\documentclass{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \newtheorem{thm}{Theorem} \newtheorem{pr}{Problem} \newtheorem{re}[thm]{Remark} \newtheorem{lem}[thm]{Lemma} \newtheorem{ex}[thm]{Example} \newtheorem{df}[thm]{Definition} \newtheorem{cor}[thm]{Corollary} \newtheorem{con}[thm]{Conjecture} \newcommand\mathR{{\mathbb R}} \newcommand\mathQ{{\mathbb Q}} \newcommand{\RRR}{\mathR^{\mathR}} \newcommand{\co}{{\mathfrak c}} \newcommand{\bo}{{\mathfrak b_{\kkk}}} \newcommand{\dd}{{\mathfrak d_{\kkk}}} \newcommand{\rr}{{\mathfrak r_{\kkk}}} \newcommand{\MM}{{\mathfrak M}} \newcommand{\aaa}{{\alpha}} \newcommand{\bbb}{{\beta}} \renewcommand{\ggg}{{\gamma}} \newcommand{\ddd}{{\delta}} \newcommand{\fff}{{\varphi}} \newcommand{\kkk}{{\kappa}} \renewcommand{\lll}{{\lambda}} \newcommand{\ooo}{{\omega}} \newcommand{\eee}{{\varepsilon}} \newcommand{\card}{\mbox{\/\rm card}\,} \newcommand{\add}{\mbox{\/\rm add}\,} \newcommand{\cov}{\mbox{\/\rm cov}\,} \newcommand{\non}{\mbox{\/\rm non}\,} \newcommand{\shr}{\mbox{\/\rm shr}\,} \newcommand{\LIM}{\mbox{\/\rm LIM}\,} \newcommand{\LIMK}{\mbox{\/\rm LIM}_{\kkk}} \newcommand{\cl}{\mbox{\/\rm cl}\,} \newcommand{\cf}{\mbox{\/\rm cf}\,} \newcommand{\F}{{\cal F}} \newcommand{\I}{{\cal I}} \newcommand{\B}{{\cal B}} \newcommand{\M}{{\cal M}} \newcommand{\T}{{\cal T}} \newcommand{\C}{{\cal C}} \newcommand{\A}{{\cal A}} \newcommand{\Q}{{\cal Q}} \newcommand{\J}{{\cal J}} \renewcommand{\L}{{\cal L}} \newcommand{\D}{{\cal D}} \renewcommand{\P}{{\cal P}} \newcommand{\N}{{\cal N}} \renewcommand{\int}{\mbox{\/\rm int}\,} \newcommand{\nwd}{{\rm NWD}} \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}