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
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
or to
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.