  We show that the inertia stack of a topological stack is again a topological
stack. We further observe that the inertia stack of an orbispace is again an
orbispace. We show how a U(1)-banded gerbe over an orbispace gives rise to a
flat line bundle over its inertia stack. Via sheaf theory over topological
stacks it gives rise to the twisted delocalized cohomology of the orbispace.
  With these results and constructions we generalize concepts, which are
well-known in the smooth framework, to the topological case. In the smooth case
we show, that our sheaf-theoretic definition of twisted delocalized cohomology
of orbispaces coincides with former definitions using a twisted de Rham
complex.
