Hasse diagrams and orbit class spaces

Let X be a topological space and G be a group of homeomorphisms of X. Let G ˜ be an equivalence relation on X defined by x G ˜ 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.