Abstract
Let G subset of 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/(G) over tilde 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/(G) over tilde, 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 R or to 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 of Homeo(+) (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.