\documentclass{amsart}

\newcommand{\UpdateDate}{January 8, 2004}
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\title[On additive almost continuous functions under \psmPrGame]
{On additive almost continuous functions under \psmPrGame}



\author[K. Ciesielski]{Krzysztof Ciesielski}
\address{Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA;
e-mail: K\_Cies@math.wvu.edu; web page: {\tt http://www.math.wvu.edu/\~{}kcies}}
\author[J. Pawlikowski\ \ \ \ \ \ \UpdateDate]{Janusz Pawlikowski}
\address{Department of Mathematics, University of  Wroc\l aw, pl. Grunwaldzki 2/4, 50-384 Wroc\l aw, Poland;
e-mail: pawlikow@math.uni.wroc.pl}


\thanks{The work of the first author was partially supported by 
2002/03 West Virginia University Senate Research Grant.
The second
author wishes to thank West Virginia University for its hospitality
in years 1998-2001, where the results presented here were obtained. 
}

\keywords{Additive, almost continuous, Hamel basis, Covering Property Axiom, CPA}

\subjclass{Primary  26A15, 26A30; Secondary 03E35.}


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% characteristic function
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\begin{document}
 
\maketitle



\begin{abstract}
We prove that the Covering Property Axiom \psmPrGame,
which holds in the iterated perfect set model, implies 
that there exists an additive discontinuous
almost continuous function $f\colon\real\to\real$ whose graph
is of measure zero. We also show that, under \psmPrGame,
there exists a Hamel basis $H$ for which, $E^+(H)$, the set 
of all linear combinations of elements from 
$H$ with positive rational coefficients, 
is of measure zero. 
The existence of both of these examples follows from 
Martin's axiom, while it is unknown whether 
either of them can be constructed in ZFC. 

As a tool for the constructions we will show that
\psmPrGame\ implies its seemingly stronger version,
in which $\omega_1$-many games are played simultaneously. 
\end{abstract}


\section{Preliminaries and axiom \psmPrGame}


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 $X$ symbol $\perf(X)$
will stand for a 
collection of all subsets of $X$ homeomorphic to the Cantor set $\Cantor$.
For a fixed $0<\alpha<\omega_1$ and $0<\beta\leq\alpha$
a symbol $\pi_\beta$ 
will stand for the projection  
from $\Cantor^{\alpha}$ onto~$\Cantor^\beta$.
In what follows we will consider $\real$ as a linear space over
$\rational$. 
For $Z\subset\real$ its linear span
with respect to this structure will be denoted by 
$\lin(Z)$. 
A subset $H$ of $\real$ is a {\em Hamel basis}\/
provided it is a linear basis of $\real$ over $\rational$,
that is, it is linearly independent and $\lin(H)=\real$.  



Axiom \psmPrGame\ was introduced by the authors in~\cite{CP83},
where it is shown that it holds in the iterated perfect set model. 
Also, \psmPrGame\ is a simpler version of the axiom
\psm\ which is described in a monograph~\cite{CPAbook}. 
For the reader's convenience, we will 
restate \psmPrGame\ in the next few paragraphs. 

For $0<\alpha<\omega_1$ let  
$\Phi_{\rm prism}(\alpha)$
be the family of all continuous injections
$f\colon\Cantor^\alpha\to\Cantor^\alpha$ 
with the property that
\[
f(x)\restriction\beta=f(y)\restriction\beta\ \Equi\ 
x\restriction\beta=y\restriction\beta
\ \ \ \ \ \mbox{ for all $\beta\in \alpha$ and $x,y\in \Cantor^\alpha$}.
\]
Functions $\Phi_{\rm prism}(\alpha)$ are called
{\em projection-keeping homeomorphisms}. (Compare~\cite{Ka}.) 
Let $\mathPerf_\alpha=\{\range(f)\colon f\in \Phi_{\rm prism}(\alpha)\}$
and $\mathPerf_{\omega_1}=\bigcup_{0<\alpha<\omega_1}\mathPerf_\alpha$.
We will refer to elements of 
$\mathPerf_{\omega_1}$ as 
{\em iterated perfect sets}. (In~\cite{Zap} the elements of $\mathPerf_\alpha$ are called 
$I$-perfect, where $I$ is the ideal of countable sets.)
The simplest elements of 
$\mathPerf_\alpha$ are {\em perfect cubes}, that is, 
the sets of the form 
$C=\prod_{\beta<\alpha} C_\beta$, where $C_\beta\in\perf(\Cantor)$ 
for each $\beta<\alpha$. 

\claim{claim1}{Let $0<\alpha<\omega_1$. 
If $G$ is a Borel second category subset of $\Cantor^\alpha$
then $G$ contains a perfect cube. 
In particular, if $\G$ is a Borel countable cover of $\Cantor^\alpha$
then there is a $G\in\G$ which contains an $E\in\mathPerf_\alpha$. 
}

An argument for the claim 
can be found in~\cite[claim~3.2]{CP88} or~\cite[claim~1.1.5]{CPAbook}.


The only properties of the iterated perfect sets that we will use 
in this paper are listed in the next three lemmas. 



\lem{lem:dual1simple}{
For every $E\in\mathPerf_{\omega_1}$,
a Polish space $X$, 
and a continuous function $f\colon E\to X$ 
there exists a $P\in\mathPerf_{\omega_1}$ such that $P\subset E$
and $f[P]$ is either a singleton or it is homeomorphic to the Cantor set.}

Lemma~\ref{lem:dual1simple} follows immediately 
from~\cite[lemma~3.2.5]{CPAbook}
(see also~\cite[lemma~1.1]{CP96} or \cite[lemma~2.4]{CP84})
which is a particular case of~\cite[thm.~20]{Ka}.

The next two lemmas will allow us to construct the elements of 
$\mathPerf_{\omega_1}$ by fusion arguments. 
They can be found, respectively, in~\cite[lemmas~4.3 and~4.4]{CP83}
or in~\cite[lemmas~3.1.1 and~3.1.2]{CPAbook}.
Here, for a fixed $0<\alpha<\omega_1$ and $k<\omega$
we define $A_k=\{\la \beta_i,n_i\ra\colon i< k\}$, where 
$\{\la \beta_k,n_k\ra\colon k<\omega\}$ is a fixed enumeration of
$\alpha\times\omega$.


\lem{FusionSequenceLemma}{
Let $0<\alpha<\omega_1$ and for $k<\omega$ let
$\E_k=\left\{E_s\in\mathPerf_\alpha\colon s\in 2^{A_k}\right\}$.  
Assume that for every $k<\omega$, $s,t\in 2^{A_k}$, 
and $\beta<\alpha$ we have: 
\begin{itemize}
\item[\rm (i)] the diameter of $E_s$ is less than  or equal to $2^{-k}$,
\item[\rm (ii)] if $i<k$ then $E_s\subset E_{s\restriction i}$, 
\item[\rm (ag)] (agreement) if
  $s\restriction(\beta\times\omega)=t\restriction(\beta\times\omega)$
    then $\pi_{\beta}[E_s]=\pi_{\beta}[E_t]$, 
\item[\rm (sp)] (split) if
  $s\restriction(\beta\times\omega)\neq t\restriction(\beta\times\omega)$
    then $\pi_{\beta}[E_s]\cap\pi_{\beta}[E_t]=\emptyset$.
\end{itemize}
Then $Q=\bigcap_{k<\omega}\bigcup\E_k$ 
belongs to $\mathPerf_\alpha$.
}

For a family $\E\subset\mathPerf_\alpha$ 
we say that an $\E_0\subset\mathPerf_\alpha$ is a {\it refinement}\/
of $\E$ provided $\E_0=\{P_E\colon E\in\E\}$, where $P_E\subset E$ 
for every $E\in\E$. 
A family $\D\subset[\mathPerf_\alpha]^{<\omega}$ 
is {\it closed under refinements}\/
if for each $\E\in\D$ every refinement of $\E$ also belongs to $\D$. 

\lem{ELemmaNew}{
Let $0<\alpha<\omega_1$ and $k<\omega$. If
$\E_k=\left\{E_s\in\mathPerf_\alpha\colon s\in 2^{A_k}\right\}$
satisfies (ag) and (sp) then
\begin{itemize}
\item[\rm (A)] there exists an 
$\E_{k+1}=\left\{E_s\in\mathPerf_\alpha\colon s\in 2^{A_{k+1}}\right\}$
such that (i), (ii), (ag), and (sp)
hold for all $s,t\in 2^{A_{k+1}}$ and $r\in 2^{A_k}$.
\end{itemize}
Moreover, if 
$\D\subset[\mathPerf_\alpha]^{<\omega}$ is a family 
of pairwise disjoint sets such that
$\emptyset\in\D$,  $\D$ is 
closed under refinements, and
\begin{itemize}
\item[\rm ($\dagger$)] for every 
$\E\in\D$ and $E\in\mathPerf_\alpha$ which is disjoint with
$\bigcup\E$ there exists an $E'\in\mathPerf_\alpha\cap\P(E)$ such that
$\{E'\}\cup\E\in\D$ 
\end{itemize}
then 
\begin{itemize}
\item[\rm (B)] there exists a refinement $\E'_k\in\D$ of $\E_k$
satisfying (ag) and (sp),
\item[\rm (C)] there exists 
     an $\E_{k+1}$ as in (A) such that $\E_{k+1}\in\D$.
\end{itemize}
}


To state \psmPrGame\ we need a few more definitions. 
For a fixed Polish space $X$ 
let $\Fpr$ stand for the family of all continuous 
injections from an $E\in\mathPerf_{\omega_1}$ onto perfect subsets of $X$. 
Each such injection $f$ is called a {\em prism}\ and is
considered as a coordinate system imposed on~$P=\range(f)$.%
\footnote{In a language of forcing
a coordinate function $f$ is simply a nice
name for an element from~$X$.}
We will usually abuse this terminology and 
refer to $P$ itself as a {\em prism}\ (in $X$)
and to $f$ as a {\em witness function}\ for $P$. 
A function $g\in\Fpr$ is {\em subprism}\ of $f$ provided 
$g\subset f$. 
In the above spirit we call 
$Q=\range(g)$ a {\em subprism of a prism}\/ $P$.
Thus, when we say that {\em $Q$~a subprism of a prism $P\in\perf(X)$}\/ 
we mean that $Q=f[E]$, where $f$ is a witness function for $P$ and 
$E\subset\dom(f)$ is an iterated perfect set. 
Using the fact that $\Phi_{\rm prism}(\alpha)$
is closed under the composition it is easy to see that we can always assume that 
a witness function of a prism is always defined on the 
entire space $\Cantor^\alpha$ for an appropriate $\alpha$. 




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 prisms}, that is,
with the constant coordinate 
function from $\Cantor^\alpha$
onto the singleton. In particular, a subprism of a constant prism is 
the same singleton. 


Consider the following game
${\rm GAME}_{\rm prism}(X)$
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 prism $P_\xi\in\perf^*(X)$
and Player~II must respond with a subprism $Q_\xi$ 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.

A strategy for Player~II
is any function $S$
such that
$S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)$
is a subprism 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
$f_\eta$ making each $P_\eta$ a prism.)
A game $\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
is played according to a strategy $S$ for Player~II when
$Q_\xi=S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)$
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.

\medskip

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

\medskip

In what follows we will use the following 
prism density fact, which proof can be found 
in~\cite[lemma~2.1]{CP92} or in \cite[lemma~5.1.5]{CPAbook}.

\lem{LemRelLinInd}{Let $M\subset\real$ 
be a sigma-compact and linearly independent.
Then for every prism $P$ in $\real$ there exist a subprism $Q$ of $P$
and a compact subset $R$ of $P\setminus M$ such that 
$M\cup R$ is a maximal  linearly independent subset of $M\cup Q$. 
}

We will also use the following fact.

\fact{FactCofM}{\psmPrGame\ implies that $\cf(\M)=\omega_1$,
where $\cf(\M)$ is the cofinality of the ideal of meager sets.}

\proof It is proved in \cite[cor.~4.3]{CP88} 
(see also \cite[cor.~1.1.3]{CPAbook}) that \psmC\ implies
that $\cf(\NN)$, the cofinality of the ideal of measure zero sets,
is equal to $\omega_1$, while it is well known 
that $\cf(\NN)=\omega_1$  implies that $\cf(\M)=\omega_1$.
To finish the argument, it is enough to recall that 
\psmPrGame\ implies \psmC. (See e.g. \cite{CP83} or \cite{CPAbook}.)
\qed



\section{Multi-games} 

For a non-empty collection $\X$ of pairwise disjoint Polish spaces 
consider the following ``simultaneous'' 
two-player game ${\rm GAME}_{\rm prism}(\X)$
of length $\omega_1$. 
At each stage $\xi<\omega_1$ of the game Player~I can play 
a prism $P_\xi\in\perf^*(X)$ from an arbitrarily chosen $X\in\X$.
Player~II responds with a subprism $Q_\xi$ of $P_\xi$. 
The game $\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
is won by Player~I provided
\[
\bigcup_{\xi<\omega_1}Q_\xi=\bigcup\X;
\]
otherwise the game is won by Player~II. 
Thus, for any Polish space $X$
the games ${\rm GAME}_{\rm prism}(X)$
and ${\rm GAME}_{\rm prism}(\{X\})$ are identical. 





\thm{th:multGames}{
Let $\X$ of size $\leq\omega_1$ be 
a non-empty collection of pairwise disjoint Polish spaces.
Then \psmPrGame\ is equivalent to 
\begin{description}
\item[{\bf \psmPrGame$(\X)$}] 
Player~II has no winning strategy
in ${\rm GAME}_{\rm prism}(\X)$.
\end{description}
}

\proof We will leave the implication ``\psmPrGame$(\X)$ implies \psmPrGame''
without a proof, since it will not be used in the sequel. 
Its proof can be found in \cite{CPAbook}.

To see the converse implication assume that \psmPrGame\ holds and let 
$I=[0,1]$. Let $L=\{x_\xi\colon \xi<\omega_1\}$ 
be a Luzin set in $I$, that is, such that 
$|L\cap N|\leq\omega$ for every nowhere dense subset $N$ of $I$.
The existence of such a set under \psmPrGame\ 
follows from Fact~\ref{FactCofM}.

Let $\kappa=|\X|\leq\omega_1$ and let 
$\{X_\eta\colon \eta<\kappa\}$ be an enumeration of $\X$. 
We will identify each $X_\eta$, $\eta<\kappa$, with a $G_\delta$ subset of
$\{x_\eta\}\times I^\omega$ homeomorphic to~it. 

Now, let $S_0$ be a Player~II strategy
in the game ${\rm GAME}_{\rm prism}(\X)$.
We will modify it to a Player~II strategy $S$ 
in the game ${\rm GAME}_{\rm prism}(I\times I^\omega)$
in the following way. 
First, for every prism $P$ in $I\times I^\omega$ 
let $R(P)$ be its subprism such that 
\begin{center}
either $R(P)\subset X_\eta$ for some $\eta<\kappa$ or 
$R(P)\cap\bigcup\X=\emptyset$. 
\end{center}
To choose such $R(P)$ first choose subprism $R$ of $P$ such that
its first coordinate projection $\pi[R]$ is nowhere dense in $I$.
(This can be done, for example, applying Lemma~\ref{lem:dual1simple}.)
So, $\pi[R]$
contains at most countably many points $x_\eta$.
Thus, by Claim~\ref{claim1}, 
there is a subprism $R_1$ of $R$ 
such that either $\pi[R_1]$ 
is disjoint with $L$ or there is an $\eta<\kappa$
such that $\pi[R_1]=\{x_\eta\}$.
In the first case we put $R(P)=R_1$. 
In the second case we use Claim~\ref{claim1}
to find a subprism $R(P)$ of $R_1$ such that either $R(P)\subset X_\eta$ 
or $R(P)\cap X_\eta=\emptyset$.

Now strategy $S$ is defined by induction 
on $\xi$, the step level of the game. 
Thus, if a sequences 
$\bar T=\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra$
represents a ``legal'' sequence (a sequence that 
could have been produced by $S$ defined so far) 
we define $S(\bar T,P_\xi)$ as follows. 
If $R(P_\xi)\cap \bigcup\X=\emptyset$
we just put $S(\bar T,P_\xi)=R(P_\xi)$. 
For the other case, define 
$J=\{\eta<\xi\colon R(\P_\eta)\subset X\mbox{ for some } X\in\X\}$
and let
\[
S(\bar T,P_\xi)=
S_0(\la\la R(P_\eta),Q_\eta\ra\colon \eta\in J\ra,R(P_\xi)),
\]
where $\la\la R(P_\eta),Q_\eta\ra\colon \eta\in J\ra$ is identified with
$\la\la R(P_{i(\eta)}),Q_{i(\eta)}\ra\colon \eta<\alpha\ra$,
while $i$ is an order isomorphism between an ordinal $\alpha$ and $J$. 

Since, by \psmPrGame, 
$S$ is not winning in ${\rm GAME}_{\rm prism}(I\times I^\omega)$
for Player~II there is a game
$\la\la P_\xi,Q_\xi\ra\colon\xi<\omega_1\ra$
played according to $S$ in which Player~I wins.
To finish the proof put 
$K=\{\xi<\omega_1\colon R(\P_\xi)\subset\bigcup\X\}$
and notice that 
$\la\la R(P_\xi),Q_\xi\ra\colon\xi\in K\ra$
is a game in ${\rm GAME}_{\rm prism}(\X)$ 
played according to $S_0$ in which Player~I wins.
Thus, $S_0$ is not winning for Player~II.
\qed


\section{Additive almost continuous discontinuous function with measure zero graph}

The construction presented here can be viewed as a ``model
example'' of how some CH
proofs can be 
modified to the proofs from \psmPrGame. 



Recall that a function $f\colon\real\to\real$ is
{\em almost continuous}\/ 
provided any open subset $U$ of $\real^2$ which contains
the graph of $f$ contains also a graph of a continuous function
from $\real$ to $\real$. It is known that 
if $f$ is almost continuous then its 
graph is connected in
$\real^2$ (i.e., $f$ is a connectivity function)
and that $f$ has the intermediate value property 
(i.e., $f$ is Darboux).
(See e.g. \cite{natkaniec} or \cite{cie-jas}.)
Recall also that a function 
$f\colon\real\to\real$ is
{\em additive}\/ provided $f(x+y)=f(x)+f(y)$ for every $x,y\in\real$.
It is well known that every function defined on a Hamel basis
can be uniquely extended to an additive function.
(See e.g. \cite[thm.~7.3.2]{CiBook}.)


Our next goal will be to construct an additive 
discontinuous
almost continuous function $f\colon\real\to\real$ whose graph
is of measure zero. 
In fact, we will show that, under \psmPrGame,
such an $f$ can be found inside a set 
$(\real\times G)\cup(G\times\real)$
for every $G_\delta$ subset $G$ of $\real$ with $0\in G$. 
A first construction of such a function, 
under Martin's axiom,
was given by
K.~Ciesielski in~\cite{KC59}. 
Although it can be shown that such a function
(i.e., with graph being a subset of $(\real\times G)\cup(G\times\real)$)
does not exist in the Cohen model
it is unknown whether the existence of 
an additive discontinuous 
almost continuous function with graph
of measure zero can be proved in ZFC alone. 

Now we are ready to state the theorem. 

\thm{thm:smallGraph0}{
\psmPrGame\ implies that 
for every dense $G_\delta$ set $G\subset\real$ 
such that $0\in G$ 
there exists an additive discontinuous almost continuous
function $f\colon\real\to\real$ whose graph is
a subset of $(\real\times G)\cup(G\times\real)=(G^c\times G^c)^c$. 
}

Using Theorem~\ref{thm:smallGraph0} with $G$ of measure zero
we obtain immediately the following corollary. 

\cor{cor:smallGraph0}{
\psmPrGame\ implies that 
there exists a discontinuous, almost continuous,
additive function $f\colon\real\to\real$
whose graph is of measure zero.
}

Notice that if $L_m$, for $0<m<\omega$,
is the collection of all functions 
$\ell\colon\real^m\to\real$ given by a formula
\begin{equation}\label{eq:linFunc}
\mbox{$\ell(x_0,\ldots,x_{m-1})=\sum_{i<m} q_i x_i$,
where $q_i\in\rational\setminus\{0\}$ for all $i<m$} 
\end{equation}
then
\[
\lin(Z)=\bigcup_{0<m<\omega} \ \bigcup_{\ell\in L_m} \ell[Z^m].
\]
Also $Z\subset\real$ is linearly independent 
(over $\rational$) provided 
$\ell(x_0,\ldots,x_{m-1})\neq 0$
for every $\ell\in L_m$, $0<m<\omega$, and 
$\{x_0,\ldots,x_{m-1}\}\in[Z]^m$. 

Recall also that a function $f\colon \real\to\real$ 
is almost continuous if and only if it intersects every 
{\em blocking set\/}, that is,
a closed set $K \sq \real^2$ which meets every 
continuous function from
$\C(\real)$ and is disjoint with at least one function 
from $\real^{\real}$. The domain of every blocking set 
contains a non-degenerate connected set. (See \cite{kg}
or \cite{natkaniec}.) 
It is important for us that 
every blocking set contains a graph of a continuous function 
$g\colon G\to\real$, where $G$ is a dense $G_\delta$ subset
of some non-trivial interval. 
(See~\cite{kellum}. This follows from the fact that
for every closed bounded set $B$ with domain $I$, the mapping 
$I\ni x\mapsto \inf\{y\colon \la x,y\ra\in B\}$
is of first Baire class,
so it is continuous when restricted to a dense $G_\delta$ subset.) 
Thus, in order to make sure that a function
is almost continuous it is enough
to insure that its graph intersects every function
from the family 
\begin{equation}\label{def:K}
\K=\bigcup\{\C(G)\colon 
\mbox{ $G$ is a $G_\delta$ second category subset of }\real\}.
\end{equation}




In what follows we will use the following notation 
for $G,P\subset\real$:
\begin{equation}\label{eqGL}
G[P]=\{x\in\real\colon x-P\subset G\}=\bigcap_{p\in P}(p+G).
\end{equation}
It is also convenient to note that
\[
G[P]^c=G^c+P.
\]
We start with noticing some simple properties of 
this operation.  

\fact{fact:BasicGL}{
Let $P,S,G,G'\subset\real$.
\begin{itemize}
\item[\rm (a)] If $P\subset S$ and $G'\subset G$ then $G[P]\supset G'[S]$.
\item[\rm (b)] If $P$ is compact and $G$ is open
               then $G[P]$ is open. 
\item[\rm (c)] If $P=\bigcup_{i<\omega} P_i$ and 
               $G=\bigcap_{n<\omega} G_n$ then
               $G[P]=\bigcap_{i,n<\omega} G_n[P_i]$.
\item[\rm (d)] If $P$ is sigma compact and $G$ is a $G_\delta$ set 
               then $G[P]$ is also a $G_\delta$ set.
\item[\rm (e)] If $G[P_n]$ is a dense $G_\delta$ set for every $n<\omega$
               then so is $G[\bigcup_{n<\omega}P_n]$.
\item[\rm (f)] $G[P][S]=G[P+S]$.
\end{itemize}
}

\proof (a) follows immediately from the second part of (\ref{eqGL})
while (b) from its first part. 
To see (c) notice that, by (\ref{eqGL}),
\[
G[P]=\bigcap_{i<\omega}\{x\in\real\colon x-P_i\subset G\}=\!
\bigcap_{i,n<\omega}\{x\in\real\colon x-P_i\subset G_n\}=\!
\bigcap_{i,n<\omega} G_n[P_i]. 
\]
So, (d) follows immediately from (b),
while (e) is an easy consequence of (c).  
Note also that
\[
G[P][S]^c=G[P]^c+S=G^c+P+S=G[P+S]^c
\]
so (f) holds. \qed






Recall that for a Polish space $X$ the space $\C(X)$
of continuous functions from $X$ into $\real$ is considered 
with the metric of uniform convergence. 

\lem{lem:smallGraph0}{Let $X$ be a Polish space and 
$\bar x\in\bar K\in\perf(X)$. 
For every dense $G_\delta$-set 
$G\subset\real$ 
and a prism $P$ in $\C(X)$ there exist a subprism $Q$ of $P$ and 
a $K\in\perf(\bar K)$  with $\bar x\in K$ 
such that $G[\lin(R_K(Q))]$ 
is a dense $G_\delta$ subset of $\real$, where 
$R_K(Q)=\{h(x)\colon h\in Q\ \&\ x\in K\}$.
}



\proof Let $\U$ be a countable family of open subsets of $\real$ 
with the property that 
$G=\bigcap\U$ and fix a countable basis
$\B$ for $\real$. For $0<m<\omega$ let $L_m$ 
be the set of all functions $\ell$ 
defined as in (\ref{eq:linFunc})
and put $L=\bigcup_{0<m<\omega}L_m$. 
In what follows for $\ell\colon X^m\to\real$ from $L$ 
and $Z\subset X$  
we will write $\ell[Z]$ in place of $\ell[Z^m]$. 

Fix an enumeration $\{\la U_k,\ell_k,B_k\ra\colon k<\omega\}$
of $\U\times L\times\B$ and let  
$h\in\Fpr(\C(X))$ be such that
$P=h[\Cantor^\alpha]$.
By induction on $k<\omega$ we will construct
the sequences
$\la\E_k\colon k<\omega\ra$
and $\la\K_k\colon k<\omega\ra$ such that for every $k<\omega$
\begin{itemize}
\item[(a)] $\K_k$ is a family 
     $\{K_t\in\perf(\bar K)\colon t\in 2^k\}$
     of pairwise disjoint sets such that $\bar x\in\bigcup\K_k$,
\item[(b)] $K_s\subset K_t$ for each $t\in 2^k$
           and $t\subset s\in 2^{k+1}$,
\item[(c)] $\E_k=\left\{E_s\in\mathPerf_\alpha\colon s\in 2^{A_k}\right\}$,
\item[(d)] $\E_k$ and $\E_{k+1}$ satisfy conditions 
         (i), (ii), (ag), and (sp) from Lemma~\ref{FusionSequenceLemma}
         for every $s,t\in 2^{A_{k+1}}$ and $r\in 2^{A_k}$,
\item[(e)] if 
 $R_k=\left\{h(g)(x)\colon g\in\bigcup\E_k\ \&\ x\in\bigcup\K_k\right\}$
    then $U_k(\ell_k[R_k])\cap B_k\neq\emptyset$.
\end{itemize}

Before we construct such sequences, note how this will complete the proof.
Clearly, by (a) and (b), sequence $\la\E_k\colon k<\omega\ra$
satisfies the assumptions of Lemma~\ref{FusionSequenceLemma}.
Thus, $E=\bigcap_{k<\omega}\bigcup\E_k$ belongs to $\mathPerf_\alpha$,
so $Q=h[E]$ is a subprism of $P$.
Also, if $K=\bigcap_{k<\omega}\bigcup\K_k$
then $\bar x\in K\in\perf(\bar K)$. 
To see that $G[\lin(R_K(Q))]$ is a dense $G_\delta$
notice that $R_K\subset R_k$ for all $k<\omega$.
So, by (e), we have $U_k(\ell_k[R_K])\cap B_k\neq\emptyset$.
In particular, $U(\ell[R_K])$ is dense and open for every 
$U\in\U$ and $\ell\in L$. 
Thus, for every $U\in\U$ the set
\[
\bigcap_{\ell\in L}U[\ell[R_K(Q)]]=
U\left[\bigcup_{\ell\in L}\ell[R_K(Q)]\right]=
U\left[\lin(R_K(Q))\right]
\]
is a dense $G_\delta$-set, and so is 
$G[\lin(R_K(Q))]=\bigcap_{U\in\U}U\left[\lin(R_K(Q))\right]$, 
as desired.

To choose 
$\E_0=\{E_\emptyset\}$ and  $\K_0=\{K_\emptyset\}$
pick $g_0\in\Cantor^\alpha$, put $y=h(g_0)(\bar x)$,  
and let $\{z\}=\ell_0[\{y\}]$. 
Clearly
$U_0[\{z\}]=\{x\in\real\colon x-\{z\}\subset U_0\}$
is open and dense, so there is a $b_0\in B_0$
such that $b_0-\{z\}\subset U_0$. 
Let $\e>0$ be such that 
$b_0-(z-\e,z+\e)\subset U_0$.
Find a number $\delta>0$ such that 
$\ell_0[(y-2\delta,y+2\delta)]\subset (z-\e,z+\e)$
and a clopen subset $K_\emptyset$ of $\bar K$
containing $\bar x$ for which 
$h(g_0)[K_\emptyset]\subset(y-\delta,y+\delta)$.
Also, let $\delta_0>0$ be such that 
the diameter of $h[B_\alpha(g,\delta_0)]$ is less than $\delta$
and put $E_\emptyset=B_\alpha(g_0,\delta_0)$. 
We just need to check (e).
But for every $g\in E_\emptyset$ and $x\in K_\emptyset$
we have 
$|h(g)(x)-y|\leq |h(g)(x)-h(g_0)(x)|+|h(g_0)(x)-h(g_0)(\bar x)|<2\delta$.
So, $R_0\subset(y-2\delta,y+2\delta)$ and
$b_0-\ell_0[R_0]\subset b_0-(z-\e,z+\e)\subset U_0$.
Thus, $b_0\in U_0(\ell_0[R_0])\cap B_0$.

To make an inductive step assume that for some $k<\omega$
families $\E_k$ and $\K_k$ are already constructed. 
We will find appropriate $\E_{k+1}$ and $\K_{k+1}$. 
First use Lemma~\ref{ELemmaNew}(A) to 
pick an 
$\E_{k+1}'=\left\{E_s'\in\mathPerf_\alpha\colon s\in 2^{A_{k+1}}\right\}$
such that (d) holds. 
For any $s\in 2^{A_{k+1}}$ choose a $g_s\in E_s'$
such that the family $\left\{\{g_s\}\colon s\in 2^{A_{k+1}}\right\}$
satisfies condition (ag). 
Also, for every $r\in 2^{k+1}$ choose an $x_r\in K_{r\restriction k}$
such that all points in 
$\bar X=\left\{x_r\colon r\in 2^{k+1}\right\}$ are distinct
and $\bar x\in \bar X$. Put 
$Y=\bigcup\left\{h(g_s)[\bar X]\colon s\in 2^{A_{k+1}}\right\}$  
and $Z=\ell_{k+1}[Y]$. 
Clearly
$U_{k+1}[Z]=\{x\in\real\colon x-Z\subset U_{k+1}\}=
\bigcap_{z\in Z}(z+U_{k+1})$
is open and dense since $Z$ is finite.  
Thus there is a $b_{k+1}\in B_{k+1}$
such that $b_{k+1}-Z\subset U_{k+1}$. 
Let $\e>0$ be such that 
$b_{k+1}-B(Z,\e)\subset U_{k+1}$,
where $B(Z,\e)$ is the set of all $x\in\real$ 
with distance from $Z$ less than $\e$. 
Since $Y$ is finite, $\ell_{k+1}$ is continuous, and 
$Z=\ell_{k+1}[Y]$, we can find 
a $\delta>0$ such that 
$\ell_{k+1}[B(Y,2\delta)]\subset B(Z,\e)$.
Also, for every $r\in 2^{k+1}$ find 
a clopen subset $K_r$ of $K_{r\restriction k}$
containing $x_r$ such that 
$h(g_s)[K_r]\subset B(Y,\delta)$ for every $s\in 2^{A_{k+1}}$
and $\K_{k+1}=\{K_r\colon r\in 2^{k+1}\}$ is pairwise disjoint. 
This ensures (a) and (b). 
Let $\delta_0>0$ be such that 
for every $s\in 2^{A_{k+1}}$ 
the diameter of $h[B_\alpha(g_s,\delta_0)]$ is less than $\delta$
and put $E_s=B_\alpha(g_s,\delta_0)\cap E_s'$. 
It is easy to see that with 
$\E_{k+1}=\left\{E_s\colon s\in 2^{A_{k+1}}\right\}$
conditions (c) and (d) are satisfied. 
We just need to check (e).
To see it notice that $R_{k+1}\subset B(Y,2\delta)$ since for every 
$h(g)(x)\in R_{k+1}$ there are $s\in 2^{A_{k+1}}$ and $r\in 2^{k+1}$
such that 
\[
|h(g)(x)-h(g_s)(x_r)|\leq 
|h(g)(x)-h(g_s)(x)|+|h(g_s)(x)-h(g_s)(x_r)|<2\delta,
\]
while $h(g_s)(x_r)\in Y$. 
So, 
$b_{k+1}-\ell_{k+1}[R_{k+1}]\subset b_{k+1}-B(Z,\e)\subset U_{k+1}$.
Thus, $b_{k+1}\in U_{k+1}(\ell_{k+1}[R_{k+1}])\cap B_{k+1}$.
\qed

As a corollary, needed in the proof but also interesting
on its own, we conclude the following.

\lem{lem:smallGraph0xx}{For every dense $G_\delta$ subset $G$ of $\real$ 
and for every prism $P$ in $\real$ there exists
a subprism $Q$ of $P$ such that
$G[\lin(Q)]$ 
is a dense $G_\delta$ subset of~$\real$.
}



\proof Let $f\in\Phi_{\rm prism}(\alpha)$ be such that
$P=f[\Cantor^\alpha]$ and let $h\colon\real\to\C(\real)$
be given by $h(r)(x)=r+x$. Then $h[P]$ is a prism in 
$\C(\real)$ witnessed by $h\circ f$. 
By Lemma~\ref{lem:smallGraph0}
there exist a subprism $Q_0=h\circ f[E]$ of $h[P]$ and 
a $K\in\perf(\real)$ with $0\in K$ 
such that $Z=G[\lin(\{g(x)\colon g\in Q_0\ \&\ x\in K\})]$ 
is dense in $\real$.
But then $Q=f[E]=h^{-1}(Q)$ is a subprism of $P$ and, since $0\in K$, 
\begin{eqnarray*}
Z & = & G[\lin(\{h(r)(x)\colon r\in Q\ \&\ x\in K\})]\\
  & = & G[\lin(\{r+x\colon r\in Q\ \&\ x\in K\})]\\
  & \subset & G[\lin(\{r\colon r\in Q\})]\\
  & = & G[\lin(Q)].
\end{eqnarray*}
So, $G[\lin(Q)]$ is dense. It is $G_\delta$ 
by Fact~\ref{fact:BasicGL}(d) since $\lin(Q)$ is sigma compact. 
\qed 

We will also need the following fact about perfect sets.  

\lem{lem:smallGraph1xx}{Let $G$ be a 
proper dense $G_\delta$-subset of
$\real$, $W$ a second category $G_\delta$-subset of $\real$,
and let $M$ be an $F_\sigma$-subset of
$\real$ such that $G[\lin(M)]$ is a dense $G_\delta$-subset of $\real$. 
Then there exists a linearly independent 
set $K\in\perf(W)$ 
such that
$G[\lin(M\cup K)]$ is dense,
$\lin(M)\cap\lin(K)=\{0\}$, 
and $\lin(M\cup K)\setminus \lin(M)\subset G$. 

In particular, if $M$ is linearly independent then so is $M\cup K$. 
}



\proof First note that the density of $G[\lin(M)]$
implies $\lin(M)\neq\real$.
So, $\lin(M)$ must be of first category. 


Replacing $G$ with 
$\bigcap\{q\, G\colon {q\in\rational\setminus\{0\}}\}$, if
necessary, we can assume that 
$q\, G = G \mbox{ for every } q\in\rational\setminus\{0\}$.
Notice that then for every $q\in\rational\setminus\{0\}$
and linear subspace $V$ of $\real$ 
we also have
\[
q\, G[V] 
\!=\!\{q\, x\colon x - V\subset G\}
\!=\!\{y\colon (y/q) - V\subset G\}
\!=\!\{y\colon y - q\, V\subset q\, G\}
\!=\!G[V].
\]

Let $J$ be a non-empty open interval such that
$W$ is dense in $J$ and let $\la G_k\colon k<\omega\ra$ 
and $\la W_k\colon k<\omega\ra$  be the decreasing sequences
of open subsets of $\real$ such that 
$G=\bigcap_{k<\omega}G_k$ and 
$W\cap J=\bigcap_{k<\omega}W_k$. 
Choose an increasing sequence
$\la M_k\colon k<\omega\ra$ 
of compact sets such that $\lin(M)=\bigcup_{k<\omega}M_k$, 
let $\RR$ be a family of all triples 
$\la \ell,m,n\ra$ such that 
$m,n<\omega$, $n>0$, and $\ell\in L_{m+n}$, where 
$L_i$'s are as in (\ref{eq:linFunc}), and
fix a sequence $\la \la \ell_k,m_k,n_k\ra\in\RR\colon k<\omega\ra$
with each triple appearing infinitely many times. 
We will construct, by induction on $k<\omega$,
a sequence $\la U_s\colon s\in 2^k\ \&\ k<\omega\ra$ 
of non-empty open subsets of $\real$ such that $U_\emptyset=J$
and for every $0<k<\omega$ and $s\in 2^{k-1}$ 
the following inductive conditions hold. 
\begin{itemize}
\item[\rm (a)] 
$\cl(U_{s\hat{\ }0})$ and $\cl(U_{s\hat{\ }1})$ are disjoint 
subsets of $U_s\cap W_k$. 
\item[\rm (b)] 
$\ell_k(\bar a,x_1,\ldots,x_{n_k})\in G_k\setminus M_k$
for every $\bar a\in (M_k)^{m_k}$ and 
$x_j$ chosen from different $U_t$ with $t\in 2^k$.
\end{itemize}
To see that such a sequence can be built 
assume that for some $0<k<\omega$ the sets 
$\{U_s\colon s\in 2^{k}\}$ have been already constructed. 
Let $\{t_i\colon i<2^{k}\}$ be an enumeration of $2^{k}$
and by induction on $i$ choose 
\[
x_{t_i}\in U_{t_i\restriction k-1}\cap 
(W\setminus\lin(M\cup\{x_{t_j}\colon j<i\})\cap
\bigcap_{y\in\lin\{x_{t_j}\colon j<i\}}(y+G[\lin(M)]).
\]
The choice can be made since
$U_{t_i\restriction k-1}$ is non-empty and open while 
the remaining sets are dense $G_\delta$'s 
in $U_{t_i\restriction k-1}\subset J$. 
Notice that the choice guarantees that 
\begin{equation}\label{ConBHamel} 
\mbox{(b) holds for $x_j$ 
chosen as different elements of $\{x_t\colon t\in 2^k\}$.}
\end{equation}

To see it first notice that 
clearly $\{x_{t_i}\colon i<2^k\}$ is linearly 
independent and  that 
$\lin(M)\cap\lin(\{x_{t_i}\colon i<2^k\})=\{0\}$. 
Also, 
$q\, x_{t_i}-\lin\{x_{t_j}\colon j<i\}\subset G[\lin(M)]$,
that means,
$q\, x_{t_i}-\lin\{x_{t_j}\colon j<i\}-\lin(M)\subset G$
for every $q\in\rational\setminus\{0\}$ and $i<2^k$.  
But if $\bar a\in (M_k)^{m_k}$
and $\{x_1,\ldots,x_{n_k}\}\in [\{x_{t_i}\colon i<2^k\}]^{n_k}$
then for appropriate $q\in\rational\setminus\{0\}$ and $i<2^k$
we have 
\[
\ell_k(\bar a,x_1,\ldots,x_{n_k})
\in q\, x_{t_i}-\lin\{x_{t_j}\colon j<i\}-\lin(M)
\subset G\setminus \lin(M).
\]
So, (\ref{ConBHamel}) is proved. 


Now, by the compactness of $M_k$ and continuity of $\ell_k$, 
the set 
\[
Z=\{\la x_1,\ldots,x_{n_k}\ra\colon
(\exists \bar a\in (M_k)^{m_k})\ 
\ell_k(\bar a,x_1,\ldots,x_{n_k})\in G_k\setminus M_k\}
\] 
is open and, by (\ref{ConBHamel}),
contains all one-to-one sequences $\bar s$ of 
points from the set $\{x_t\colon t\in 2^k\}$.
Since there is only finitely many such sequences $\bar s$
we can find disjoint basic clopen neighborhoods 
$U_t$ of $x_t$ such that (a) and (b) hold.
This finishes the inductive construction.

Let $K_0=\bigcap_{k<\omega}\bigcup_{t\in 2^k} U_t$.
By (a), $K_0$ is a perfect subset of $W$. 
Notice also that, by condition (b), 
\[
T=\!\!\bigcup_{m,n<\omega}\!
\{\ell(\bar a,x_0,\ldots,x_{n})\colon 
\bar a\in M^m\ \&\ \{x_0,\ldots,x_{n}\}\in[K_0]^{n+1} 
\ \&\ \ell\in L_{n+m}\}
\]
is a subset of $G\setminus\lin(M)$. 
Clearly $0\in\lin(M)$, so $0\notin T$. Thus
$K_0$ is linearly independent
and $\lin(M)\cap\lin(K_0)=\{0\}$.
So, $\lin(M\cup K_0)\setminus\lin(M)=T\subset G$. 

Now fix an $x\in K_0$ and $K\in\perf(K_0\setminus\{x\})$.
Then for every $q\in\rational\setminus\{0\}$ and 
$v\in\lin(M\cup K)$ we have 
$q\, x - v\in \lin(M\cup K_0)\setminus\lin(M)\subset G$.
Thus, 
$q\, x - \lin(M\cup K)\subset G$
and so, $G[\lin(M\cup K)]$ contains a set 
$\{q\, x\colon q\in\rational\setminus\{0\}\}$,
which is clearly dense. 
Thus, $K$ is as desired. \qed

\ignore{ 
notice that, for any $\la \ell,m,n\ra\in\RR$,
$\bar a\in M^m$, and 
$x_1,\ldots,x_n\in[P]^n$ there is $k<\omega$
such that 
$\la \ell,m,n\ra=\la \ell_k,m_k,n_k\ra$,
$0\in M_k$, $\bar a\in (M_k)^m$,
and $x_i$'s belong to disjoint sets from $\{U_t\colon t\in 2^k\}$.
But then, by (b), 
$\ell(\bar a,x_1,\ldots,x_{n})\neq 0$.
}

We will also need the following strengthening of \cite[thm.~1.1]{CP92}.
(See also \cite[thm.~5.1.7]{CPAbook}.)


\prop{prop:DisHamSmallGr}{
\psmPrGame\ implies that for every 
dense $G_\delta$ subset $G$ of $\real$ there is a family 
$\HH$ of compact 
pairwise disjoint sets such that 
$H=\bigcup\HH$ is a Hamel basis and 
for every non-meager $G_\delta$ subset $B$ of $\real$ 
and every countable $\HH_0\subset\HH$  
there exists an uncountable $H\in\HH\setminus \HH_0$  
such that $H\subset B$ and 
$\lin(H\cup\bigcup\HH_0)\setminus\lin(\bigcup\HH_0)\subset G$. 
}


\proof First notice that if $\G_\delta$
stands for the family of all $G_\delta$ second 
category subsets of $\real$ then, assuming \psmPrGame, 
there exists a $\B\in[\G_\delta]^{\omega_1}$
coinitial with $\G_\delta$, that is, such that
\begin{equation}\label{eqDHF}
\mbox{for every $G\in\G_\delta$ there exists a
$B\in\B$ such that $B\subset G$.}
\end{equation}
Indeed, since \psmPrGame\ implies $\cf(\M)=\omega_1$
(see \cite[cor.~4.3]{CP88} or \cite[cor.~1.3.3]{CPAbook})
there exists a decreasing sequence
$\la G_\xi\colon \xi<\omega_1\ra$ of dense $G_\delta$ 
subsets of $\real$ such that 
for every dense $G_\delta$-set $W\subset\real$
there exists a $\xi<\omega_1$ with $G_\xi\subset W$.
It is easy to see that
$\B=\{G_\xi\cap(p_0,p_1)\colon \xi<\omega_1\ \&\
p_0,p_1\in\rational\ \&\ p_0<p_1\}$
satisfies (\ref{eqDHF}). 

Decreasing set $G$, if necessary, we can assume that $G\neq\real$. 
Fix  a sequence 
$\la B_\xi\in\B\colon \xi<\omega_1\ra$ 
in which each $B\in\B$ is listed $\omega_1$-many times.  
For a sequence $\la P_\xi\colon \xi<\omega_1\ra$
of prisms in $\real$ representing potential play of Player~I
construct a sequence
$\la \la Q_\xi,R^0_\xi,R^1_\xi\ra\colon \xi<\omega_1\ra$
such that the following inductive conditions hold for every 
$\xi<\omega_1$, where 
$R_\xi=\bigcup_{\eta<\xi}(R^0_\eta\cup R^1_\eta)$. 
\begin{itemize}
\item[(i)] Sets 
$\{R^i_\eta\colon \eta\leq\xi\ \&\ i<2\}$ 
are compact, pairwise disjoint. 

\item[(ii)] $R_{\xi+1}=\bigcup\{R^i_\eta\colon \eta\leq\xi\ \&\ i<2\}$ 
is linearly independent over $\rational$. 

\item[(iii)] $Q_\xi$ is a subprism of $P_\xi$
             and $Q_\xi\subset \lin(R_{\xi+1})$. 

\item[(iv)] $R^0_\xi\in\perf(B_\xi)$ and 
     $\lin(R_\xi\cup R^0_\xi)\setminus\lin(R_\xi)\subset G$.

\item[(v)] $G[\lin(R_{\xi+1})]$ is a dense $G_\delta$ in $\real$. 
\end{itemize}
To make an inductive step assume that
for some $\xi<\omega_1$ the required sequence
$\la \la Q_\zeta,R^0_\zeta,R^1_\zeta\ra\colon \zeta<\xi\ra$
is already constructed. So, $R_\xi$ is already defined and,
by the inductive assumption, $R_\xi$ is clearly 
linearly independent. Next notice that 
\begin{center}
$G[\lin(R_\xi)]$ is a dense $G_\delta$.
\end{center}
If $\xi=\eta+1$ then it follows 
from (v) for $\eta$. 
On the other hand, if $\xi$ is a limit ordinal then
$G\left[\lin(R_\xi)\right] =
G\left[\bigcup_{\eta<\xi}\lin(R_{\eta+1})\right]=
\bigcap_{\eta<\xi} G\left[\lin(R_{\eta+1})\right]$ 
so it follows from the inductive assumption as well. 

We define $R^0_\xi$ as a $K$ from Lemma~\ref{lem:smallGraph1xx}
applied to $W=B_\xi$ and $M=R_\xi$. 
This guarantees (iv), $R_\xi\cap R^0_\xi=\emptyset$, 
density of $G[\lin(R_\xi\cup R^0_\xi)]$, 
and linear indpendence of $R_\xi\cup R^0_\xi$. 

Next use Lemma~\ref{lem:smallGraph0xx}
to prism $P_\xi$ and $G[\lin(R_\xi\cup R^0_\xi)]$
to find a subprism $Q'$ of $P_\xi$
such that 
\begin{eqnarray*}
G[\lin(R_\xi\cup R^0_\xi)][\lin(Q')]
& = &
G[\lin(R_\xi\cup R^0_\xi)+\lin(Q')]\\
& = &
G[\lin(R_\xi\cup R^0_\xi\cup Q')]
\end{eqnarray*}
is a dense $G_\delta$,
where the first equation follows from Fact~\ref{fact:BasicGL}(f).  
Further, apply Lemma~\ref{LemRelLinInd}
to $M=R_\xi\cup R^0_\xi$ and prism $P=Q'$ 
to find a subprism $Q_\xi$ of $Q'$ and 
a compact $R_\xi^1$ subset of $Q'\setminus M$ such that 
$M\cup R_\xi^1$ is a maximal linearly independent subset 
of $M\cup Q_\xi$.

The maximality immediately implies 
$Q_\xi\subset \lin(M\cup R_\xi^1)=\lin(R_{\xi+1})$ 
so (iii) holds. We also clearly have (i) and (ii).
Condition (v) follows from the density of 
$G[\lin(R_\xi\cup R^0_\xi\cup Q')]$ 
and the fact that $R_\xi^1\subset Q'$.
This finishes the inductive construction. 

Now, if $S$ is a Player~II strategy 
associated with our construction, then by 
\psmPrGame, there exists a game
$\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
played according to $S$ in which 
$\real=\bigcup_{\xi<\omega_1}Q_\xi$. 
Let $\la \la R^0_\xi,R^1_\xi\ra\colon \xi<\omega_1\ra$
be a sequence associated with this game. 
Then 
$\HH=\{R^i_\xi\colon \xi<\omega_1\ \&\ i<2\}$
is as desired. \qed  

\bigskip 




\noindent{\sc Proof of Theorem~\ref{thm:smallGraph0}.}
Let $\X=\{\C(B)\colon B\in\B\}$ where
$\B$ is as in (\ref{eqDHF}).
We will play ${\rm GAME}_{\rm prism}(\X)$
in which, by Theorem~\ref{th:multGames},
Player~II has no winning strategy. 
Notice that since each $g\in \K$, 
where $\K$ is defined as in (\ref{def:K}),
contains some function from $\bigcup\X$,
every function $f$ intersecting each $g\in\bigcup\X$
is almost continuous. 

Let $\HH=\{H_\xi\colon\xi<\omega_1\}$ 
be as in Proposition~\ref{prop:DisHamSmallGr}. 
We also fix a sequence $\bar P=\la P_\xi\colon \xi<\omega_1\ra$
such that each $P_\xi$ represents a prism in 
some $\C(B)\in\X$. 
Sequence $\bar P$ represents potential 
play for Player~I in ${\rm GAME}_{\rm prism}(\X)$
and we will 
construct, by induction, a strategy $S$ for Player~II
which will describe a game played
according to $S$ in response to $\bar P$.
To make $S$ a legitimate strategy its 
value at stage $\xi<\omega_1$ will 
depend only on $\bar P_\xi=\la P_\eta\colon \eta\leq\xi\ra$.

So, construct a sequence 
$\la \la H^0_\xi,H^1_\xi,Q_\xi,K_\xi,R_\xi,Y_\xi\ra\colon \xi<\omega_1\ra$
of subsets of $\real$ 
such that for every $\xi<\omega_1$ 
the following inductive conditions are satisfied,
where $B_\xi\in\B$ is such that $P_\xi\subset\C(B_\xi)$
and $\HH_\xi=\{H^i_\eta\colon\eta<\xi\ \&\ i<2\}$. 
\begin{itemize}
\item[(I)] $H^0_\xi$ and $H^1_\xi$ are distinct elements of 
$\HH\setminus\bigcup\HH_\xi$.

\item[(II)] $H^0_\xi\in[B_\xi]^\continuum$ 
and $\lin(H^0_\xi\cup\bigcup\HH_\xi)\setminus\lin(\bigcup\HH_\xi)\subset G$. 

\item[(III)] $H_\xi\in\{H^i_\eta\colon\eta\leq\xi\ \&\ i<2\}$.
\end{itemize}
We can choose such $H^0_\xi$ and $H^1_\xi$
since $\HH$ was taken from Proposition~\ref{prop:DisHamSmallGr}.
Also, if $F_\xi=\bigcup_{\eta<\xi}(R_\eta\cup Y_\eta)$
and $U_\xi=G[\lin(F_\xi)]$ then 
\begin{itemize}
\item[(IV)] $U_\xi$ is a dense $G_\delta$ in $\real$, 

\item[(V)] $K_\xi\in\perf(H_\xi^0)$, $Q_\xi$ is a subprism
           of $P_\xi$, $R_\xi=\{h(x)\colon h\in Q_\xi,\ x\in K_\xi\}$,
           and $U_\xi[\lin(R_\xi)]$ is dense $G_\delta$ in $\real$, and

\item[(VI)] $Y_\xi\in\perf(\real)$ is a linearly independent set 
            such that $G[\lin(F_{\xi+1})]$ is dense,
            $\lin(F_\xi\cup R_\xi)\cap\lin(Y_\xi)=\{0\}$, 
            and 
            $\lin(F_\xi\cup R_\xi\cup Y_\xi)\setminus \lin(F_\xi\cup R_\xi)$
            is a subset of $G$.
\end{itemize}
Assuming that (IV) holds 
the possibility of a choice of $Q_\xi$, $K_\xi$, and $R_\xi$
as in (V) follows directly from Lemma~\ref{lem:smallGraph0}.
Next, since by Fact~\ref{fact:BasicGL}(f)
\begin{eqnarray*}
U_\xi[\lin(R_\xi)]
& = & G[\lin(F_\xi)][\lin(R_\xi)]\\
& = & G[\lin(F_\xi)+\lin(R_\xi)]\\
& = & G[\lin(F_\xi\cup R_\xi)]
\end{eqnarray*}
we can apply Lemma~\ref{lem:smallGraph1xx} to our $G$, $W=\real$, and 
$M=F_\xi\cup R_\xi$ to find a linearly independent $K\in\perf(\real)$ 
for which 
$G[\lin(F_\xi\cup R_\xi\cup K)]$ is a dense $G_\delta$ subset of $\real$,
$\lin(F_\xi\cup R_\xi\cup K)\setminus \lin(F_\xi\cup R_\xi)\subset G$, and 
$\lin(F_\xi\cup R_\xi)\cap\lin(K)=\{0\}$.
Then put $Y_\xi=K$ and notice that 
(VI) is satisfied, since 
$F_{\xi+1}=F_\xi\cup R_\xi\cup Y_\xi$. 

To finish the construction it is enough to argue that (IV) is preserved. 
But if $\xi=\eta+1$ is a successor ordinal then it follows immediately
from (VI) for $\eta$. 
But if $\xi$ is a limit ordinal then (IV) 
follows easily from the density of sets $U_\eta$ for $\eta<\xi$
since 
$U_\xi=G\left[\bigcup_{\eta<\xi}
\lin\left(\bigcup_{\zeta<\eta}(R_\zeta\cup\{y_\zeta\})\right)\right]
=\bigcap_{\eta<\xi}U_\eta$. 
This finishes the inductive construction
of the sequence. 

We define a strategy $S$ for Player~II  by
$S(\la\la P_\eta,Q_\eta\ra\colon \eta<\xi\ra,P_\xi)=Q_\xi$.
By Theorem~\ref{th:multGames} this is not a winning strategy, 
so there exists a game 
$\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
played according to $S$ in which $\bigcup\X=\bigcup_{\xi<\omega_1}Q_\xi$. 
We will use the sequence
$\la \la H^0_\xi,H^1_\xi,Q_\xi,K_\xi,R_\xi,y_\xi\ra\colon \xi<\omega_1\ra$
associated with this game
to construct the desired function $f$.

Since, by (I) and (III), 
$\{H^i_\xi\colon \xi<\omega_1\ \&\ i<2\}=\HH$,
it is enough to define $f$ on each $H^i_\xi$
and extend it to a unique additive function.
So, for each $\xi<\omega$ 
define $f$ on $H_\xi^1$
as a one-to-one function with values in $Y_\xi$.
On each 
$H^0_\xi$ we define $f$ such that $f[H^0_\xi]\subset R_\xi$
and $f$ intersects every $g\in Q_\xi$ on a set $K_\xi$.
It remains to prove that $f$ is as advertised. 

Certainly $f$ is additive. It is also not difficult
to see that $f$ defined that way cannot be continuous. 
To see that it is almost continuous it is enough to notice that 
every $g\in\bigcup\X$ belongs to some $Q_\xi$,
so it is intersected by $f$.
To finish the proof it is enough to show that
$f\subset (\real\times G)\cup(G\times\real)$. 
So, define  $f_\xi$ as $f\restriction \lin\left(\bigcup\HH_\xi\right)$. 
Since $f=\bigcup_{\xi<\omega_1}f_\xi$, it is enough to 
prove that 
\begin{equation}\label{eq:SGfxi}
f_\eta\subset (\real\times G)\cup(G\times\real)
\end{equation}
for every $\eta<\omega_1$. This will be proved by induction.

Clearly $f_0=\{\la 0,0\ra\}\subset (\real\times G)\cup(G\times\real)$
since $0\in G$. 
So assume that for some $0<\eta<\omega_1$
condition (\ref{eq:SGfxi}) holds for every $\zeta<\eta$. 
If $\eta$ is a limit ordinal then 
$f_\eta=\bigcup_{\zeta<\eta}f_\zeta$ so (\ref{eq:SGfxi}) clearly holds.
So assume that $\eta=\xi+1$ and notice that
\begin{center}
$f_\eta^*=f_\eta\restriction \lin(H^0_\xi\cup\bigcup\HH_\xi)$
is a subset of $(\real\times G)\cup(G\times\real)$.
\end{center}
This is the case since 
$f_\xi\subset (\real\times G)\cup(G\times\real)$
by the inductive assumption while
$f_\eta^*\setminus f_\xi\subset(G\times\real)$
since
$\dom(f_\eta^*\setminus f_\xi)=
\lin(H^0_\xi\cup\bigcup\HH_\xi)\setminus\lin(\bigcup\HH_\xi)
\subset G$
is guaranteed by (II).  

Thus, to finish the proof, it is enough to show that
\begin{center}
$f_\eta\setminus f_\eta^*\subset(\real\times G)$.
\end{center}

To see it first note that, from our construction,  
$\range(f_\eta^*)\subset \lin(F_\xi\cup R_\xi)$.
Now, if $x\in\dom(f_\eta\setminus f_\eta^*)=
\lin(\bigcup\HH_{\xi+1})\setminus\lin(H^0_\xi\cup\bigcup\HH_\xi)$
then $x=v+w$ for some $v\in\lin(H_\xi^1)\setminus\{0\}$
and $w\in \lin(H^0_\xi\cup\bigcup\HH_\xi)$.
Hence, by the definition of $f$ and condition (VI),
\begin{eqnarray*}
f_\eta(x)=f_\eta(v)+f_\eta(w)
& \in &
(\lin(Y_\xi)\setminus\{0\})+\lin(F_\xi\cup R_\xi)\\
& = &
\lin(F_\xi\cup R_\xi\cup Y_\xi)\setminus \lin(F_\xi\cup R_\xi)\subset G.
\end{eqnarray*}
This completes the proof. \qed

\section{Hamel basis} 


For a subset $A$ of $\real$ we define $E^+(A)$ as
\[
E^+(A)=\left\{\sum_{i=0}^k q_ia_i\colon k<\omega\ \&\
a_i\in A\ \&\ q_i\in\rational\cap[0,\infty)
\mbox{ for every } i\leq k\right\}.
\]
In~\cite{Erdos} P.~Erd\H os
proved that under the continuum 
hypothesis
there exists a Hamel basis $H$ for which
$E^+(H)$ is a Luzin set. In particular,
such $E^+(H)$ is of measure zero. 
K.~Muthuvel~\cite{Muth},
answering a question of 
H.~Miller~\cite{HMiller},
generalized Erd\H os'
result by proving that, under Martin's axiom,
there exists a Hamel basis $H$ for which
$E^+(H)$ is simultaneously of measure zero and 
first category. 
However, it is unknown whether there is a ZFC example of 
a Hamel basis $H$ for which $E^+(H)$ is of measure zero.
In what follows we show that the existence of such a Hamel basis is 
a consequence of \psmPrGame. 


\thm{thm:OnEplus}{\psmPrGame\ implies that for every 
dense $G_\delta$ subset $G$ of $\real$ with $0\in G$
there exists an $A\subset \real$
such that $\lin(A)=\real$ and $E^+(A)\subset G$.}



Using Theorem~\ref{thm:OnEplus} with $G$ of measure zero
and the fact that every set $A$ spanning $\real$
contains a Hamel basis
we obtain immediately the following corollary. 

\cor{cor:OnEplus}{\psmPrGame\ implies that 
there exists a Hamel basis $H$ such that $E^+(H)$ has measure zero.
}


\noindent{\sc Proof of Theorem~\ref{thm:OnEplus}.}
Decreasing $G$, if necessary, we can assume that
$q G=G$ for every non-zero $q\in\rational$. 
Since $G$ has a Polish metric, we can use 
\psmPrGame\ for ${\rm GAME}_{\rm prism}(X)$
with $X=G$. 

Fix a sequence $\bar P=\la P_\xi\colon \xi<\omega_1\ra$
such that each $P_\xi$ represents a prism in~$X$. 
Sequence $\bar P$ represents a potential 
play for Player~I. We will 
construct, by induction, a strategy $S$ for Player~II
which will describe a game played
according to $S$ in response to $\bar P$.
The value of $S$ at stage $\xi<\omega_1$ will 
depend only on $\bar P_\xi=\la P_\eta\colon \eta\leq\xi\ra$.

For this, we will construct a sequence 
$\la \la Q_\xi,A_\xi\ra\colon \xi<\omega_1\ra$
of pairs of sigma-compact subsets of $\real$ 
such that for every $\zeta\leq\xi<\omega_1$ 
\begin{itemize}
\item[(I)] $Q_\xi$ is a subprism of $P_\xi$,

\item[(II)] $A_\zeta\subset A_\xi$ and 
   $\bigcup_{\eta\leq\xi}Q_\eta\subset\lin(A_\xi)$,

\item[(III)] set $G[E^+(A_\xi)]$ is dense and 
             $E^+(A_\xi)\subset G$.
\end{itemize}
Assume that for some $\xi<\omega_1$ the desired sequence
$\la \la Q_\eta,A_\eta\ra\colon \eta<\xi\ra$
is already constructed.
Let $B_\xi=\bigcup_{\eta<\xi}A_\eta$. Then 
$E^+[B_\xi]=\bigcup_{\eta<\xi}E^+[A_\eta]$
is sigma-compact and 
$G_\xi=G[E^+(B_\xi)]=\bigcap_{\eta<\xi}G[E^+(A_\eta)]$
is a dense $G_\delta$. 
Thus, by Lemma~\ref{lem:smallGraph0xx}, 
we can find a subprism $Q_\xi$ of $P_\xi$
such that $G_\xi[\lin(Q_\xi)]$ 
is a dense $G_\delta$ subset of $\real$.
Since
\[
G_\xi[\lin(Q_\xi)]=G[E^+(B_\xi)][\lin(Q_\xi)]
=G[E^+(B_\xi)+\lin(Q_\xi)]
\] 
there exists an $x\in\real$ such that
$x+E^+(B_\xi)+\lin(Q_\xi)\subset G$.
Therefore, we have also $qx+E^+(B_\xi)+\lin(Q_\xi)\subset G$
for every non-zero $q\in\rational$. 
Let us define $C_\xi=x+\lin(Q_\xi)$
and put $A_\xi=B_\xi\cup C_\xi$. 
This clearly ensures (II). 
To see $E^+(A_\xi)\subset G$
notice that every element of $E^+(A_\xi)$
either belongs to $E^+(B_\xi)\subset G$
or to $qx+E^+(B_\xi)+\lin(Q_\xi)\subset G$
for some positive $q\in\rational$. 
The density of $G[E^+(A_\xi)]$ follows from 
\begin{eqnarray*}
G[E^+(A_\xi)]
& = & G[E^+(B_\xi)+E^+(C_\xi)]\\
& = & G[E^+(B_\xi)][E^+(C_\xi)]\\
& = & G_\xi[E^+(C_\xi)]\\
& = & 
\mbox{$G_\xi\left[\bigcup_{q\in\rational^+}(qx+\lin(Q_\xi))\right]$}\\
& = & 
\mbox{$\bigcap_{q\in\rational^+} G_\xi\left[qx+\lin(Q_\xi)\right]$}\\
& = & 
\mbox{$\bigcap_{q\in\rational^+}(qx+G_\xi\left[\lin(Q_\xi)\right])$},
\end{eqnarray*}
where $\rational^+=\rational\cap(0,\infty)$,
since $G_\xi\left[\lin(Q_\xi)\right]$ is a dense $G_\delta$.
This finishes the inductive construction. 

Let $S$ be a strategy of Player~II 
given by the above inductive construction. 
Since $S$ is not winning, 
there is a game 
$\la\la P_\xi,Q_\xi\ra\colon \xi<\omega_1\ra$
played according to $S$ in which $G=X=\bigcup_{\xi<\omega_1}Q_\xi$. 
Thus, for $A=\bigcup_{\xi<\omega_1} A_\xi$ condition (III)
implies that $E^+(A)\subset G$,
while by (II) we have 
$\real=\lin(G)=\lin
\left(\bigcup_{\xi<\omega_1}Q_\xi\right)
\subset\lin(A)$. 
\qed



\begin{thebibliography}{22}



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

\bibitem{KC59} Ciesielski, K.,
Some additive Darboux-like functions, {\it J. Appl. Anal.} {\bf 4}(1) (1998),
43--51. (Preprint$^\star$ available.%
\footnote{Preprints marked by $^\star$ are available in electronic form
%. They can be accessed accessible
from {\it Set Theoretic Analysis Web Page:}
http://www.math.wvu.edu/homepages/kcies/STA/STA.html})



\bibitem{cie-jas} Ciesielski, K., and Jastrz{\c{e}}bski, J.
Darboux-like functions within the classes of Baire one, Baire two,
and additive functions,
{\it Topology Appl.} {\bf 103} (2000), 203--219. 
(Preprint$^\star$ available.)

\bibitem{CP88} Ciesielski, K., and Pawlikowski, J. 
Covering Property Axiom CPA$_{\rm cube}$ and its consequences, 
{\it Fund. Math.} {\bf 176}(1) (2003), 63--75.
(Preprint$^\star$ available.)

\bibitem{CP83} Ciesielski, K., and Pawlikowski, J. 
Crowded and selective ultrafilters under the Covering Property Axiom, 
{\it J. Appl. Anal.} {\bf 9}(1) (2003), 19--55. (Preprint$^\star$ available.)

\bibitem{CP96} Ciesielski, K., and Pawlikowski, J. 
Uncountable intersections of open sets under \psmP, 
{\it Proc. Amer. Math. Soc.}, to appear. 
(Preprint$^\star$ available.)



\bibitem{CP84} Ciesielski, K., and Pawlikowski, J. 
Small coverings with smooth functions  
under the Covering Property Axiom, 
preprint$^\star$.



\bibitem{CP92} Ciesielski, K., and Pawlikowski, J. 
Nice Hamel bases under the Covering Property Axiom, 
{\it Acta Math. Hungar.}, to appear. 
(Preprint$^\star$ available.)



\bibitem{CPAbook} Ciesielski, K., and Pawlikowski, J. 
{\it Covering Property Axiom CPA. A combinatorial core of the iterated perfect set model}, 
version of January 2003, work in progress$^\star$.


\bibitem{Erdos}  
Erd\H{o}s, P. On some properties of Hamel bases, 
{\it Colloq. Math.}  {\bf 10} (1963), 267--269.

\bibitem{Ka}  Kanovei, V. Non-Glimm--Effros equivalence relations 
at second projective level, {\it Fund. Math.} {\bf 154} (1997), 1--35.




\bibitem{kg} Kellum, K.R. 
Sums and limits of almost continuous functions, 
{\it Colloq. Math.} {\bf 31} (1974), 125--128.

\bibitem{kellum} Kellum, K.R. 
Almost Continuity and connectivity - sometimes it's as easy as to 
prove a stronger result,
{\it Real Anal. Exchange} {\bf 8} (1982-83), 244--252.

\bibitem{HMiller}
Miller, H. On a property of Hamel bases, 
{\it Boll. Un. Mat. Ital A(7)} {\bf 3} (1989), 39--43.

\bibitem{Muth}
Muthuvel, K. Some results concerning Hamel bases, 
{\it Real Anal. Exchange} {\bf 18}(2) (1992-93), 571--574.



\bibitem{natkaniec} Natkaniec, T. Almost Continuity,
{\it Real Anal. Exchange} {\bf 17} (1991-92), 462--520.


\bibitem{Zap} Zapletal, J.
{\it Descriptive Set Theory and Definable Forcing},
Mem. Amer. Math. Soc. {\bf 167}, 2004. 



\end{thebibliography}

\end{document}

