A FEW REMARKS ON LOCALLY COMPACT TOPOLOGIES AND HAAR SYSTEMS.

We start from the question raised by Williams (Proc. Am. Math. Soc. 2016): Must a second countable, locally compact, transitive groupoid G have open range map? If the answer is positive, the topology of G is in fact locally transitive (in the sense of [Seda, 1976]). We prove that even if the answer is negative, we can replace the original topology of G with a local transitive topology so that the topologies of the r-fibres are not affected. The new topology is locally compact Hausdorff but not necessary second countable. However its full C*-algebra (introduced in [Renault, 1980]) is still isomorphic to C*(H)⊗K(L2(μ)), where H is the isotropy group at a unit u and μ is a positive Radon measure on the unit space. We also present a few remarks concerning the Haar systems on locally compact groupoids and for every locally compact groupoid having paracompact unit space and second countable r-fibres, we prove the existence of a pre-Haar system bounded on the compact sets. [ABSTRACT FROM AUTHOR]