  We study and compare the gap and the Riesz topologies of the space of all
unbounded regular operators on Hilbert C*-modules. We show that the space of
all bounded adjointable operators on Hilbert C*-modules is an open dense subset
of the space of all unbounded regular operators with respect to the gap
topology. The restriction of the gap topology on the space of all bounded
adjointable operators is equivalent with the topology which is generated by the
usual operator norm. The space of regular selfadjoint Fredholm operators on
Hilbert C*-modules over the C*-algebra of compact operators is path-connected
with respect to the gap topology, however, the result may not be true for some
Hilbert C*-modules.
