Quasi-orbit spaces associated to $T_0$-spaces

Volume 211 / 2011

C. Bonatti, H. Hattab, E. Salhi Fundamenta Mathematicae 211 (2011), 267-291 MSC: 54F65, 54H20. DOI: 10.4064/fm211-3-4


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.


  • C. BonattiIMB, UMR 5584 du CNRS
    9 av. Alain Savary
    21000 Dijon, France
  • H. HattabInstitut Supérieur
    d'Informatique et du Multimedia
    Route de Tunis km 10
    B.P. 242, Sfax 3021, Tunisia
  • E. SalhiDépartement de Mathématiques
    Faculté des Sciences de Sfax
    Route de Soukra km 3.5
    B.P. 802, Sfax 3018, Tunisia

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