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\author{Tomasz Natkaniec}

\title{The density topology can be not extraresolvable}




\begin{document}
{
\renewcommand{\thefootnote}{}
 \footnotetext{AMS Subject Classification (1991): 28A05; 03E35; 54A25}
} {\renewcommand{\thefootnote}{} \footnotetext{Key words: density
topology; extraresolvable space; Covering Property Axiom CPA.} }

\date{Feb. 10, 2004}

\maketitle

\begin{abstract}
We note that in some models of ZFC the density topology can be not
extraresolvable.
\end{abstract}

A topological space $X$ is called {\it extraresolvable} if there
exists a family $\D$ of dense subsets of $X$ such that
$|\D|>
\Delta(X)$, where $\Delta(X)$  is the dispersion character of $X$,
and $D\cap D'$ is nowhere dense whenever $D,D'\in\D$ and $D\neq
D'$~\cite{GMT}.

Assuming Martin Axiom (MA) A.~Bella proved that the real line
$\mathR$ with the density topology $\T_d$ is
extraresolvable~\cite{Be}. He also asked whether this fact can be
proved in ZFC. In this note we answer this question in negative.

We use standard terminology. In particular, $|X|$ denotes the
cardinality of a set $X$. For a topological space $X$ the
dispersion character of $X$ is the smallest cardinality of a
non-empty subset of $X$. If $\J$ is an ideal of subsets of a set
$X$ then $\cf(\J)$ is the smallest cardinality of a basis of $\J$,
i.e., the family $\J_0\subset\J$ such that each $J\in\J$ is
contained in some $J_0\in\J_0$. $\shr(\J)$ is the minimal cardinal
$\kkk$ such that in each set $A\subset X$, $A\not\in\J$, there is
$A_0\subset A$ with $|A_0|\le\kkk$ and $A_0\not\in\J$.

$\T_d$ denotes the (Lebesgue) density topology on $\mathR$. It is
well known that a set $A\subset\mathR$ is nowhere dense in the
density topology iff it has a measure null. (See \cite{Ox}.) The
ideal of null sets in $\mathR$ is denoted by $\N$.

We start with the following strengthening of \cite[Proposition
3]{Be}.
\begin{lem}\label{l1}
If $X$ is a topological space, $\nwd$ is the ideal of nowhere
dense sets in $X$ and $|X|^{\shr(\nwd)}\le\Delta(X)$, then $X$ is
not extraresolvable.
\end{lem}
\proof Suppose $X$ is extraresolvable. Let $\{ D_{\aaa}\colon
\aaa<\kkk\}$ be a family of dense sets in $X$ such that
$\Delta(X)<\kkk$ and $D_{\aaa}\cap D_{\bbb}\in\nwd$ whenever
$\aaa<\bbb<\kkk$. For each $\aaa<\kkk$ choose a set
$E_{\aaa}\subset D_{\aaa}$ such that $|E_{\aaa}|\le \shr(\nwd)$
and $E_{\aaa}\not\in\nwd$. Since $|\{ E_{\aaa}\colon
\aaa<\kkk\}|\le |X|^{\shr(\nwd)}\le\Delta(X)<\kkk$, there are
$\aaa<\bbb<\kkk$ for which $E_{\aaa}=E_{\bbb}$. Then
$E_{\aaa}\subset D_{\aaa}\cap D_{\bbb}$, so $D_{\aaa}\cap
D_{\bbb}\not\in\nwd$, a contradiction. \qed

\begin{thm}
There exists a model of ZFC in which the space $(\mathR,\T_d)$ is
not extraresolvable.
\end{thm}
\proof We will work in ZFC with three additional axioms:
\begin{description}
\item[(A1) ] $\co=\ooo_2$;
\item[(A2) ] $2^{\ooo_1}=\ooo_2$;
\item[(A3) ] $\cf(\N)=\ooo_1$.
\end{description}

It is known that the axioms (A1), (A2), (A3) are consistent with
ZFC. In fact, (A1) and (A3) are consequences of so called {\it
Covering Property Axiom} CPA, introduced by K.~Ciesielski and
J.~Pawlikowski. Moreover, it is known that under CPA $2^{\ooo_1}$
can be arbitrarily large. (See \cite{CP}.)

First observe that (A3) implies that  $\shr(\N)=\ooo_1$. In fact,
let $\{ H_{\aaa}\colon\aaa<\ooo_1\}$ be a basis of the ideal $\N$,
i.e., for each $N\in\N$ there is $\aaa<\ooo_1$ with $N\subset
H_{\aaa}$. Assume $A\not\in\N$. For each $\aaa<\ooo_1$ choose
$x_{\aaa}\in A\setminus \bigcup_{\bbb<\aaa}H_{\aaa}$. Then $A_0=\{
x_{\aaa}\colon \aaa<\ooo_1\}$ is a subset of $A$ and
$A_0\not\in\N$.

It is clear that $\Delta(X)=\co$, thus (A1) implies
$\Delta(X)=\ooo_2$. Now we have
$|X|^{\shr(\nwd)}=\co^{\shr(\N)}=\co^{\ooo_1}=2^{\ooo_1}$, so (A2)
yields $|X|^{\shr(\nwd)}=\ooo_2\le\Delta(X)$, and,  by
Lemma~\ref{l1}, $X$ is not extraresolvable. \qed

\begin{thebibliography}{M-N}
\bibitem[Be]{Be}
 A.~Bella, {\it The density topology is extraresolvable}, Atti Sem. Mat. Fis. Univ. Modena, 48 (2000), 495-498.
\bibitem[CP]{CP}
K.~Ciesielski, J.~Pawlikowski, {\it Covering Property Axiom CPA},
to appear in Cambridge Tracts in Mathematics, Cambridge Univ.
Press. (Available in electronic form from {\it Set Theoretic
Analysis Web Page:}
http://www.math.wvu.edu/\~{}kcies/STA/STA.html)
\bibitem[GMT]{GMT}
S.~Garcia-Ferreira, V.I.~Malykhin, A.H.~Tomita, {\it
Extraresolvable spaces}, Topology Appl. {\bf 101} (2000),
257--271.
\bibitem[Ox]{Ox}
J.C. Oxtoby, {\it Measure and Category}, Springer-Verlag, New
York, 1971.
\end{thebibliography}

\bigskip
\noindent \sc Tomasz Natkaniec,\\
 Institute of Mathematics, Gda\'{n}sk University,\\
 Wita Stwosza 57, 80--952 Gda\'{n}sk, Poland.\\
 E-mail: mattn@math.univ.gda.pl

\end{document}
