  In this paper we introduce and investigate top (bi)comodules} of corings,
that can be considered as dual to top (bi)modules of rings. The fully coprime
spectra of such (bi)comodules attains a Zariski topology, defined in a way dual
to that of defining the Zariski topology on the prime spectra of (commutative
rings. We restrict our attention in this paper to duo (bi)comodules (satisfying
suitable conditions) and study the interplay between the coalgebraic properties
of such (bi)comodules and the introduced Zariski topology. In particular, we
apply our results to introduce a Zariski topology on the fully coprime spectrum
of a given non-zero coring considered canonically as duo object in its category
of bicomodules.
