill\'an%
\thanks{
This work was completed when the second author was a Ph.D. student at
West Virginia University. Some part of this work is likely to 
be included in his dissertation, written under the supervision of K.~Ciesielski .}\\
{\footnotesize Department of Mathematics,}
{\footnotesize West Virginia University,} \\
{\footnotesize Morgantown, WV 26506-6310, USA}; %\\
{\footnotesize e-mail: amillan@math.wvu.edu} %\\
\and
Janusz Pawlikowski\thanks{
The work of the third author was partially supported by
??? %KBN
Grant ??? %2 P03A 031~14.
}\\
{\footnotesize Department of Mathematics,}
{\footnotesize University of  Wroc\l aw,} \\
{\footnotesize pl. Grunwaldzki 2/4, 50-384 Wroc\l aw, Poland;} %\\
{\footnotesize e-mail: pawlikow@math.uni.wroc.pl}\\
}

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\begin{document}

\maketitle



\begin{abstract}
In the paper we formulate a Covering Property Axiom \psmCGame,
which holds in the iterated perfect set model,
and show that it implies
the existence of uncountable strong $\gamma$-sets in $\real$
(which are strongly meager)
as well as uncountable $\gamma$-sets in $\real$ which 
are not strongly meager. These sets must be
of cardinality $\omega_1<\continuum$, since
every $\gamma$-set is universally null, while
\psmCGame\ implies that every universally null
has cardinality less than $\continuum=\omega_2$.


We will also show that \psmCGame\ implies the existence of
a partition of $\real$ into $\omega_1$ null compact sets.
\end{abstract}

\section{Axiom \psmCGame\ and other preliminaries}
Our set theoretic terminology is standard and follows
that of~\cite{CiBook}.
In particular, $|X|$ stands for the cardinality of a set $X$
and $\continuum=|\real|$.
The Cantor set $2^\omega$ will be denoted by a symbol $\Cantor$.
We use term {\em Polish space}\/
for a complete separable metric space {\tt without
isolated points}.
For a Polish space $X$, the symbol $\perf(X)$ will denote the 
collection of all subsets of $X$ homeomorphic to $\Cantor$.
We will consider $\perf(X)$ as ordered by inclusion.

Axiom \psmCGame\ was first formulated by Ciesielski and
Pawlikowski in~\cite{CP83}. (See also~\cite{CPAbook}.)
It is a simpler version of a Covering Property Axiom
CPA which holds in the iterated perfect set model.
(See \cite{CP83} or~\cite{CPAbook}.)
In order to formulate \psmCGame\ we need the following
terminology and notation.
A subset $C$ of a product 
$\Cantor^\omega$ of the Cantor set is
said to be a {\em perfect cube}\/
if
$C=\prod_{n\in\omega} C_n$, where $C_n\in\perf(\Cantor)$ for each $n$.
For a fixed Polish space $X$
let $\Fcube$
stand for the family of all continuous
injections from a perfect cube $C\subset\Cantor^\omega$ onto a set $P$
from $\perf(X)$.
We consider each function $f\in\Fcube$ from $C$ onto $P$
as a coordinate system imposed on~$P$.
We say that $P\in\perf(X)$ is a {\em cube}\/ if
we consider it with (implicitly given)
witness function $f\in\Fcube$ onto $P$,
and $Q$ is a {\em subcube of a cube}\/ $P\in\perf(X)$
provided $Q=f[C]$, where $f\in\Fcube$ is 
the witness function for $P$
and $C\subset\dom(f)\subset\Cantor^\omega$
is a perfect cube.
Here and in what follows symbol
$\dom(f)$
stands for the domain of $f$.


We say that a family $\E\subset\perf(X)$ is {\em cube dense}\/
in $\perf(X)$
provided every cube $P\in\perf(X)$ contains a subcube
$Q\in\E$.
More formally,
$\E\subset\perf(X)$ is cube dense provided
\begin{equation}\label{eqDefFdense}
\forall f\in\Fcube\ \exists g\in\Fcube\
(g\subset f\ \&\ \range(g)\in\E).
\end{equation}
It is easy to see that the notion of
cube density is a generalization of
a notion of density with respect to $\la \perf(X),\subseteq\ra$,
that is, if $\E$ is cube dense in $\perf(X)$
then $\E$ is dense in $\perf(X)$.
On the other hand, the converse implication is not true, as shown by the
following simple example.

\ex{exCubeDens}{{\rm (\cite{CP88,CPAbook})}
Let $X=\Cantor\times\Cantor$ and let $\E$ be the family
of all $P\in\perf(X)$ such that either
all vertical sections of $P$ are countable, or else
all horizontal sections of $P$ are countable.
Then $\E$ is dense in $\perf(X)$, but it is not cube dense in $\perf(X)$.
}

It is also worth to notice that in order to check
that $\E$ is cube dense it is enough to consider
in condition (\ref{eqDefFdense})
only functions $f$ defined on the entire space $\Cantor^\omega$,
that is

\fact{factFcube}{{\rm (\cite{CP83,CP88,CPAbook})}
$\E\subset\perf(X)$ is cube dense if and only if
\begin{equation}\label{eqDefFdenseWeak}
\forall f\in\Fcube,\
\dom(f)=\Cantor^\omega,\
\exists g\in\Fcube\
(g\subset f\ \&\ \range(g)\in\E).
\end{equation}
}

Let $\perf^*(X)$ stand for the family of
all sets $P$ such that either $P\in\perf(X)$ or $P$ is a singleton in $X$.
In what follows we will consider singletons 
as {\em constant cubes}, that is,
with the constant coordinate 
function from $\Cantor^\omega$
onto the singleton. In particular, a subcube of a constant cube is 
the same singleton. 


Consider the following game
${\rm GAME}_{\rm cube}(X)$\label{PageGameCube}
of length $\omega_1$.
The game has two players, Player~I and Player~II.
At each stage $\xi<\omega_1$ of the game Player~I can play an
arbitrary cube $P_\xi\in\perf^*(X)$
and Player~II must respond with a subcube $Q_\xi$ of $P_\xi$.
The game $\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
is won by Player~I provided
\[
\bigcup_{\xi<\omega_1}Q_\xi=X;
\]
otherwise the game is won by Player~II.

By a strategy for Player~II
we will understand any function $S$
such that
$S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)$
is a subcube of $P_\xi$, where
$\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra$ is any partial game.
(We abuse here slightly the notation, since function $S$
depends also on the implicitly given coordinate functions
$f_\eta\colon \Cantor^\omega\to P_\eta$ making each $P_\eta$ a cube.)
A game $\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
is played according to a strategy $S$ for Player~II provided
$Q_\xi=S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)$
for every $\xi<\omega_1$.
A strategy $S$ for Player~II is a {\em winning strategy}\/
for Player~II provided Player~II wins any game
played according to the strategy $S$.

Here is the axiom.

\begin{description}
\item[{\bf \psmCGame:}] $\continuum=\omega_2$ and for any Polish space $X$
Player~II has no winning strategy
in the game ${\rm GAME}_{\rm cube}(X)$.
\end{description}

Notice that

\prop{prop:CGImplC}{{\rm (\cite{CP83,CPAbook})}
Axiom \psmCGame\ implies
\begin{description}
\item[{\bf \psmC:}] $\continuum=\omega_2$ and
for every Polish space $X$ and every cube dense
family $\E\subset\perf(X)$ there is an $\E_0\subset\E$
such that $|\E_0|\leq\omega_1$ and $|X\setminus\bigcup\E_0|\leq\omega_1$.
\end{description}
}

In \cite{CP83} (see also \cite{CPAbook}) it was proved that
\psmC\ (so, also \psmCGame) implies that
$\cof(\NN)=\omega_1$
and that all perfectly meager sets
and all universally null sets have cardinality at most $\omega_1$.

In what follows we will also use the following simple fact.
Its proof can be found in \cite{CP88} and~\cite{CPAbook}.

\claim{claim1}{Consider $\Cantor^\omega$ with standard
topology and standard product measure.
If $G$ is a Borel subset of $\Cantor^\omega$ which is either
of second category or of positive measure
then $G$ contains a perfect cube $\prod_{i<\omega}P_i$.
}



\section{Disjoint coverings by $\omega_1$ null compacts}

\thm{th:DisjPart1}{Assume that \psmCGame\ holds and let
$X$ be a Polish space. If
$\D\subset\perf(X)$ is $\Fcube$-dense and it is
closed under perfect subsets then there exists a
partition of $X$ into $\omega_1$ disjoint sets from
$\D\cup\{\{x\}\colon x\in X\}$.
}

In the proof we will use the following easy lemma.

\lem{lem:DisjPart1}{Let $X$ be a Polish space
and let $\P=\{P_i\colon i<\omega\}\subset\perf^*(X)$.
For every cube $P\in\perf(X)$ there exists 
a subcube $Q$ of $P$
such that either $Q\cap\bigcup_{i<\omega}P_i=\emptyset$ or
$Q\subset P_i$ for some $i<\omega$.}

\proof Let $f\in\Fcube$ be such that $f[\Cantor^\omega]=P$.

If $P\cap\bigcup_{i<\omega}P_i$ is meager in $P$
then, by  Claim~\ref{claim1},
we can find a subcube $Q$ of $P$ such that
$Q\subset P\setminus \bigcup_{i<\omega}P_i$.

If $P\cap\bigcup_{i<\omega}P_i$ is not meager in $P$
then there exists an $i<\omega$ such that
$P\cap P_i$ has a non-empty interior in $P$.
Thus, there exists a basic clopen set $C$ in $\Cantor^\omega$,
which is a perfect cube, such that $f[C]\subset P_i$.
So, $Q=f[C]$ is a desired subcube of $P$. \qed

\noindent{\sc Proof of Theorem~\ref{th:DisjPart1}.}
For a cube $P\in\perf(X)$ and a countable family $\P\subset\perf^*(X)$
let $D(P)\in\D$ be a subcube of $P$
and $Q(\P,P)\in\D$ be as in Lemma~\ref{lem:DisjPart1}
used with $D(P)$ in place of $P$.
For a singleton $P\in\perf^*(X)$
we just put $Q(\P,P)=P$.

Consider the following strategy $S$ for Player~II:
\[
S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)=
Q(\{Q_\eta\colon\eta<\xi\},P_\xi).
\]
By \psmCGame\ strategy $S$ is not a winning strategy for Player~II.
So there exists a game
$\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
played according to $S$
in which Player~II looses, that is,
$X=\bigcup_{\xi<\omega_1}Q_\xi$.

Notice that for every $\xi<\omega_1$ either
$Q_\xi\cap\bigcup_{\eta<\xi}Q_\eta=\emptyset$ or there is
an $\eta<\omega_1$ such that $Q_\xi\subset Q_\eta$.
Let
\[
\F=\left\{Q_\xi\colon \xi<\omega_1\ \&\
Q_\xi\cap\bigcup_{\eta<\xi}Q_\eta=\emptyset\right\}.
\]
Then $\F$ is as desired.
\qed

Since a family of all measure zero perfect subsets of
$\real^n$ is $\Fcube$-dense we get the following corollary.

\cor{cor:DisjPart1}{\psmCGame\ implies that
there exists a
partition of $\real^n$ into $\omega_1$ disjoint closed nowhere dense measure
zero sets.
}

Note that the conclusion of Corollary~\ref{cor:DisjPart1}
does not follow from the fact that $\real^n$ can be covered
by $\omega_1$ perfect measure zero subsets.
(See \cite[thm.~6]{Miller0}.)

\section{Uncountable $\gamma$-sets}\label{secGamma}

In this subsection we will prove that \psmCGame\ implies
the existence of an uncountable $\gamma$-set.
Recall that a subset $T$ of a Polish space $X$ is
a {\em $\gamma$-set}\/ provided for
every open $\omega$-cover $\U$ of $T$ there is
a sequence $\la U_n\in\U\colon n<\omega\ra$ such that
$T\subset\bigcup_{n<\omega}\bigcap_{i>n} U_i$,
where $\U$ is an {\em $\omega$-cover}\/ of $T$
if for every finite set $A\subset T$ there a $U\in\U$ with $A\subset U$.

$\gamma$-sets were introduced by Gerlits and Nagy~\cite{GN}.
They were studied by Galvin and Miller~\cite{GM},
Rec\l aw~\cite{Rec}, Bartoszy\'nski, Rec\l aw~\cite{BR}, and others.
It is known that under the Martin's axiom
there are $\gamma$-sets of cardinality continuum~\cite{GM}.
On the other hand, every $\gamma$-set is strong measure zero~\cite{GN},
so it is consistent with ZFC that every
$\gamma$-set is countable.
Moreover, \psmCGame\ implies that every $\gamma$-set has
cardinality at most $\omega_1<\continuum$,
since every strong measure zero is universally null
and under \psmCGame\ every universally null has cardinality $\leq\omega_1$.

In what follows we will use the characterization of $\gamma$-sets due to
Rec\l aw~\cite{Rec}. To formulate it we need to fix some
terminology.
Thus, in what follows we will consider $\P(\omega)$ as a Polish
space by identifying it with $2^\omega$ via characteristic functions.
For $A,B\subset\omega$ we will write $A\subseteq^*B$ when $|A\setminus B|<\omega$. 
We say that a family $\A\subset\P(\omega)$ is {\em centered}\/
provided $\bigcap\A_0$ is infinite for every finite $\A_0\subset\A$;
and $\A$ {\em has a pseudointersection}\/ provided there exists
a $B\in[\omega]^\omega$ such that $B\subseteq^*A$ for every
$A\in\A$. In addition for the rest of this section
$\K$ will stand for the family of all continuous functions
from $\P(\omega)$ to $\P(\omega)$ and for
$A\in\P(\omega)$ we put $A^*=\{B\in\P(\omega)\colon B\subseteq^*A\}$.

\prop{prop:Rec}{{\bf (Rec\l aw~\cite{Rec})}
For $T\subset\P(\omega)$ the following conditions are equivalent.
\begin{itemize}
\item[{\rm (i)}] $T$ is a $\gamma$-set.
\item[{\rm (ii)}] For every $f\in\K$ if $f[T]$ is centered than
$f[T]$ has a pseudointersection.
\end{itemize}
}

In the proof that follows we will apply axiom 
\psmCGame\ to the cubes from the space $\K$.
The fact that the subcubes given by the axiom cover $\K$ will 
allow us to use the above characterization 
to conclude that the constructed set is indeed a $\gamma$-set.
It is also possible to construct an uncountable $\gamma$-set
by applying axiom \psmCGame\ to the space $\Y$ 
of all $\omega$-covers of $\P(\omega)$,%
\footnote{More precisely, if $\B_0$ is a countable base for 
$\P(\omega)$ and $\B$ is the collection of all
finite unions of elements from $\B_0$
then we can define $\Y$ as $\B^\omega$ considered with the product topology, where 
$\B$ is taken with discrete topology.}
similarly as in Section~\ref{sec5}. 
However, we believe that greater diversification of spaces
to which we apply \psmCGame\ makes the paper more interesting. 


In what follows we will need the following two lemmas.

\lem{lemGamma0}{For every countable set $Y\subset\P(\omega)$ the set
\[
\K_Y=\{f\in\K\colon f[Y] \mbox{ is centered}\}
\]
is Borel in $\K$.
}

\proof Let $Y=\{y_i\colon i<\omega\}$ and note that
\[
\K_Y=\bigcap_{n,k<\omega}\bigcup_{m\geq k}\bigcap_{i<n}
\{f\in\K\colon m\in f(y_i)\}.
\]
So, $\K_Y$ is a $G_\delta$ set, since each set
$\{f\in\K\colon m\in f(y_i)\}$ is open in $\K$. \qed

\lem{lemGamma}{Let $Y\subset\P(\omega)$ be countable and such that
$[\omega]^{<\omega}\subset Y$.
For every $W\in[\omega]^\omega$ and a compact set
$Q\subset\K_Y$
there exist $V\in[W]^\omega$
and a continuous function $\varphi\colon Q\to[\omega]^\omega$ such that
$\varphi(f)$ is a pseudointersection of $f[Y]\cup f[V^*]$
for every $f\in Q$.

Moreover, if $\J$ is an infinite family
of non-empty pairwise disjoint finite subsets of $W$ then we can choose $V$
such
that it contains infinitely many $J$'s from $\J$.
}

\proof First notice that there exists a continuous
$\psi\colon Q\to[\omega]^\omega$ such that
$\psi(f)$  is a pseudointersection of $f[Y]$ for every $f\in Q$.

Indeed, let $Y=\{y_i\colon i<\omega\}$ and for every $f\in Q$ let
$\psi(f)=\left\{n_i^f\colon i<\omega\right\}$, where
$n_0^f=\min f(y_0)$ and
$n_{i+1}^f=\min\left\{n\in\bigcap_{j\leq i}f(y_j)\colon n>n_i^f\right\}$.
The set in the definition of $n_{i+1}^f$ is non-empty, since
$f[Y]$ is centered, as $f\in Q\subset\K_Y$.
It is easy to see that $\psi$ is continuous and
that $\psi(f)$ is as desired.

We will define
a sequence $\la J_i\in\J\colon  i<\omega\ra$
such that $\max J_i<\min J_{i+1}$ for every $i<\omega$.
We are aiming for $V=\bigcup_{i<\omega}J_i$.

A set $J_0\in\J$ is chosen arbitrarily.
Now, if $J_i$ is already defined for some $i<\omega$ we
define $J_{i+1}$ as follows. Let $w_i=1+\max J_i$. Thus $J_i\subset w_i$.
For every $f\in Q$ define
\[
m_i^f=\min\left(\psi(f)\cap\bigcap f[\P(w_i)] \right).
\]
The set $\psi(f)\cap\bigcap f[\P(w_i)]$ is infinite,
since $\psi(f)$ is a pseudointersection of $f[Y]$ while
$\P(w_i)\subset Y$.
Let $k_i^f=\min K_i^f$, where
\[
K_i^f=\left\{k\geq w_i\colon m_i^f\in f(a)
\mbox{ for all $a\subset\omega$ with $a\cap k\subset w_i$}\right\}.
\]
The fact that $K_i^f\neq\emptyset$ follows from the continuity of $f$
since $m_i^f\in f(a) $ for all $a\subset w_i$.
Notice that, by the continuity of $\psi$ and the assignment of $k_i^f$,
for every $p<\omega$ the set
$U_p=\{f\in Q\colon k_i^f<p\}$ is open in $Q$.
Since sets $\{U_p\colon p<\omega\}$ form an increasing
cover of $Q$, compactness of
$Q$ implies the existence of $p_i<\omega$ such that
$Q\subset U_{p_i}$.
Thus, $w_i\leq k_i^f<p_i$ for every $f\in Q$.
We define $J_{i+1}$ as an arbitrary element of $\J$ disjoint with $p_i$
and notice that
\begin{center}
$m_i^f\in f(a)$ for every $f\in Q$ and $a\subset\omega$ with
$a\cap \min J_{i+1}\subset w_i$.
\end{center}
This finishes the inductive construction.

Let $V=\bigcup_{i<\omega}J_i\subset W$ and $\varphi(f)=\{m_i^f\colon
i<\omega\}$.
It is easy to see that $\varphi$ is continuous
(though, we will not use this fact).
To finish the proof it is enough to show that
$\varphi(f)$ is a pseudointersection of $f[Y]\cup f[V^*]$
for every $f\in Q$.

So, fix an $f\in Q$.  Clearly $\varphi(f)\subset\psi(f)$ is a
pseudointersection of $f[Y]$ since so was $\psi(f)$.
To see that $\varphi(f)$ is a
pseudointersection of $f[V^*]$ take an $a\subseteq^* V$.
Then for almost all $i<\omega$ we have
$a\cap \min J_{i+1}\subset w_i$,
so that $m_i^f\in f(a)$.
Thus $\varphi(f)\subseteq^* f(a)$.
\qed


\thm{thmGammaSet}{\psmCGame\ implies that there exists an uncountable
$\gamma$-set in $\P(\omega)$.}

\proof For $\alpha<\omega_1$ and an $\subseteq^*$-decreasing
sequence $\V=\{V_\xi\in[\omega]^\omega\colon\xi<\alpha\}$
let $W(\V)\in[\omega]^\omega$ be such that
$W(\V)\subsetneq^* V_\xi$ for all $\xi<\alpha$.
Moreover, if $P\in\perf^*(\K)$
is a cube then we
define a subcube $Q=Q(\V,P)$ of $P$
and an infinite subset $V=V(\V,P)$ of $W=W(\V)$
as follows.
Let $Y=\V\cup[\omega]^{<\omega}$ and
choose a subcube $Q$ of $P$ such that
either $Q\cap \K_Y=\emptyset$ or $Q\subset\K_Y$.
This can be done by Claim~\ref{claim1} since
$\K_Y$ is Borel.
If $Q\cap \K_Y=\emptyset$ we put $V=W$.
Otherwise we apply Lemma~\ref{lemGamma} to find $V$.

Consider the following strategy
$S$ for Player~II:
\[
S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)=
Q(\{V_\eta\colon\eta<\xi\},P_\xi),
\]
where sets $V_\eta$ are defined inductively by
$V_\eta=V(\{V_\zeta\colon\zeta<\eta\},P_\eta)$.
In other words, Player~II remembers (recovers)
sets $V_\eta$ associated with the cubes $P_\eta$ played so far, and he uses them
(and Lemma~\ref{lemGamma}) to get the next answer
$Q_\xi=Q(\{V_\eta\colon\eta<\xi\},P_\xi)$,  while remembering
(or recovering each time)
the set $V_\xi=V(\{V_\eta\colon\eta<\xi\},P_\xi)$.

By \psmCGame\ strategy $S$ is not a winning strategy for Player~II.
So there exists a game
$\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
played according to $S$
in which and Player~II loses, that is,
$\K=\bigcup_{\xi<\omega_1}Q_\xi$.
Let $\V=\{V_\xi\colon\xi<\omega_1\}$ be a sequence
associated with this game, which is
strictly $\subseteq^*$-decreasing, and let $T=\V\cup[\omega]^{<\omega}$.
We claim that $T$ is a $\gamma$-set.

In the proof we use Lemma~\ref{lemGamma0}.
So, let $f\in\K$ be such that $f[T]$ is centered.
There exists an $\alpha<\omega_1$ such that $f\in Q_\alpha$.
Since $f[\{V_\xi\colon \xi<\alpha\}\cup[\omega]^{<\omega}]\subset f[T]$
we must have applied Lemma~\ref{lemGamma}
in the choice of $Q_\alpha$ and $V_\alpha$.
Therefore, the family
$f[\{V_\xi\colon \xi<\alpha\}\cup[\omega]^{<\omega}\cup V_\alpha^*]$
has a pseudointersection.
So, $f[T]$ has a pseudointersection too, since
$T\subset \{V_\xi\colon \xi<\alpha\}\cup[\omega]^{<\omega}\cup V_\alpha^*$.
\qed

Since $\P(\omega)$ embeds into any Polish space,
we conclude that, under \psmCGame,
any Polish space contains an uncountable $\gamma$-set.
In particular, there exists an uncountable $\gamma$-set $T\subset\real$.

\section{$\gamma$-sets in $\real$ which are not strongly meager}

Recall (see e.g.~\cite[p.~437]{BJ})
that a subset $X$ of $\real$ is {\em strongly meager}\/
provided $X+G\neq\real$ for every measure zero subset $G$ of $\real$.
This is 
a notion which is dual to a strong measure zero subset of $\real$, since
Galvin, Mycielski, and Solovay proved (see e.g.~\cite[p.~405]{BJ}) that:
$X\subset\real$ is strong measure zero if and only if
$X+M\neq\real$ for every meager subset $M$ of $\real$.

Now, although every $\gamma$-set is strong measure zero,
under the Martin's axiom Bartoszy\'nski and  Rec\l aw~\cite{BR} constructed
a $\gamma$-set $T$ in $\real$ which is not strongly meager.
In what follows we will show that the existence of such a set
follows also from \psmCGame.
The construction is a generalization of that
used in the proof of Theorem~\ref{thmGammaSet}.

In the proof we will use the following notation.
For $A,B\subset\omega$
we define $A+B$ as the symmetric difference between $A$ and $B$.
Upon identification of a set $A\subset\omega$ with its characteristic
function $\charf A\in 2^\omega$ this definition is motivated
by the fact that $\charf{A+B}(n)=\charf{A}(n)+_2\charf{B}(n)$,
where $+_2$ is the addition modulo~$2$.
Also, let $\bar\J=\{J_n\in[\omega]^{2^n}\colon n<\omega\}$
be a family of pairwise disjoint sets and
let $\tilde G$ be the family of all $W\subset\omega$ which are disjoint
with infinitely many $J\in\bar\J$.
Notice that $\tilde G$ has measure zero with respect to the
standard measure on $\P(\omega)$ induced by the product measure on
$2^\omega$.



\lem{lemGamma2}{If $\J\in[\bar\J]^\omega$ and
$P$ is a cube in $\P(\omega)$ then
there exists a subcube $Q$ of $P$ and
a set $V\subset\bigcup\J$ containing infinitely many $J\in\J$
such that $V+Q\subset \tilde G$.
}

\proof 
Let $D=\bigcup\J$ and 
\begin{eqnarray*}
H & = & \{\la U,W\ra\in\P(D)\times\P(\omega)\colon
(U+W)\cap J=\emptyset\mbox{ for infinitely many } J\in\J\}\\
& \subseteq & \{\la U,W\ra\in\P(D)\times\P(\omega)\colon U+W\in \tilde G\}.
\end{eqnarray*}
Note that $H$ is a $G_\delta$ subset of $\P(D)\times\P(\omega)$
since $H_J=\{\la U,W\ra\colon(U+W)\cap J=\emptyset\}$
is open for every $J\in\J$. Moreover horizontal sections of $H$
are dense in $\P(D)$. So,
$\bar H=H\cap(\P(D)\times P)$ is a dense $G_\delta$ subset of
$\P(D)\times P$, as all its horizontal sections are dense.
Thus, by Kuratowski-Ulam theorem, there is a dense $G_\delta$ subset
$\K_0$ of $\P(D)$ such that for every $U\in\K_0$
the vertical section $\bar H_U$ of $\bar H$ is dense
in $P$.
Now, since
\[
\K_1=\{U\in\P(D)\colon J\subset U\mbox{ for infinitely many } J\in\J\}
\]
is a dense $G_\delta$ there is a $V\in\K_0\cap\K_1$.
In particular, $V$ contains infinitely many $J\in\J$ and
$\bar H_V$ is a dense $G_\delta$ subset of $P$. 
So, by Claim~\ref{claim1},
there exists a subcube $Q$ of $P$ contained in $\bar H_V$.
Thus, $Q\subset\bar H_V\subset\{W\in P\colon V+W\in \tilde G\}$
and so $V+Q\subset \tilde G$. \qed

\thm{thmGammaSet2}{\psmCGame\ implies that there exists a
$\gamma$-set $T\subset\P(\omega)$ such that $T+\tilde G=\P(\omega)$.}

\proof We will use \psmCGame\ for the space
$X=\K\cup\P(\omega)$, a direct sum of $\K$ and $\P(\omega)$.



For $\alpha<\omega_1$ and an $\subseteq^*$-decreasing
sequence $\V=\{V_\xi\in[\omega]^\omega\colon\xi<\alpha\}$
such that each $V_\xi$ contains infinitely many
$J\in\bar\J$
let $W(\V)\in[\omega]^\omega$ be such that
$\J=\{J\in\bar\J\colon J\subset W(\V)\}$ is infinite and
$W(\V)\subsetneq^* V_\xi$ for all $\xi<\alpha$.
For a cube $P\in\perf^*(\K)$
we define a subcube $Q=Q(\V,P)$ of $P$
and an infinite subset $V=V(\V,P)$ of $W=W(\V)$
as follows. By Claim~\ref{claim1} we can find subcube $P'$ of $P$
such that either $P'\subset\K$ or $P'\subset\P(\omega)$.

If $P'\subset\K$ we proceed as in the proof of Theorem~\ref{thmGammaSet}.
We put $Y=\V\cup[\omega]^{<\omega}$ and we use Claim~\ref{claim1}
to find a subcube $Q$ of $P'$ such that
either $Q\cap \K_Y=\emptyset$ or $Q\subset\K_Y$.
If $Q\cap \K_Y=\emptyset$ we put $V=W$.
Otherwise we apply Lemma~\ref{lemGamma} to find $V$.
If $P'\subset\P(\omega)$ we use Lemma~\ref{lemGamma2}
to find $Q$ and $V$.

Consider the following strategy
$S$ for Player~II:
\[
S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)=
Q(\{V_\eta\colon\eta<\xi\},P_\xi),
\]
where sets $V_\eta$ are defined inductively by
$V_\eta=V(\{V_\zeta\colon\zeta<\eta\},P_\eta)$.
By \psmCGame\ strategy $S$ is not a winning strategy for Player~II.
So there exists a game
$\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
played according to $S$
in which and Player~II loses, that is,
$X=\bigcup_{\xi<\omega_1}Q_\xi$.
Let $\V=\{V_\xi\colon\xi<\omega_1\}$ be a sequence
associated with this game, which is
strictly $\subseteq^*$-decreasing, and let $T=\V\cup[\omega]^{<\omega}$.
We claim that $T$ is as desired.

The argument that $T$ is a $\gamma$-set is the same as in the proof of
Theorem~\ref{thmGammaSet}. To see that
$\P(\omega)\subset T+\tilde G$ notice that for every
$A\in\P(\omega)$ there is an $\alpha<\omega_1$ such that
$A\in Q_\alpha$. But then at step $\alpha$ we used
Lemma~\ref{lemGamma2}
to find $Q_\alpha$ and $V_\alpha$.
In particular, $V_\alpha+Q_\alpha\subset \tilde G$.
So, $A\in Q_\alpha\subset V_\alpha+\tilde G\subset T+\tilde G$.
\qed

\cor{corGammaNotSM}{\psmCGame\ implies that there exists a
$\gamma$-set $X\subset\real$ which is not strongly meager.}

\proof This is the argument from~\cite{BR}.
Let $T$ be as in
Theorem~\ref{thmGammaSet2} and let $f\colon\P(\omega)\to[0,1]$,
$f(A)=\sum_{i<\omega}2^{-(i+1)}\charf A(i)$.
Then $f$ is continuous, so $X=f[T]$
is a $\gamma$-set. Let $H=\bigcap_{m<\omega}\bigcup_{n>m} f[J_n]$.
Then $H$ has measure zero and it is easy to see that
$[0,1]=f[\P(\omega)]\subset f[T]+H=X+H$.
Then $\bar G=H+\rational$ has measure zero
and $X+\bar G=\real$.
\qed

\section{Uncountable strongly meager $\gamma$-sets in $\real$}\label{sec5}

Let $X$ be a Polish space with topology $\tau$. 
We say that $\U\subset\tau$ is a cover of $Z\subset[X]^{<\omega}$ provided
for every $A\in Z$ there is a $U\in\U$ with $A\subset U$. 
Following \cite{GM} we say that
a subset $S$ of $X$ is a
{\em strong $\gamma$-set}\/ provided there exists
an increasing sequence $\la k_n<\omega\colon n<\omega\ra$
such that for every sequence $\la J_n\subset\tau\colon n<\omega\ra$,
where each $J_n$ is a cover of $[X]^{k_n}$,
there exists a sequence $\la D_n\in J_n\colon n<\omega\ra$
with $X\subset\bigcup_{n<\omega}\bigcap_{m>n} D_m$.
It is proved in \cite{GM} that every strong $\gamma$-set
$X\subset\real$ is strongly meager.
The goal of this section is to construct, under \psmCGame,
an uncountable strong $\gamma$-set in $\P(\omega)$.
So, after identifying $\P(\omega)$ with its homeomorphic copy in $\real$, 
this will become an uncountable $\gamma$-set in $\real$ which is strongly meager.
Under Martin's axiom a strong $\gamma$-set in $\P(\omega)$ 
of cardinality continuum exists, see~\cite{GM}.

Let $\B_0$ be a countable basis for the topology of 
$\P(\omega)$ and let $\B$ be the collection of all
finite unions of elements from $\B_0$. 
Since every open cover of $[\P(\omega)]^k$, $k<\omega$, contains 
a refinement from $\B$, in the definition of strong $\gamma$-set
it is enough to consider only sequences
$\la J_n\colon n<\omega\ra$ with $J_n\subset\B$.

Now, consider $\B$ with the discrete topology. Since
$\B$ is countable, the space $\B^\omega$, considered with 
the product topology, is a Polish space
and so is $\X=(\B^\omega)^\omega$. 
For $J\in\X$ we will write $J_n$ in place of $J(n)$. 
It is easy to see that a subbasis for the topology of $\X$ is given for the clopen sets
\[
\{J\in\X\colon J_{n}(m)=B\},
\]
where $n,m<\omega$ and $B\in\B$. 

For the reminder of this section fix 
an increasing sequence 
$\la k_n<\omega\colon n<\omega\ra$ such that  $k_n\geq n\; 2^n +n$ for every $n<\omega$.  
Then we have the following lemma.

\lem{lemA1}{Let $X\in{[\omega]^\omega}$ and let $F$ be a countable subset of $\P(\omega)$ 
such that $[\omega]^{<\omega}\subset F$.
Assume that $P$ is a compact subset of $\X$ such that
for every $J\in P$ and $n<\omega$ the family $J_n[\omega]=\{J_n(m)\colon m<\omega\}$
covers $[F]^{k_{n}}$. Then there exists a set $Y\in[X]^\omega$ and for each $J\in P$ a sequence
$\la D_n^J\in J_n\colon n<\omega\ra$ such that
$F\cup{Y^{*}}\subset{\bigcup_{n<\omega}\bigcap_{m>n}D_{m}^{J}}$.
}

\proof 
Let $\{F_n\colon n<\omega\}$ be an enumeration of $[\omega]^{<\omega}$ 
such that  $F_n\subset n$ for all $n<\omega$ and let $F=\{f_n\colon n<\omega\}$.
We will construct inductively the sequences
$\la s_n\in X\colon n<\omega\ra$ and 
$\la \{D^J_n\in J_n[\omega]\colon J\in P\}\colon n<\omega\ra$ such that
for every $n<\omega$, $J\in P$, and $A\subset\omega$ we have 
\begin{description}
\item{(i)} $\{f_i\colon i<n\}\subset D^J_n$ and $s_n<s_{n+1}$; 
\item{(ii)} if $i<j\leq n+1$ and $(A\cap{s_{n+1}})\setminus{\{s_0,\ldots,s_n\}}=F_i$ then $A\in D_{j}^{J}$. 
\end{description}

We chose $s_0\in X$ and $\{D^J_n\in J_n[\omega]\colon J\in P\}$ arbitrarily. 
Then conditions (i) and (ii) are trivially satisfied.
Next, assume that the sequence $\{s_i\colon i\leq n\}$ is already constructed.
We will construct $s_{n+1}$ and sets $D^J_{n+1}$ as follows. 

Let
\[
Q=\{q\in[\omega]^{<\omega}\colon q\setminus{\{s_0,\dots,s_{n}\}}=F_{i}\ \mbox{ for some }\ i\leq n\}.
\]
Then $|Q|\leq {(n+1)\;2^{n+1}}$ and $\vert{Q\cup{\{f_0,\dots,f_{n}\}}}\vert\leq{k_{n+1}}$.

Fix $J\in{P}$. 
Since $J_{n+1}[\omega]$ covers $[F]^{\leq{k_{n+1}}}$, there exists a $\bar D_{n+1}^J\in J_{n+1}[\omega]$ containing 
$Q\cup{\{f_{0},\dots,f_{n}\}}$. Since $\bar D_{n+1}^J$ is open and covers finite set $Q$, there is 
an  $s_{n+1}^{J}>s_{n}$ in $X$ such that for every $q\in Q$
\[
\{x\subset{\omega}\colon x\cap{s_{n+1}^{J}}=q\cap{s_{n+1}^{J}}\}\subset\bar D_{n+1}^J.
\]
Notice that
\begin{description}
\item{($*$)} for every $A\subset\omega$ and $\bar s_{n+1}\geq s_{n+1}^{J}$ condition (ii) holds. 
\end{description}

Indeed, assume that $(A\cap{\bar s_{n+1}})\setminus{\{s_0,\ldots,s_n\}}=F_i$ for some $i<j\leq n+1$.
If $j\leq n$ then $n\geq 1$ and since $F_i\subset i\subset s_{n-1}$ we have
\[
(A\cap{s_{n}})\setminus{\{s_0,\dots,s_{n-1}\}}=(A\cap \bar s_{n+1})\setminus{\{s_0,\dots,s_{n}\}}=F_{i}.
\]
So, by the inductive assumption, $A\in D_{j}^{J}$. 
If $j=n+1$ then $q=A\cap \bar s_{n+1}\in Q$. 
So $A\in{\{x\subset{\omega}\colon x\cap{\bar s_{n+1}}=q\cap{\bar s_{n+1}}\}}\subset 
{\{x\subset{\omega}\colon x\cap{s_{n+1}^{J}}=q\cap{s_{n+1}^{J}}\}}\subset \bar D_{n+1}^J$,
finishing the proof of ($*$). 

For each $J\in P$ let $m^J<\omega$ be such that $J_{n+1}(m^J)=\bar D_{n+1}^J$
and define $U_J=\{K\in\X\colon K_{n+1}(m^J)=\bar D_{n+1}^J\}$. 
Then $U_J$ is an open neighborhood of $J$.
In particular, $\{U_J\colon J\in P\}$ is an open cover of a compact set $P$,
so there exists a finite $P_0\subset P$ such that
$P\subset\bigcup\{U_{\bar J}\colon {\bar J}\in P_0\}$. 
Choose $s_{n+1}\in X$ such that $s_{n+1}\geq\max\{s_{n+1}^{\bar J}\colon {\bar J}\in P_0\}$. 
Moreover, for every $J\in P$ choose ${\bar J}\in P_0$ such that $J\in U_{\bar J}$
and define $D_{n+1}^J=\bar D_{n+1}^{\bar J}$. 
It is easy to see that, by ($*$), conditions (i) and (ii) are preserved. 
This completes the inductive construction. 


Put $Y=\{s_n\colon n<\omega\}$. To see that it satisfies the 
lemma pick an arbitrary $J\in{P}$. We will show that 
$F\cup{Y^{*}}\subset{\bigcup_{n<\omega}\bigcap_{m>n}D_{m}^{J}}$.

Clearly $F\subset{\bigcup_{n<\omega}\bigcap_{m>n}D_{m}^{J}}$ since, by (i),
$f_n\in D_{m}^{J}$ for every $m>n$. So, fix an $A\in{Y^{*}}$. 
Then $A\setminus{Y}=F_{i}$ for some $i<\omega$. 
Let $n<\omega$ be such that $i<n$ and $s_n>\max F_i$. 
Then for every $m>n$ we have $i<m\leq m+1$ and 
$(A\cap{s_{m+1}})\setminus{\{s_0,\ldots,s_m\}}=F_i$.  
So, by (ii), we have $A\in D_m^{J}$ for every $m>n$.
Thus, $A\in\bigcap_{m>n}D_{m}^J$. \qed
 

\lem{lemA2}{If $F\subset\P(\omega)$ is countable then the set
\[
\X_{F}=\{J\in\X\colon J_{n}[\omega]\mbox{ covers }\ [F]^{k_{n}}\   \mbox{ for every }\ 
n<\omega\}
\] 
is Borel in $\X$.}

\proof This follows from the fact that 
\[
\X_{F}=\bigcap_{n<\omega}\bigcap_{A\in{[F]^{k_{n}}}}
\bigcup_{m<\omega}\bigcup_{A\subset B\in\B}\{J\in\X\colon J_{n}(m)=B\}
\]
since each set $\{J\in\X\colon J_{n}(m)=B\}$ is clopen in $\X$. Thus, $\X_{F}$
is a $G_\delta$-set. \qed




\thm{thmA3}{\psmCGame\ implies that there exists 
an uncountable strong $\gamma$-set in $\P(\omega)$.}

\proof 
For $\alpha<\omega_1$ and an $\subseteq^*$-decreasing
sequence $\V=\{V_\xi\in[\omega]^\omega\colon\xi<\alpha\}$
let $W(\V)\in[\omega]^\omega$ be such that
$W(\V)\subsetneq^* V_\xi$ for all $\xi<\alpha$.
Moreover, if $P\in\perf^*(\X)$ is a cube then we
define a subcube $Q=Q(\V,P)$ of $P$
and an infinite subset $Y=V(\V,P)$ of $X=W(\V)$ as follows.
Let $F=\V\cup[\omega]^{<\omega}$ and
choose a subcube $Q$ of $P$ such that
either $Q\cap \X_F=\emptyset$ or $Q\subset\X_F$.
This can be done by Claim~\ref{claim1} since
$\X_F$ is Borel.
If $Q\cap\X_F=\emptyset$ we put $Y=X$.
Otherwise we apply Lemma~\ref{lemA1} to find $Y$.

Consider the following strategy $S$ for Player~II:
\[
S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)=
Q(\{V_\eta\colon\eta<\xi\},P_\xi),
\]
where sets $V_\eta$ are defined inductively by
$V_\eta=V(\{V_\zeta\colon\zeta<\eta\},P_\eta)$.
By \psmCGame\ strategy $S$ is not a winning strategy for Player~II.
So there exists a game
$\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
played according to $S$
in which and Player~II loses, that is,
$\X=\bigcup_{\xi<\omega_1}Q_\xi$.
Let $\V=\{V_\xi\colon\xi<\omega_1\}$ be a sequence
associated with this game, which is
strictly $\subseteq^*$-decreasing, and let $T=\V\cup[\omega]^{<\omega}$.
We claim that $T$ is a strong $\gamma$-set. 

Indeed, let 
$\la\U_n\subset\B\colon n<\omega\ra$ be such that 
$\U_n$ covers $[T]^{k_n}$ for every $n<\omega$. 
Then there is a $J\in\X$ such that $J_n[\omega]=\U_n$ for every $n<\omega$. 
Let $\alpha<\omega_1$ be such that $J\in Q_\alpha$. 
Then $J\in\X_{\{V_{\eta}\colon\eta<\alpha\}\cup{[\omega]^{<\omega}}}$,
so we must have used Lemma~\ref{lemA1} to get $Q_\alpha$.
In particular, there is a sequence 
$\la D_{n}^{J}\in J_n[\omega]=\U_n\colon n<\omega\ra$
such that 
$\left([\omega]^{<\omega}\cup\{V_{\eta}\colon\eta<\alpha\}\right)\cup(V_\alpha)^*
\subset{\bigcup_{n<\omega}\bigcap_{m>n}D_{m}^{J}}$.
So, $T\subset{\bigcup_{n<\omega}\bigcap_{m>n}D_{m}^{J}}$, as
$\{V_{\eta}\colon\alpha\leq \eta<\omega_1\}\subset(V_\alpha)^*$. 
\qed

Since every homeomorphic image of a strong $\gamma$-set is 
evidently a strong $\gamma$-set,
we obtain immediately the following conclusion. 

\cor{corA4}{\psmCGame\ implies that there exists 
an uncountable $\gamma$-set in $\real$ which is strongly meager.}

It is worth to mention that a construction 
of an uncountable strong $\gamma$-set in $\P(\omega)$
under \psmCGame\ can be also done in a formalism similar
to that used in Section~\ref{secGamma}. 
In order to do it, we need the following definitions and facts.
For a fixed sequence $\bar k=\la k_n<\omega\colon n<\omega\ra$ 
we say that $\A\subset(\P(\omega))^\omega$ is
{\em $\bar k$-centered}\/ provided for every $n<\omega$ any
$k_n$-many sets from $\{A(n)\colon A\in\A\}$ have a common point;
$B\in\omega^\omega$ is a {\em quasi-intersection}\/ of
$\A\subset(\P(\omega))^\omega$ provided 
for every $A\in\A$ there is infinitely many $n<\omega$ with $B(n)\in A(n)$.
Now, if $\K^*$ is a family of all continuous functions from
$\P(\omega)$ to $(\P(\omega))^\omega$ then the following is true:
\begin{quote}
A set $X\subset\P(\omega)$ is a strong $\gamma$-set if and only if
there exists an increasing sequence $\bar k=\la k_n<\omega\colon n<\omega\ra$
such that for every $f\in\K^*$ 
if $f[X]$ is $\bar k$-centered then $f[X]$ has a quasi-intersection.
\end{quote}
With this characterization in hand we can construct 
an uncountable strong $\gamma$-set in $\P(\omega)$
by applying \psmCGame\ to the space $\K^*$. 





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\end{thebibliography}

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