Quasi-orbit spaces associated to T₀-spaces

Abstract

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/ e G be the space of classes of orbits, called the quasi-orbit space. We show that everyThe regular part X0 of a T0-space X is the union of open subsets homeomorphic to R or to S 1 . 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.2010 Mathematics Subject Classification: 54F65, 54H20.

On quasi-orbital space

“…The study of this space is difficult: just consider the example of a group generated by an irrational rotation on the circle; indeed the orbit space does not verify the weaker separation axioms, as the T 0 separation axiom. For this reason [8,1,2,7] consider an intermediary quotient, called the quasi-orbit space.…”

“…Thus E/ G is a good representative of E/G. According to [8,1], the space E/ G keeps information on the initial dynamical system.…”

“…In [1], the authors asked the following problem: under which conditions a T 0 -space is quasi-orbital? In [2] the authors showed that a finite T 0 -space is quasi-orbital.…”

“…In [2] the authors showed that a finite T 0 -space is quasi-orbital. Note that, according to [1,Example 3.4], if X is a non quasicompact space then E is not in general compact.…”

Hasse diagrams and orbit class spaces

“…Roughly speaking, the vertices will correspond to the orbit classes and there is an arc joining the vertex "class of x" to the vertex "class of y" if the closure of the orbit of x contains the closure of the orbit of y. In our last example (Y , G), the associated graph Γ consists of three vertices: [0], [1] and [ 1 2 ], and two arcs: [ 1 2 ] → [0] and [ 1 2 ] → [1]. In particular, if the orbit class space is finite, the associated graph is also finite and it can be described as the Hasse diagram related to a canonical order defined on X/G.…”