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Topologies on the graph of the equivalence relation associated to a groupoid. (English) Zbl 1100.22003

Summary: Let \(G\) be a topological groupoid, \(r\) and \(d\) be the range map, respectively the domain map of \(G\). The relation \(u\sim v\) if there is an \(x\) such that \(r(x)=u\) and \(d(x)=v\) is an equivalence relation on the unit space \(G^{(0)}\). The graph of this equivalence relation can be regarded as a groupoid \(R\), and can be endowed with different topologies. We shall prove that if the restriction of the range map to the isotropy group bundle of \(G\) is open then we can endow \(R\) with a locally compact topology such that the existence of a Haar system on \(G\) is equivalent to the existence of a Haar system on \(R\).

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
43A05 Measures on groups and semigroups, etc.
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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