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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-3-4,
author = {C. Bonatti and H. Hattab and E. Salhi},
title = {Quasi-orbit spaces associated to T0-spaces},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {267-291},
zbl = {1218.54025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-3-4}
}
C. Bonatti; H. Hattab; E. Salhi. Quasi-orbit spaces associated to T₀-spaces. Fundamenta Mathematicae, Tome 215 (2011) pp. 267-291. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-3-4/