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/G̃ be the space of classes of orbits, called the quasi-orbit space. We show that every second countable T₀-space Y is a quasi-orbit space E/G̃, where E is a second countable metric space. The regular part X₀ of a T₀-space X is the union of open subsets homeomorphic to ℝ or to 𝕊¹. 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.