Quasi-orbit spaces associated to $T_0$-spaces

C. Bonatti ¹   ; H. Hattab ²   ; E. Salhi ³

¹ IMB, UMR 5584 du CNRS 9 av. Alain Savary 21000 Dijon, France
² Institut Supérieur d'Informatique et du Multimedia Route de Tunis km 10 B.P. 242, Sfax 3021, Tunisia
³ Département de Mathématiques Faculté des Sciences de Sfax Route de Soukra km 3.5 B.P. 802, Sfax 3018, Tunisia

Fundamenta Mathematicae, Tome 211 (2011) no. 3, pp. 267-291

Résumé

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.

Détail

DOI : 10.4064/fm211-3-4

Keywords: subset mbox homeo group homeomorphisms topological space class orbit union orbits having closure widetilde space classes orbits called quasi orbit space every second countable space quasi orbit space widetilde where second countable metric space regular part space union subsets homeomorphic mathbb mathbb characterization spaces finite singular part x x which quasi orbit spaces countable groups subset mbox homeo mathbb finally every finite space singular part quasi leaf space codimension foliation closed three manifold