Characterization of quasi-orbit spaces (Q1944381)

If \(X\) is a \(T_0\)-topological space, then its regular part \(X_0\subseteq X\) is defined as the union of its open sets which are homeomorphic to \(\mathbb{R}\) or to \(S^1\). Its singular part is \(X\setminus X_0\). The space \(X\) is partially ordered by setting \(x\leq y\) if \(\overline{\{x\}}\subseteq \overline{\{y\}}\). Let \(M\) be a one-dimensional manifold and \(G\) a subgroup of the group \(H_+(M)\) of all orientation preserving homeomorphisms on \(M\). By a quasiorbit space \(M/\widetilde G\) is understood the quotient space, where \(\widetilde G\) is the partition of the \(G\)-orbits into classes defined by: two orbits are in the same class of their closures in \(M\) are identical. In terms of the order relation on \(X\) and assuming that \(X\) has finite singular part, the author finds necessary and sufficient conditions for \(X\) to be homeomorphic to \(M/\widetilde G\) for some \(M\) where the group is Abelian and finitely generated. He gives these conditions on \(X\) for \(M= \mathbb{R}\), \(m= S^1\) and \(M= [0,\infty)\).