  Dana P. Williams raised in [Proc. Am. Math. Soc., Ser. B, 2016] 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.
