  We investigate cases where the finite dual coalgebra of a twisted tensor
product of two algebras is a crossed product coalgebra of their respective
finite duals. This is achieved by interpreting the finite dual as a topological
dual; in order to prove this, we show that the continuous dual is a strong
monoidal functor on linearly topologized vector spaces whose open subspaces
have finite codimension. We describe a sufficient condition for the result on
finite dual coalgebras to be applied, and we this condition to particular
constructions including Ore extensions, smash product algebras, and crossed
product bialgebras.
