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Let G ⊂ 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/ e G be the space of classes of orbits, called the quasi-orbit space. We show that everyThe regular part X0 of a T0-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 − X0 which are the quasi-orbit spaces of countable groups G ⊂ Homeo+(R).Finally we show that every finite T0-space is the singular part of the quasi-leaf space of a codimension one foliation on a closed three-manifold.2010 Mathematics Subject Classification: 54F65, 54H20.

Let X be a topological space and G be a group of homeomorphisms of X. LetG be an equivalence relation on X defined by xG y if the closure of the G-orbit of x is equal to the closure of the G-orbit of y. The quotient space X/G is called the orbit class space and is endowed with the natural order inherited from the inclusion order of the closure of the classes, so that, if such a space is finite, one can associate with it a Hasse diagram. We show that the converse is also true: any finite Hasse diagram can be realized as the Hasse diagram of an orbit class space built from a dynamical system (X, G) where X is a compact space and G is a finitely generated group of homeomorphisms of X.

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