A reconstruction theorem for locally moving groups acting on completely metrizable spaces

Edmund Ben-Ami¹

¹ Department of Mathematics Ben Gurion University of the Negev Beer Sheva, Israel

Fundamenta Mathematicae, Tome 209 (2010) no. 1, pp. 1-8

Résumé

Let $G$ be a group which acts by homeomorphisms on a metric space $X$. We say the action of $G$ is locally moving on $X$ if for every open $U \subseteq X$ there is a $g \in G$ such that $g {\restriction} X \neq {\rm Id}$ while $g {\restriction} (X \setminus U) = {\rm Id}$. We prove the following theorem:Theorem A. Let $X,Y$ be completely metrizable spaces and let $G$ be a group which acts on $X$ and $Y$ with locally moving actions. If the orbits of the action of $G$ on $X$ are of the second category in $X$ and the orbits of the action of $G$ on $Y$ are of the second category in $Y$, then $X$ and $Y$ are homeomorphic. A particular case of Theorem A gives a positive answer to a question of M. Rubin and J. van Mill who asked whether $X$ and $Y$ are homeomorphic whenever $G$ is strongly locally homogeneous on $X$ and $Y$.

Détail

DOI : 10.4064/fm209-1-1

Keywords: group which acts homeomorphisms metric space say action locally moving every subseteq there restriction neq while restriction setminus prove following theorem theorem completely metrizable spaces group which acts locally moving actions orbits action second category orbits action second category homeomorphic particular theorem gives positive answer question rubin van mill who asked whether homeomorphic whenever strongly locally homogeneous