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\begin{document}


\begin{center}
{\Large\bf ON SIERPI\'{N}SKI-ZYGMUND BIJECTIONS AND THEIR INVERSES}%
\end{center}

{\small\noindent Krzysztof Ciesielski,%
\footnote{1991 {\it Mathematics Subject Classification}.
Primary 26A15; secondary 03E35, 03E75
\newline \hspace*{1pc}
{\it Key words and phrases}. Sierpi{\'n}ski-Zygmund function; Blumberg
Theorem; inverse function; covering of category.
\newline \hspace*{1pc}
This work was partially supported by NSF Cooperative
Research Grant INT-9600548 with its Polish part being
financed by Polish Academy of Science PAN. The results were obtained
during a visit of the second author at West Virginia University. }
Department of Mathematics, West
 Virginia University, Morgantown, WV 26506-6310, USA
 (kcies@wvnvms.wvnet.edu)}

\medskip
{\small\noindent
Tomasz Natkaniec,$^1$ Department of Mathematics, Gda{\'n}sk
 University, Wita Stwosza 57, 80-952 Gda{\'n}sk, Poland
(mattn@ksinet.univ.gda.pl)}

\vspace{.25in}

\begin{abstract}
In the paper we will examine when the inverses of one-to-one
Sierpi\'{n}ski-Zygmund partial functions from $\real$ to $\real$ are
also
of Sierpi\'{n}ski-Zygmund type. We show that the existence of
a partial Sierpi\'{n}ski-Zygmund function $f$ with $f^{-1}$ being also
Sierpi\'{n}ski-Zygmund is independent of ZFC
axioms of set theory. However, there exists
a one-to-one Sierpi\'{n}ski-Zygmund injection $f\colon\real\to\real$
such that $f^{-1}$ is not Sierpi\'{n}ski-Zygmund.
This work is related to the investigation of
algebraic properties of the Sierpi\'{n}ski-Zygmund functions
discussed in~\cite{CN}.
\end{abstract}

\section{Preliminaries}


We will use the standard terminology and notation as
in~\cite{Ci:book}.
In particular, the functions will be identified with their graphs.
The family of all functions from a set $X$ into $Y$ will be denoted
by $Y^X$.
Ordinal numbers will be identified with the set of their predecessors
and
cardinal numbers with the initial ordinals.
Symbol $|X|$ will stand for the cardinality of a set $X$.
For a cardinal number $\kappa$
we will write $\kappa^+$ for its cardinal successor
while symbol $[X]^{\kkk}$ will denote the family of all subsets
$Y$ of $X$ with $|Y|=\kkk$.
The cardinality of the set $\mathR$ of real numbers is denoted by
$\co$.
%%For partial functions $f$ and $g$ from subsets of $\real$ into
$\real$
%%we will use symbol $[f=g]$ to denote the set
%%$\{x\in\dom(f)\cap\dom(g)\colon f(x)=g(x)\}$.
%Likewise for $[f>g]$, $[f \neq g]$, etc.

%     Similarly we define $[X]^{<\kkk}$ and $[X]^{\leq\kkk}$.
%$\cf(\kappa)$ for the cofinality of $\kappa$.
%Recall that a cardinal number $\kappa$ is regular, if
%$\kappa=\cf(\kappa)$.  For $f,g\in\RRR$ the notation $[f=g]$ means
the set
%$\{x\in\real\colon f(x)=g(x)\}$. Likewise for $[f>g]$, $[f \neq g]$,
etc.

For $X\subset\real$ we say that a function
$f\colon X\to\mathR$ is of {\it Sierpi\'{n}ski-Zygmund type\/}
(shortly, an $\sz$-function) if its restriction
$f|M$ is discontinuous for any set $M\in[X]^{\co}$.
The class of Sierpi\'{n}ski-Zygmund functions was introduced
in \cite{SZ}
in connection with a discussion around the Blumberg theorem
which says that for every function $f\colon\real\to\real$ there exists
a dense set $D\subset\real$ such that $f|D$ is continuous.
(For more information on this discussion see survey~\cite[Sec.
2]{KC:STAsurvey}.)
The family of all $\sz$-functions from a set $X\in[\real]^{\co}$ into
$\real$
will be denoted by $\sz$. The symbol $\C$ will stand for the family of
all continuous functions $f\colon\real\to\real$
and $\Cd$ for the family of all continuous
functions defined on $G_{\delta}$ subsets  of $\mathR$.
%with $|G|=\co$.
%We will sometimes abuse this notation by writing
%$f\in SZ$ and $f\in\C$ for partial functions
%$f\colon X\to\mathR$ with $X\subset\mathR$.

Recall that a function for $X\in[\real]^\co$ a function
$f\colon X\to\real$ is an \sz-function
if and only if $|f\cap g|<\co$ %$|[f=g]|<\co$
for every $g\in\Cd$. (See e.g.~\cite{SZ}.)
We will also use the following easy
characterization of one-to-one \sz-functions,
where symbol $\Cds$ stands for the class of nowhere constant functions
$g\in\Cd$.

\begin{Le}\label{L1}{\rm (\cite[Lemma 4.24]{CN})}
Assume that $f\colon X\to\real$ is one-to-one, where
$X\in[\real]^\co$.
Then $f\in\sz$ if and only if  $|f\cap g|<\co$ for every $g\in\Cds$.
%continuous nowhere constant function $g$ defined on a $G_{\delta}$
set
%\Qed
\end{Le}

Lemma~\ref{L1} is a consequence of a fact that for every continuous
function
$f\colon G\to\real$ with $G\subset\real$
there exists an open set $U\subset\real$ such that
$f|(G\cap U)$ is locally constant while
$f|(G\setminus U)$ is nowhere constant.

\section{The results}

It is easy to construct a one-to-one \sz-function
$f\colon\real\to\real$. On the other hand the existence of
an \sz-bijection $f\colon\real\to\real$ is not provable in ZFC, since
in the iterated perfect set %(Sacks)
model
there is no \sz-function from $\mathR$ {\it onto} $\mathR$.
(See~\cite{BCN}.)
In particular in this model there are no bijections
$f\colon\mathR\to\mathR$ such that $f$ and $f^{-1}$ are both of an
\sz-type.
Therefore in our investigation which one-to-one \sz-functions
$f\colon \real\to\real$ have $\sz$ inverses we are forced
either to consider partial functions or to work with some additional
set theoretical assumptions. Below we will use both of these
approaches.

We will start with noticing that there are ZFC examples of
$\sz$ injections $f\colon\real\to\real$ for which $f^{-1}\notin\sz$.

\begin{Th}\label{T1}
There exists a one-to-one \sz-function $f\colon [0,1]\to [0,1]$ such
that
$f^{-1}\colon f[0,1]\to\mathR$ is continuous. Moreover, if $C\subset
[0,1]$
is nowhere dense then so is $f[C]$.
\end{Th}
\pf
Let $g\colon [0,1]\to[0,1]$ be a continuous nowhere constant function
with
$|g^{-1}(y)|=\co$ for every $y\in [0,1]$.
(For a construction of such a function see e.g. \cite[pp. 148--
150]{AB}.)
Fix enumerations $\{g_{\alpha}\colon\alpha <\co\}$ of $\Cds$
and $\{x_{\alpha}\colon\alpha <\co\}$ of $[0,1]$.
By transfinite induction for every $\alpha<\co$ choose
\[
y_{\alpha}\in g^{-1}(x_{\aaa})\setminus
\left(\{ y_{\bbb}\colon \bbb<\aaa\}\cup \{
g_{\bbb}(x_{\alpha})\colon\bbb\leq\aaa\}\right)
\]
and put $f(x_{\aaa})=y_{\aaa}$. Then
$f$ is one-to-one and, by Lemma~\ref{L1}, it is $\sz$. Moreover,
$f^{-1}\subset g$. Thus $f^{-1}$ is continuous.


To see the the additional property
suppose that $C\subset [0,1]$ is nowhere dense.  Then
$g^{-1}(C)$ is a closed nowhere dense subset of $[0,1]$,
since continuous functions map connected sets into connected sets
and $g$ is nowhere constant.
Thus $f[C]\subset g^{-1}(C)$ is nowhere dense.
\Qed


\begin{Co} \label{Co1}
There exists a one-to-one \sz-function $g\colon\real\to\real$ such
that
$g^{-1}\notin\sz$.
\end{Co}

\pf Let $h$ be a homeomorphism between $\real$ and $(0,1)$
and $f$ be from Theorem~\ref{T1}. Then $g=f\circ h$
has the desired properties. \Qed

\begin{Rem}
If $f\colon\mathR\to\mathR$ is an $\sz$ injection then
$f^{-1}$ is not
continuous on any perfect set $P\subset f[\real]$.
In particular, if $f$ is the function from Theorem~\ref{T1}
then $f[0,1]$ does not contain any perfect set.
\end{Rem}

\pf
If $f^{-1}$ is continuous on a perfect set $P\subset f[\real]$
then it is a homeomorphism between $P$ and a perfect set
$T=f^{-1}(P)$. Then $f|T$ is continuous, contrary to $f\in\sz$.
\Qed

Next we will show that the function $f$ from Corollary~\ref{Co1}
can be a bijection under appropriate set theoretical assumption
guaranteeing existence of \sz-surjection $f\colon\real\to\real$.
For this we will use the following lemma.

%Note that in the analogous way we can prove the following lemma.

\begin{Le}\label{L2}
Assume that $\real$ cannot be covered by less than $\co$
many meager sets.
Then for every residual set
$G\subset\mathR$ and every $Y\in [\mathR]^{\co}$ there exists an
\sz-bijection $g\colon G\to Y$.
\end{Le}

\pf
Let $\{g_{\alpha}\colon\alpha <\co\}$,
$\{y_{\alpha}\colon\alpha <\co\}$, and
$\{x_{\alpha}\colon\alpha <\co\}$ be the enumerations
of $\Cds$, $Y$ and $G$, respectively. By induction on
$\alpha<\co$ we will construct the sequences
$\{\{a_\alpha,b_\alpha\}\in[G]^2\colon\alpha<\co\}$ and
$\{\{c_\alpha,d_\alpha\}\in[Y]^2\colon\alpha<\co\}$
aiming for defining $g$ by $g(a_\alpha)=c_\alpha$ and
$g(b_\alpha)=d_\alpha$.
The construction is done maintaining the following inductive
conditions
for every $\alpha<\co$.
\begin{description}
\item{(i)}  $a_\alpha=x_\alpha$ if
$x_\alpha\not\in\bigcup_{\xi<\alpha}\{a_\xi,b_\xi\}$.
Otherwise $a_\alpha$ is an arbitrary element of
$G\setminus \bigcup_{\xi<\alpha}\{a_\xi,b_\xi\}$.

\item{(ii)} $d_\alpha=y_\alpha$
if $y_\alpha\not\in\bigcup_{\xi<\alpha}\{c_\xi,d_\xi\}$.
Otherwise $d_\alpha$ is an arbitrary element of
$Y\setminus \bigcup_{\xi<\alpha}\{c_\xi,d_\xi\}$.

\item{(iii)}
$b_{\aaa}\in
G\setminus\left(\{a_\alpha\}\cup\bigcup_{\xi<\alpha}\{a_\xi,b_\xi\}
\cup \bigcup_{\xi<\alpha}g_\xi^{-1}(d_\alpha)\right)$.

\item{(iv)}
$c_{\aaa}\in Y\setminus
\left(\{d_\alpha\}\cup\bigcup_{\xi<\alpha}\{c_\xi,d_\xi\}
\cup  \{g_\xi(a_\alpha)\colon \xi<\alpha\}\right)$.
\end{description}
The choice as in (iii) can be made since the set
$\bigcup_{\xi<\alpha}g_\xi^{-1}(d_\alpha)$
is a union of less
than continuum many nowhere dense sets, so
it cannot cover $G$.
It is easy to see that $g$ constructed that way
is a well defined one-to-one function from $G$ onto $Y$.
It is $\sz$ by Lemma~\ref{L1}. Indeed, let
$g_\alpha\in\Cds$ and $x\in\real$ be such that $g(x)=g_\alpha(x)$.
Then there exist $\xi<\co$ such that $x\in\{ a_{\xi}, b_{\xi}\}$.
To finish the argument it is enough to notice that
$\xi\leq\aaa$. But if $\xi>\aaa$ then, by (iv),
$g(a_{\xi})=c_\xi\neq g_\alpha(a_{\xi})$
and, by (iii), $g(b_{\xi})=d_\xi\neq g_\alpha(b_{\xi})$
implying that $g(x)\neq g_\alpha(x)$.
\Qed


\begin{Co}\label{C3}
If $\real$ cannot be covered by less than $\co$
many meager sets then there exists an \sz-bijection
$h\colon\mathR\to\mathR$ such that $h^{-1}$ is not an \sz-function.
\end{Co}
\pf
Let $f$ be as in Theorem~\ref{T1} and let $C\subset [0,1]$ be a
Cantor set.
Apply Lemma~\ref{L2} for $G=\mathR\setminus C$ and $X=\mathR\setminus
f[C]$
to find an \sz-bijection $g\colon G\to X$.
Then $h=g\cup f|C$ is an SZ bijection
and $h^{-1}|f[C]$ is continuous. Thus $h^{-1}$ is not an \sz-function.
\Qed

The next theorem shows in particular
that under appropriate set theoretical assumptions
there exist also \sz-bijections $f\colon\real\to\real$ such that
$f^{-1}$ is also an \sz-function.



\begin{Th}\label{T2}
Assume that $\real$ cannot be covered by less than $\co$
many meager sets. Then
%for every residual set $X\subset \mathR$
there exists an
$\sz$ bijection $f\colon\mathR\to\mathR$ such that $f^{-1}=f$.
\end{Th}
\pf
Let $\{g_{\alpha}\colon\alpha <\co\}$ and $\{r_{\alpha}\colon\alpha
<\co\}$
be the enumerations of $\Cds$ and $\real$, respectively.
We will construct by induction on $\alpha<\co$
a family of pairwise disjoint %two-element
sets
$\{\{a_{\aaa},b_{\aaa}\}\in [\real]^2\colon \aaa<\co\}$ such that
the following conditions hold for every
$\alpha<\co$:

\begin{description}
\item{(i)}
$r_{\aaa}\in\bigcup_{\xi\leq\aaa}\{ a_{\xi},b_{\xi}\}$; and,
\item{(ii)}
$\la a_{\aaa},b_{\aaa}\ra\notin
\bigcup_{\xi\leq\aaa}(g_{\xi}\cup g_{\xi}^{-1})$,
 where $g_{\xi}^{-1}=\{\la y,x\ra\colon y=g_{\xi}(x)\}$.
\end{description}
The construction is aimed to define $f$ by putting
$f(a_\alpha)=b_\alpha$ and $f(b_\alpha)=a_\alpha$
for every $\alpha<\co$. This clearly will imply that $f^{-1}=f$,
while (i) will guarantee that $f$ is defined for all real numbers.

To see that the choice of such a sequence is possible, assume that
for some $\alpha<\co$ the sequence
$\{\{a_\xi,b_\xi\}\in [\real]^2\colon \xi<\aaa\}$
is already defined. Then we
define $a_{\aaa}$ as $r_{\aaa}$
if $r_{\aaa}\not\in\bigcup_{\xi<\aaa}\{a_{\xi},b_{\xi}\}$
and as an arbitrary element of the set
$\real\setminus \bigcup_{\xi<\aaa}\{a_{\xi},b_{\xi}\}$
otherwise. Now, to have (ii) it is enough to choose $b_\alpha$ such
that
\begin{equation}\label{eq1}
b_{\aaa}\in\real\setminus (\{ a_{\aaa}\}\cup \bigcup_{\xi<\aaa}\{
a_{\xi},b_{\xi}\}\cup \bigcup_{\xi\leq\aaa}(\{ g_{\xi}(a_{\aaa})\}\cup
g^{-1}_{\xi}(a_{\aaa}))),
\end{equation}
which is possible since according to our assumption
less than continuum many nowhere dense sets
$g_{\xi}^{-1}(a_{\aaa})$ does not cover $\real$.

To verify that $f\in\sz$ by Lemma~\ref{L1} it is enough to show that
$|f\cap g|<\co$ for any $g\in\Cds$. So, fix a $g\in\Cds$
and let $x\in\real$ be such that $f(x)=g(x)$.
Then there exist $\aaa,\xi<\co$ such
that $g=g_{\aaa}$ and $x\in\{ a_{\xi}, b_{\xi}\}$.
Observe that
$\xi\leq\aaa$. Indeed, if $\xi>\aaa$ then, by (\ref{eq1}),
$g_{\aaa}(a_{\xi})\neq
b_{\xi}=f(a_{\xi})$ and $g_{\aaa}(b_{\xi})\neq a_{\xi}=f(b_{\xi})$, as
$b_{\xi}\notin g_{\aaa}^{-1}(a_{\xi})$,
 a contradiction.
Therefore $|f\cap g|\leq|\alpha|<\co$.
\Qed

Theorem~\ref{T2} and Corollary~\ref{C3} show that under appropriate
set
theoretic assumptions there are $\sz$ bijections $f$ from $\real$ onto
$\real$ such that $f^{-1}$ is $\sz$ and such that
$f^{-1}$ is not $\sz$. We also know that these results cannot be
obtained in
ZFC. (See~\cite{BCN}.) On the other hand,
by Corollary~\ref{Co1}, there is a ZFC example of
an $\sz$ injection $g\colon\real\to\real$ with
$g^{-1}\notin\sz$. Thus, to complete the picture, it is
reasonable to ask the question whether there is a ZFC example of
an $\sz$ injection $g\colon\real\to\real$ for which
$g^{-1}\in\sz$. The negative answer for this question
will be deduced from the following characterization.

\begin{Th}\label{T4}
The following conditions are equivalent:
\begin{description}
\item{\rm (i)}
for each bijection $f$ from a set
$X\in[\real]^\co$ onto a set $Y\in[\real]^\co$ either $f\not\in\sz$
or $f^{-1}\not\in\sz$;
\item{\rm (ii)}
there exists a family of functions
${\cal H}\subset\Cd$ of cardinality less than $\co$
such that
%with $|{\cal H}|<{\co}$ with the property that
$\mathR^2=\bigcup{\cal H}\cup \bigcup\{h^{-1}\colon h\in {\cal H}\}$,
where $h^{-1}=\{\la y,x\ra\colon y=h(x)\}$.
\end{description}
\end{Th}
\pf
(ii)$\Rightarrow$(i) Assume that there exists ${\cal H}\subset\Cd$
with $|{\cal H}|=\kappa<\co$ satisfying (ii).
First note that this implies that
\begin{equation}\label{eq2}
\co=\kappa^+.
\end{equation}
(This is an easy generalization of property $P_1$ from~\cite{Si:CH}.
See also~\cite[Thm 6.1.8]{Ci:book}.)

Indeed, by way of contradiction assume that $\co>\kappa^+$
and take $X\subset\real$ with $|X|=\kappa^+$.
Then $Y=\bigcup\{h[X]\colon h\in{\cal H}\}$
has cardinality
%less then
$\leq\kappa^+<\co$. Take $y\in\real\setminus Y$
and $x\in X\setminus\{h(y)\colon h\in{\cal H}\}$.
(This can be done, since
$|\{h(y)\colon h\in{\cal H}\}|\leq|{\cal H}|=\kappa<|X|$.)
Then $\la x,y\ra\notin\bigcup{\cal H}$ by the choice of $y$
and $\la x,y\ra\notin\bigcup\{h^{-1}\colon h\in {\cal H}\}$
by the choice of $x$, a contradiction.

To show (i) take a bijection $f\colon X\to Y$ for some
$X,Y\in [\mathR]^{\co}$.
Then, by the Pigeon Hole Principle and the regularity of $\kappa^+$,
there exists $h\in {\cal H}$ such that either
$|f\cap h|=\co$, or
$|f^{-1}\cap h|=|f\cap h^{-1}|=\co$. Thus either $f\not\in SZ$ or
$f^{-1}\not\in SZ$.

(i)$\Rightarrow$(ii) Let $\Cd=\{ g_{\aaa}\colon \aaa<\co\}$ and for
each
$\aaa<\co$ let ${\cal H}_{\aaa}=\{ g_{\bbb}\colon\bbb\leq\aaa\}$.
Suppose $\neg$(ii).
 We choose inductively a sequence
$\{\la x_{\aaa},y_{\aaa}\ra\colon{\aaa<\co}\}$ of points
such that
\begin{enumerate}
\item
$x_{\aaa}\neq x_{\bbb}$ for $\bbb<\aaa$;
\item
$y_{\aaa}\neq y_{\bbb}$ for $\bbb<\aaa$;
\item
$\la x_{\aaa},y_{\aaa}\ra \in \mathR^2\setminus \left(\bigcup{\cal
H}_{\aaa}\cup\bigcup\{ h^{-1}\colon h\in {\cal H}_{\aaa}\}\right)$.
\end{enumerate}
Such a choice is possible by the negation of (ii).

Put $X=\{ x_{\aaa}\colon \aaa<\co\}$, $Y=\{ y_{\aaa}\colon
\aaa<\co\}$ and
$f(x_{\aaa})=y_{\aaa}$. Then $X,Y\in [\mathR]^{\co}$ and $f\colon
X\to Y$ is a
bijection.
To prove that $f$ is $\sz$ function observe that for each $\aaa<\co$
we have $|f\cap g_{\aaa}|<\co$ since
$\{x\in X\colon f(x)=g_{\aaa}(x)\}\subset\{ x_{\bbb}\colon
\bbb\leq\aaa\}$.
Similarly, for every
$\aaa<\co$ we have $|f^{-1}\cap g_{\aaa}|<\co$ since
$\{\beta\colon \la y_\beta,x_\beta\ra\in g_\alpha\}\subset\alpha$. So
$f^{-1}$ is an \sz-function,
contrary to (i).
\Qed

\begin{Co}
It is consistent with ZFC that there is no bijection $f$ from a set
$X\in[\real]^\co$ onto a set $Y\in[\real]^\co$ such that
both $f\in\sz$ and $f^{-1}\in\sz$.
\end{Co}

\pf It follows from Theorem~5.1 and Corollary~3.3
of a recent paper of Stepr\={a}ns~\cite{step}
(see also Definition~2.3 and Lemma~2.2) that the condition (ii)
from Theorem~\ref{T4} is consistent with ZFC axioms. \Qed

\begin{thebibliography}{22}
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accessible
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http://www.math.wvu.edu/homepages/kcies/STA/STA.html})

\bibitem{AB}  A.~M.~Bruckner, {\it Differentiation of real
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\bibitem{Ci:book}  K.~Ciesielski,
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\bibitem{KC:STAsurvey}  K.~Ciesielski,
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\bibitem{CN}
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 class of Sierpi{\'n}ski-Zygmund functions}, Topology Appl. {\bf 79}
(1997),
75--99. (Preprint$^\star$ available.)

\bibitem{Si:CH}  W.~Sierpi\'nski, {\it Hypoth\`ese du continu},
Monografie
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\bibitem{SZ}
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\bibitem{step} J.~Stepr\={a}ns, {\it Decomposing with smooth sets},
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\end{thebibliography}

\end{document}
