A reconstruction theorem for locally moving groups acting on completely metrizable spaces
Tom 209 / 2010
Fundamenta Mathematicae 209 (2010), 1-8
MSC: Primary 54E50; Secondary 22F05.
DOI: 10.4064/fm209-1-1
Streszczenie
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 $Y$.
