A SECOND COUNTABLE LOCALLY COMPACT TRANSITIVE GROUPOID WITHOUT OPEN RANGE MAP.

Dana P. Williams raised in [Proc. Amer. Math. Soc., Ser. B 3 (2016), pp. 1-8] the following question: Must a second countable, locally compact, transitive groupoid have open range map? This paper gives a negative answer to that question. Although a second countable, locally compact transitive groupoid G may fail to have open range map, we prove that we can replace its topology with a topology which is also second countable, locally compact, and with respect to which G is a topological groupoid whose range map is open. Moreover, the two topologies generate the same Borel structure and coincide on the fibres of G. [ABSTRACT FROM AUTHOR]