Quasi-orbit spaces associated to $T_0$-spaces
Volume 211 / 2011
Abstract
Let $G\subset\mbox{Homeo}(E)$ be a group of homeomorphisms of a topological space $E$. The class of an orbit $O$ of $G$ is the union of all orbits having the same closure as $O$. Let $E/\widetilde{G}$ be the space of classes of orbits, called the quasi-orbit space. We show that every second countable $T_0$-space $Y$ is a quasi-orbit space $E/\widetilde{G}$, where $E$ is a second countable metric space. The regular part $X_0$ of a $T_0$-space $X$ is the union of open subsets homeomorphic to $\mathbb{R}$ or to $\mathbb{S}^1$. We give a characterization of the spaces $X$ with finite singular part $X-X_0$ which are the quasi-orbit spaces of countable groups $G\subset\mbox{Homeo}_+(\mathbb{R})$. Finally we show that every finite $T_0$-space is the singular part of the quasi-leaf space of a codimension one foliation on a closed three-manifold.
