% The density continuous functions are not generated. %\documentstyle[12pt,rae]{article} \documentclass{rae} \pagestyle{myheadings} %\markright{The density topology is not generated} %\input{latexmacros.tex} %Commonly used symbols. %Sets and spaces % Use blackboard bold for the normal real subsets \newfont{\amssymten}{msbm10 scaled\magstep1} \newfont{\amssymnine}{msbm9} % natural numbers \newcommand{\nats}{\mbox{{\amssymten N}}} \newcommand{\smallnats}{\mbox{{\amssymnine N}}} % rational numbers \newcommand{\rationals}{\mbox{{\amssymten Q}}} \newcommand{\smallrationals}{\mbox{{\amssymnine Q}}} % real numbers \newcommand{\reals}{\mbox{{\amssymten R}}} \newcommand{\smallreals}{\mbox{{\amssymnine R}}} \newcommand{\into}{\mbox{$\rightarrow$}} \newcommand{\e}{\mbox{$\varepsilon$}} \renewcommand{\d}{\mbox{$\delta$}} %Operations on sets \newcommand{\interior}[1]{\mbox{$\mbox{{\rm int}}(#1)$}} \newcommand{\cl}[1]{\mbox{$\overline{#1}$}} \newcommand{\dist}[2]{\mbox{{\rm dist}$(#1,#2)$}} \newcommand{\comp}[1]{\mbox{$#1^c$}} \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_#1$}} \newcommand{\Cup}{\bigcup} \newcommand{\Cap}{\bigcap} %Operations on functions \newcommand{\supp}[1]{\mbox{{\rm supp}$#1$}} %Inverse of a function \newcommand{\inv}[1]{\mbox{$#1^{-1}$}} %Types of sets and functions \newcommand{\Fs}{\mbox{$\mbox{\bf{F}}_\sigma$}} \newcommand{\Gd}{\mbox{$\mbox{\bf{G}}_\delta$}} %TeXish necessities %Macro to include labels in the margins. \newcommand{\marginlabels}[1]{\def\margintags{#1}} \marginlabels{n} \def\nolabels{n} \newcommand{\lab}[1] {\if\margintags\nolabels \label{#1} \else \label{#1}\marginpar{#1} \fi} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{example}{Example} \newtheorem{claim}{Claim} \newtheorem{definition}{Definition} \marginlabels{n} \title{The density topology is not generated} \author{Krzysztof Ciesielski\footnote{Received support from a West Virginia University Senate research grant.}, Department of Mathematics, West Virginia University, Morgantown, WV 26506 \and Lee M. Larson\footnote{Partially supported by a University of Louisville research grant.}, Department of Mathematics, University of Louisville, Louisville, KY 40292} \date{} \newcommand{\C}{\mbox{$\cal C$}} \newcommand{\CD}{\mbox{$\C_D$}} \newcommand{\SD}{\mbox{$S_D$}} \newcommand{\bstar}{\mbox{Baire*1}} \newcommand{\ambig}{\mbox{$\Fs\cap\Gd$}} \newcommand{\ind}{\mbox{$\Lambda$}} \newcommand{\cover}{\mbox{\cal S}} \begin{document} \setcounter{page}{522} \maketitle The density continuous functions, \CD, are the real functions $f\colon(\reals,\tau_d)\to(\reals,\tau_d)$ which are continuous when the density topology \[ \tau_d=\{A\subset\reals\colon \mbox{ every $a\in A$ is a density point of } A\} \] is used on both the domain and the range. (For more details on the density topology see \cite{Oxtoby:MeasCat} or \cite{Tall:DenseTop}.) It has recently been shown that the density continuous functions do not form a vector space and there are monotone, and even $C^\infty$ functions which are not density continuous \cite{CL:SpDensCont}. On the other hand, all locally convex functions are density continuous \cite{CL:SpDensCont} and density continuous functions are in the class \bstar\ \cite{CLO:DensContCont}. The purpose of this note is to answer a question posed by Krzysztof Ostaszewski \cite{KO:SemiDensity} related to the properties of the set of density continuous functions, \CD, when viewed as a semigroup. This question is: \begin{description} \item[Query] Is the density topology generated? \end{description} It turns out that this question can be answered negatively using a characterization of the level sets of a density continuous function. A topological space $(X,\tau)$ is {\em generated} if, whenever $\tau'$ is another topology on $X$, with the property that the set of continuous selfmaps \[f:(X,\tau')\into (X,\tau')\] contains the set of continuous selfmaps $f:(X,\tau)\into (X,\tau)$, then it is also true that $\tau'\supset\tau$. The generated spaces are characterized by the following theorem of Warndof \cite{JW:unique}. (Compare also \cite[Definition 2.2, p. 198]{KM:SemiSurvey}.) \begin{theorem}\lab{thm:subbase} A Hausdorff topological space $(X,\tau)$ is generated if, and only if, the class of complements of level sets of its continuous selfmaps is a subbase for $\tau$. \end{theorem} Therefore, to show that a topology is not generated, it suffices to show that the level sets of the continuous selfmaps under that topology do not form a subbase for the closed sets of that topology. Our argument is based upon the following facts \cite{CL:LevelSets}. \begin{theorem}\lab{thm:lattice} \CD\ is a lattice. \end{theorem} \begin{theorem}\lab{thm:asssets} The associated sets of density continuous functions, i.e., the sets in the form $\reals\setminus f^{-1}(a)$ for $f\in\CD$ and $a\in\reals$, are precisely the density open sets which are in $\Fs\cap\Gd$. \end{theorem} In what follows $\interior{A}$ and $\cl{A}$ stand for the interior and closure of $A\subset\reals$ with respect to the ordinary topology on \reals. \begin{lemma}\lab{lem:interior} If $f\in C_D$ and $a\in [-\infty,\infty)$, then $\interior{\{ f>a\}}$ is dense in $\{ f>a\}$. \end{lemma} Proof. Let $G=\{ f>a\}$. Assume that $\interior{G}$ is not dense in $G$. Then there is an open interval $I$ such that $I\cap G\ne\emptyset$, but $I\cap\interior{G}=\emptyset$. Since both $G$ and \comp{G}\ are \Gd\ sets according to Theorem \ref{thm:asssets}, the Baire category theorem implies that $G$ must be nowhere dense in $I$. We see that $\cl{I}\cap\cl{G}$ is a nowhere dense perfect subset of $\cl{I}$. It is clear that $G$ is dense in $\cl{I}\cap\cl{G}$. Let $J$ be a component of $\cl{I}\cap\comp{(\cl{G})}$. Since $G$ is density open, we see that $\cl{J}\subset\comp{G}$. Using the fact that \cl{G}\ is nowhere dense in $I$, this implies that \comp{G}\ is dense in $\cl{G}\cap I$. But, we have established that both $G\cap\cl{I}$ and $\comp{G}\cap\cl{I}$ are dense, disjoint, \Gd\ subsets of $\cl{G}\cap I$, which violates the Baire category theorem. This contradiction proves Lemma \ref{lem:interior}. \begin{theorem}\lab{thm:notgenerated} The density topology on \reals\ is not generated. \end{theorem} Proof. Let $\tau=\{\reals\setminus f^{-1}(0):f\in C_D\}$. It suffices to show that $\tau$ is not a subbase for the density topology. To do this, we note that since Theorem \ref{thm:lattice} shows $C_D$ is a lattice, $\tau$ is closed under finite intersections. Therefore, it suffices to show that $\tau$ is not a basis for the density topology. Let $E=\reals\setminus\rationals$. It follows at once from Lemma \ref{lem:interior} that $E$ cannot be written as a union of elements from $\tau$ because $\interior{E}=\emptyset$. But, $E$ has full measure in \reals, so it is open in the density topology. This contradiction proves the theorem. \bigskip By the above theorem the reals equipped with the density topology is an example of a completely regular not generated topological space whose semigroup of continuous selfmaps has the inner authomorphism property \cite{KO:SemiDensity}. It is the only such example known to the authors. In particular, the implication in the following theorem of Magill \cite{KM:SemiSurvey} \begin{quote} If a completely regular space $X$ is generated, then $X$ has the inner automorphism property. \end{quote} cannot be reversed. \bibliographystyle{plain} %\bibliography{nongen} \begin{thebibliography}{1} \bibitem{CL:LevelSets} Krzysztof Ciesielski and Lee Larson. \newblock Level sets of density continuous functions. \newblock {\em Proc. Amer. Math. Soc.}, to appear. \bibitem{CL:SpDensCont} Krzysztof Ciesielski and Lee Larson. \newblock The space of density continuous functions. \newblock {\em Acta Math. Hung.}, to appear. \bibitem{CLO:DensContCont} Krzysztof Ciesielski, Lee Larson, and Krzysztof Ostaszewski. \newblock Density continuity versus continuity. \newblock {\em Forum Mathematicum}, 1:1--11, 1989. \bibitem{KM:SemiSurvey} K.~D. Magill, Jr. \newblock A survey of semigroups of continuous selfmaps. \newblock {\em Semigroup Forum}, 11:189--282, 1975/76. \bibitem{KO:SemiDensity} K.~Ostaszewski. \newblock Semigroups of density continuous functions. \newblock {\em Real Anal. Exch.}, 14(1):104--114, 1988-89. \bibitem{Oxtoby:MeasCat} J.~C. Oxtoby. \newblock {\em Measure and Category}. \newblock Springer-Verlag, 1971. \bibitem{Tall:DenseTop} F.~D. Tall. \newblock The density topology. \newblock {\em Pacific Math. J.}, 62(1):275--284, 1976. \bibitem{JW:unique} Joseph~C. Warndof. \newblock Topologies uniquely determined by their continuous selfmaps. \newblock {\em Fund. Math.}, 66:25--43, 1969/70. \end{thebibliography} \end{document}