Characterization of quasi-orbit spaces

Abstract

Let G be a subgroup of the group Homeo(X) of homeomorphisms of a topological space X. The class of an orbit O of G is the union of all orbits having the same closure as O. We denote by ${X/\widetilde{G}}$ the space of classes of orbits called the quasi-orbit space. The regular part of a T ₀-space is the union of open subsets homeomorphic to ${\mathbb{R}}$ or to ${\mathbb{S}^1.}$ The complementary of the regular part is called the singular part. In this paper we give a characterization of the topological spaces with finite singular parts and which are quasi-orbit spaces with respect to countable groups of the group of homeomorphisms of a one dimensional manifold.