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.