  We study modules over the ring $\widetilde{\C}$ of complex generalized
numbers from a topological point of view, introducing the notions of
$\widetilde{\C}$-linear topology and locally convex $\widetilde{\C}$-linear
topology. In this context particular attention is given to completeness,
continuity of $\widetilde{\C}$-linear maps and elements of duality theory for
topological $\widetilde{\C}$-modules. As main examples we consider various
Colombeau algebras of generalized functions
