Abstract
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.