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\begin{document}
%\baselineskip=23pt

\title{The ideal of Sierpi{\'n}ski-Zygmund sets on the plane}
\author{Krzysztof P\l otka\thanks{This paper was written under supervision of K. Ciesielski.
The author wishes to thank him for many helpful conversations.}\\
        Department of Mathematics, West Virginia University\\
       Morgantown, WV 26506-6310, USA\\
       kplotka@math.wvu.edu \\
                 and \\
        Institute of Mathematics, Gda{\'n}sk University\\
        Wita Stwosza 57, 80-952 Gda{\'n}sk, Poland}

%\date{August 31, 2000}
\maketitle

\begin{abstract}
     We  say  that  a  set  $X  \sq \real^2$ is {\it Sierpi{\'n}ski-Zygmund\/}
(shortly  {\it SZ-set\/}) if it does not contain a partial continuous function
of  cardinality continuum $\cont$. We observe that the family of all such sets
is  $\cf(\cont)$-additive ideal. Some examples of such sets are given. We also
consider  {\it  SZ-shiftable  sets\/}, that is, sets $X \sq \real^2$ for which
there exists a function $f\colon \real \to \real$ such that $f+X$ is an SZ-set.
Some  results  are  proved about SZ-shiftable sets. In particular, we show that
the union of two SZ-shiftable sets does not have to be SZ-shiftable.
\end{abstract}

\vskip .5 in

     The  terminology  is standard and follows \cite{cie}. The symbol $\real$
stands  for  the  set  of all real numbers. The cardinality of a set $X$ we
denote  by  $|X|$.  In  particular,  $|\real|$  is denoted by $\cont$. Given a
cardinal  $\kappa$, we let $\cf(\kappa)$ denote the cofinality of $\kappa$. We
say that a cardinal $\kappa$ is regular provided that $\cf(\kappa)=\kappa$.

     A  set  $M  \sq  \real^n$  is called {\it Marczewski measurable} if every
perfect  set  $P$  has  a  perfect subset $Q$ such that $Q\sq M$ or $Q\cap M =
\emptyset$. If every perfect set $P$ has a perfect subset $Q$ such that $Q\cap
M  =  \emptyset$,  then  $M$  is called {\it Marczewski null}.

     We  consider  only  real-valued  functions  unless  stated  otherwise. No
distinction  is made between a function and its graph. For any planar set $Y$,
we  denote its $x$-projection by $\dom(Y)$. For any two partial real functions
$f,g$  we  write  $f+g$, $f-g$ for the sum and difference functions defined on
$\dom(f)  \cap  \dom(g)$. The class of all functions from a set $X$ into a set
$Y$  is denoted by $Y^X$. We write $f|A$ for the restriction of $f \in Y^X$ to
the  set $A \sq X$. For any function $g \in \real^X $, any family of functions
$F \sq \real^X$, and any set $A \sq X\times\real$ we define $g+F =\{g+f \colon
f\in  F\}$  and $g+A =\{\la x,g(x)+y \ra \colon \la x,y \ra\in~A\}$. The image
and  preimage  of  a  set $B$ under the function $h$ are denoted by $h[B]$ and
$h^{-1}[B]$, respectively.

     Let   us   recall   that  a  function  $f\colon  \real\to\real$  is  {\it
Sierpi{\'n}ski-Zygmund\/} ($f\in\sz$) if  for  every  set  $X  \sq  \real$ of cardinality
continuum $\cont$, $f|X$ is discontinuous. This definition is generalized onto
subsets of $\real^2$. (See~\cite{kplotka}.)


     \defi{defsz-set1}{A   set   $X   \subseteq   \real^2$   is   called  {\it
Sierpi{\'n}ski-Zygmund\/}  set  (shortly {\it SZ-set\/}), if for every partial
real continuous function $f$ we have $|f \cap X| < {\cont}$.}
     We  denote  the  family  of  all  SZ-sets  by $\szsets$. Since every
Sierpi{\'n}ski-Zygmund function  is  also an SZ-set we have that the family
$\szsets$ is not empty.


The next fact follows directly from the definition.

     \fact{p2-fact1}{$\szsets$  is a $\cf(\cont)$-additive ideal.} \proof 
It is obvious that $\szsets$ is closed under the operation of taking subsets.
We will show that $\szsets$ is $\cf(\cont)$-additive.
     
     Take
a  $\kappa  < \cf(\cont)$. Let $\{X_\xi \colon \xi < \kappa\} \sq \szsets$ and
$f\sq  \bigcup_{\xi  <  \kappa} X_\xi$ be a partial continuous function. Since
$X_\xi$  is  SZ-set,  we  have  that  $|f \cap X_\xi| < \cont$ for each $\xi <
\kappa$.  Consequently, $|f \cap \bigcup_{\xi < \kappa} X_\xi| = |\bigcup_{\xi
< \kappa}(f \cap X_\xi)| < \cont$. \qed


The question that one could ask here is how ``big'' an SZ-set can be. An example
of the SZ-set that can be considered ``big'' in some sense is given 
in~\cite{kplotka}.

     \lem{p2-fact2}{\mbox{\rm  \cite[Lemma  19]{kplotka}  }  There  exists  an
SZ-set  $X  \subseteq  \real^2$  such that $|\real \setminus X_x|<{\cont}$ for
every $x \in \real$, where $X_x=\{y \in \real \colon \la x,y \ra \in X\}$.}

     Observe  that the complement of every vertical section of the set $X$ has
size less than $\cont$. In particular, if MA holds then every vertical section
is  residual  in $\real$. Moreover, under CH, the complement of every vertical
section  of  $X$  is countable. It turns out that the existence of such SZ-set
(i.e., with co-countable vertical sections) is equivalent to CH. We state

\prop{p2-prop1}{ {\rm CH} is equivalent to the existence of an SZ-set $X \sq
\real^2$ with the following property $$|\real \setminus X_x|\le\omega \mbox{
for every } x \in \real .$$}
\proof The existence of the desired set under the assumption of CH follows
from the previous discussion. So we need to prove the opposite implication.
Assume, by
the way of contradiction, that the desired set $X$ exists and CH does not
hold, e.g. $\cont > \omega_1$. Since $X$ is an $\sz$-set we get
\begin{description}
\item[$(\ast)$] $X^y=\{x \in \real \colon \la x,y\ra
\in X\}$ has cardinality less than $\cont$ for every $y\in \real$.
\end{description}
We claim that there exists an $A\in[\real]^{\omega_1}$ such that
$|\bigcup_{y\in A}X^y| < \cont$. The following two cases are possible.

\noindent
{\bf Case 1.} There exists a $\kappa < \cont$ such that $Z_\kappa=\{y\colon
|X^y|=\kappa\}$ is uncountable.

Then we choose $A\in [Z_\kappa]^{\omega_1}$. Obviously, $|\bigcup_{y\in A}X^y|
=\kappa \omega_1 < \cont$.

\noindent
{\bf Case 2.}  $|Z_\kappa|\le \omega$ for every cardinal $\kappa < \cont$.

Put $Z=\{ |X^y| \colon y\in\real\}$ and observe that $\real=\bigcup_{\kappa\in
Z}Z_\kappa$. It follows from $(\ast)$ that if $\kappa\in Z$ then $\kappa
<\cont$. Consequently, since the union of less than continuum many countable
sets has size less than continuum, we conclude that $|Z|=\cont$. Let $\lambda$
be the $\omega_1$-st element of $Z$. We define $A=\{y\colon |X^y|<\lambda\}$.
Clearly, $|\bigcup_{y\in A}X^y| =|\bigcup_{\kappa < \lambda} Z_\kappa|\le
\lambda \omega < \cont$.


Now choose an $x\in\real\setminus\bigcup_{y\in A}X^y$ and notice that
$(\{x\}\times A) \cap X=\e$. So $A\sq \real \setminus X_x$. This is in
contradiction with the fact that every vertical section of $X$ is
co-countable.
\qed


     It  is  worth  remarking  here  that  SZ-sets  with the Baire property or
measurable  are  ``small.''  It means that every measurable SZ-set has measure
zero  and  every  SZ-set  with the Baire property is meager. This follows from
Fubini   Theorem  and  Kuratowski-Ulam  Theorem,  respectively.  But  do  such
``small''  SZ-sets  exist?  The  answer is positive. It is easy to construct a
Sierpi{\'  n}ski-Zygmund  function  (so  also  an  SZ-set) contained in $\real
\times  \cant$,  whose  domain is the whole real line. $\cant$ is the standard
linear  Cantor  set.  Observe  also that there are ``big'' SZ-sets in terms of
outer  measure.  The set $X$ from Lemma~\ref{p2-fact2} is of full outer measure.
To  see  this,  choose  a  closed set $F \sq \real^2\setminus X$. Based on the
properties of $X$ we conclude that every vertical section of $F$ is countable.
Hence $F$ is of measure zero. This proves that $X$ is of full outer measure.

The above discussion states that ``good'' SZ-sets (in terms of measure or
Baire property) are ``small''. However, we have the following
\rem{szmarcz}{There exists an SZ-set which is
Marczewski measurable but not Marczewski null.}

     \proof  We  claim that the set $X$ from Lemma~\ref{p2-fact2} is the desired
set.  Let us see why $X$ is Marczewski measurable but not Marczewski null. Fix
a  perfect  set  $P  \sq  \real^2$.  There are two possible cases. Either some
vertical  section  $P_a$  of  $P$  is  perfect,  or  all vertical sections are
countable.  In  the first case, there is a $Q \sq \{a\} \times P_a$ completely
contained  in $X$, because the complement of every vertical section of $X$ has
cardinality  less  than  $\cont$.  In  the  second case, we can find a partial
continuous function $f \sq P$ defined on a perfect set. To see this consider a
function  $g\colon \dom(P) \to \real $ defined by $g(x)=\sup(P_x \cap (-\infty
,0])$.  The function $g$  is  upper  semi-continuous  so also of Baire class one.
Thus, $g$ contains    a    continuous    function    defined    on    a   perfect
set. (See~\cite{oxtoby}.)

     Since $|f \cap X| < \cont$, the restriction of $f$ to some perfect subset
$R$  of $\dom(f)$ is disjoint with $X$. Note that $f|R$ is a perfect set. Thus
$P$  contains  a perfect subset disjoint with $X$.

It is obvious that $X$ contains a perfect set (every vertical section contains a
perfect set). So $X$ is not Marczewski null. This completes the proof of
our remark. \qed

Another interesting observation is that the property of being an SZ-set is not
preserved under the homeomorphic images. It is easy to see that any vertical
line is an SZ-set, but after a rotation, for example about $\frac{\pi}{4}$, it
is not an SZ-set any more. However, if $h \colon \real^2 \to \real^2$ is a
homeomorphism preserving vertical lines then $h[X]$ is an SZ-set for every $X
\in \szsets$.


\fact{p2-fact4}{Let $h \colon \real^2 \to \real^2$ be an homeomorphism such
that $h[L]$ is a vertical line for every vertical line $L$. Then $h\{\szsets\}=\{h[X] \colon
X\in\szsets \}=\szsets$.}
\proof First we show the inclusion $h\{\szsets\} \sq \szsets$. It is easy to
see that if $f \colon A \to \real$ is a partial continuous function then
$h^{-1}[f] \colon A \to \real$ is also continuous. This implies that for every
$X \in \szsets$, $h[X]$ is also in $\szsets$ since $h[X]\cap f=h[X\cap h^{-1}[f]]$.

Now to show the other inclusion, let us fix a $Y \in \szsets$. Note that
$h^{-1}$ also preserves all vertical lines. Thus, from the first part of the
proof, $X=h^{-1}[Y]~\in~\szsets$. Hence $Y=h[X]~\in~h\{\szsets\}$.
\qed

As we mentioned at the beginning of this paper, the concept of Sierpi{\'n}ski-Zygmund sets is a generalization of the concept of Sierpi{\'n}ski-Zygmund functions. One of the questions related to the family $\sz$ of Sierpi{\'n}ski-Zygmund functions is for how ``big''
families $F\sq \real^\real$ we can find a function $g\in\real^\real$ such that $g+F \sq \sz$. (See e.g. \cite{cie-nat}.) Similar question can be asked in the case of  
Sierpi{\'n}ski-Zygmund sets. This leads to the following definition.




\defi{sz-shift}{A set $X \subseteq \real^2$ is called {\it $\sz$-shiftable\/},
if there exists a function $f \colon \real \to \real$ such that $f+X$ is
SZ-set.}
We denote the family of all SZ-shiftable sets by $\szshift$. Obviously
$\szsets \sq \szshift$, so $\szshift$ is not empty.

\lem{p2-lem1}{Let $X\sq\real^2$. If for all $x \in \real$ and $A \in
[\real]^{<\cont}$ there exists an $a\in \real$ such that $(a+A)\cap
X_x=\emptyset$, then $A$ is SZ-shiftable.}
     \proof  Let  $\la  x_{\alpha}:  \alpha < \cont \ra $ and $\la f_{\alpha}:
\alpha  <  \cont \ra $ be the sequences of all real numbers and all continuous
functions  defined  on a $G_{\delta}$ subset of $\real$, respectively. We will
define a function $f \colon \real \to \real $ which shifts $X$ into $\szsets$,
using   transfinite   induction.   For   every  $\alpha  <  \cont$  we  choose
$f(x_{\alpha})  \in  \real$  with the property that  $(f(x_{\alpha})+X_{x_{\alpha}})  \cap
\{f_{\xi}(x_{\alpha})  \colon  \xi  <  \alpha \} =\emptyset$. Such a choice is
possible  because  of  the  assumptions  on  $X$. It is easy to see that $\dom
\left((f+X)  \cap  f_{\beta} \right ) \sq \{x_{\xi} \colon \xi < \beta \}$ for
each $\beta < \cont$. Thus $f+X \in \szsets$. 
\hfill \qed

     Recall  that  under  Martin's  Axiom  (MA) the union of less than $\cont$
meager  sets  is meager. Suppose that $A \in[\real]^{<\cont}$ and $B\sq \real$
is  meager.  Then  the set $B-A=\bigcup_{x\in A}(B-x)$ is meager as a union of
less  than  $\cont$  meager  sets.  Now,  if  we  choose an $a\notin B-A$ then
$(a+A)\cap B=\emptyset$. Notice that the same argument can be repeated for the
sets of measure zero.

The above discussion and Lemma~\ref{p2-lem1} immediately imply

     \cor{p2-cor1}{{\rm  (MA)}  If  each  vertical  section  of  a  set $X \sq
\real^2$ is meager or of measure zero, then $X \in \szshift$.}

     It  may  also  be  of  interest to determine whether $\szshift$ is closed
under the union operation. Fact~\ref{p2-fact1} states, in particular, that the
union  of  two  SZ-sets  is  also  an  SZ-set. Thus, the natural question that
appears  here  is whether the same is true for SZ-shiftable sets. It turns out
not to be the case.

     \ex{p2-ex1}{There  exist  $A_1,  A_2  \in  \szshift$  such that $A_1 \cup
A_2=\real^2 \not\in \szshift$.}
     \proof  Put  $A_1$ to be the set $X$ from Lemma~\ref{p2-fact2} and $A_2$ to
be  its  complement. Based on Lemma~\ref{p2-lem1} $A_2$ is SZ-shiftable. Next,
notice  that  $A_1  \in \szsets \sq \szshift$. Finally, $A_1 \cup A_2=\real^2$
and obviously $\real^2$ is not in $\szshift$. \qed


     Before we finish let us make a comment about \cite[Theorem 2
(1)]{kplotka} which says: MA implies that for every finite family $F$ of real functions there exists an almost continuous function $g$ (each open subset of $\real^2$ containing the graph of
$g$ contains also the graph of a continuous function) such that $g+f$ is Sierpi{\'n}ski-Zygmund for every $f\in F$.  Note  that  this  result  can be expressed using the notion of
SZ-sets. Under MA the following holds:
\begin{description}
\item

{\emph{\quad\quad If, for some fixed $n \in \omega$, every vertical section of
the set $X \sq \real^2$ has at most $n$ elements then there exists an almost
continuous function $f\colon \real \to \real$ such that $f+X \in \szsets$.}}
\end{description}

We generalize the above result.
\thm{p2-thm1}{{\rm (MA)} 
If every vertical section of the set $X \sq \real^2$
is finite then there exists an almost continuous function $f\colon \real \to
\real$ such that $f+X \in \szsets$.}
 
Before we prove the theorem we need to cite some lemmas and recall some properties. First let us observe that a function $f\colon \real \to \real$ is almost continuous if and only if it intersects every {\em blocking set\/}, i.e.,
a closed set $K \sq \real^2$ which meets every continuous function from $\real$ to $\real$
and is disjoint with at least one function from $\real^{\real}$. Next we give some definitions
needed to state the lemmas. (See \cite{kplotka}.)
For $X \sq \real$ by $\ccont(X)$ we denote
the family of all functions $f \colon X \to \real $ which can be represented
as a union of less than $\cont$-many partial continuous functions. The symbol $\sz(X)$ denotes the family of all partial Sierpi{\'n}ski-Zygmund functions defined on $X$.

Let $A \sq \real$ be everywhere of second 
category, that is $A \cap I $ is of second category for every nontrivial 
interval $I$. We define ${\cal F}_A$ as a family of all 
$F\subseteq \real^{\real}$ whose union $\bigcup F$ contains no function from $\ccont(A \cap B)$ for any Borel set $B$ of second category.

\lem{}{{\rm \cite[Lemma 12]{kplotka}} {\rm (MA)} Let $F \in {\cal F}_A$ be a family such that 
$|F| \le \cont $. There exists a $g \in \sz(A) $ such that every extension
$\bar{g} \colon \real \to \real$ of $g$ is almost continuous and
$g+F \subseteq \sz(A) $.}
A slight modification of the proof of the above lemma gives a little stronger result. (See \cite[Lemma 2.2.1]{kplotka_phd}.)

\lem{lemphd1}{{\rm (MA)} Let $F \in {\cal F}_A$ be a family such that $|F| \le \cont$. There exists a $g \in \sz(A) $ such that $g+F \subseteq \sz(A)$
and for every blocking set $B\sq\real^2$ there is a non-empty open interval $I_B\sq \dom(B)$ with the property that $\dom(B\cap g)$ is dense in $I_B$.}

\lem{lemphd2}{{\rm \cite[Lemma 13]{kplotka}} {\rm (MA)} Let $\{ f_i \}_1^n \subseteq \real^{\real}$, $n=1,2, \dots $.
There exists $\{ f_i^{\prime} \}_1^n \in {\cal F}_A$ such that 
$f_i | A_i \in \ccont(A_i)$, where $A_i=\{x \colon f_i(x) \ne f_i^{\prime}(x) \}$.}

Note that Lemmas~\ref{lemphd1} and ~\ref{lemphd2} imply the following.
\begin{itemize}
\item[($\star$)] {\rm (MA)} Assume that $F \subseteq \real^{\real}$ is finite and $A \sq \real $ is everywhere of second category.  
Then there exists a function
$g\colon A \to \real$ such that $g+ F \sq \sz(A)$ and
$\dom(g \cap B)$ is dense in some non-empty open interval $I_B$ for every blocking set $B$.
\end{itemize}


\proof
Let us consider the partition $\{H_n \colon n \in \omega\}$ of $\real$,
where $H_n$ is defined by $H_n=\{x \in \real \colon |X_x|=n\}$. Let
$G_n\sq\real$ be a maximal
open set such that $H_n$ is everywhere of second category in $G_n$. Such a set
can be easily constructed. Simply define $G_n$ as the interior of the set
$\real\setminus\bigcup_{I\in {\cal I}_n}I$, where ${\cal I}_n$ is the set of
all open intervals in which $H_n$ is meager.

We claim that for every $n<\omega$, there exists a function $g_n\colon
(G_n\cap H_n)\to \real$ such that $g_n + X=\{\la x,g_n(x)+y\ra\colon x\in
(G_n\cap H_n), \; \la x,y\ra\in X\} \in\szsets$ and $\bigcup_{n<\omega}g_n$
intersects every blocking set $B$.

First observe that this claim implies the conclusion of the theorem. Put
$g\colon\real\to\real$ to be an extension of $\bigcup_{n<\omega} g_n$ such that
$[g|(\real\setminus\bigcup_{n<\omega}G_n\cap H_n)]+X$ is an $\sz$-set. This
extension exists based on Corollary~\ref{p2-cor1}. Thus, $g+X$ is the union of
countable many $\sz$-sets. Consequently, $g+X \in\szsets$. Clearly, $g$
intersects every blocking set, so $g$ is almost continuous.

To complete the proof we need to show the above claim. Fix an $n<\omega$ and
put $A_n=(G_n\cap H_n)\cup\bigcup_{I\in {\cal I}_n}I$. The set $A_n$ is everywhere of
second category. Notice also that the part of $X$ contained in $(G_n\cap
H_n)\times \real$ can be covered by $n$ functions $f_1, \dots , f_n$ from $\real$ to $\real$. So, by ($\star$), there exists a function
$g_n^{\prime}\colon A_n \to \real$ such that $g_n^{\prime} + \{f_1 ,\dots ,f_n \} \sq\sz(A_n)$ and
$\dom(g_n^{\prime}\cap B)$ is dense in some non-empty open interval $I_B$ for every blocking set $B$.
Thus, if we define $g_n=g_n^{\prime}|(G_n\cap H_n)$ then $g_n + X\in\szsets$.

What remains to prove is that $\bigcup_{n<\omega}g_n$ intersects every
blocking set $B$. Notice that $I_B \cap G_n \ne \e$ for some $n$. Thus,
$g_n \cap B \ne \e$. Consequently, $\e\not=B\cap\bigcup_{n<\omega}g_n\sq B\cap g$. This finishes the proof.
\qed

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\bibitem{cie} K. Ciesielski,
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\bibitem{cie-nat} K. Ciesielski, T. Natkaniec,
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\bibitem{kk} K. Kuratowski,
{\it Topologie I}, Warszawa 1958.


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\end{document}


