  We investigate when the categories of all rational $A$-modules and of finite
dimensional rational modules are closed under extensions inside the category of
$C^*$-modules, where $C^*$ is the cofinite topological completion of $A$. We
give a complete characterization of these two properties, in terms of a
topological and a homological condition. We also give connections to other
important notions in coalgebra theory such as coreflexive coalgebras. In
particular, we are able to generalize many previously known partial results and
answer some questions in this direction, and obtain large classes of coalgebras
for which rational modules are closed under extensions as well as various
examples where this is not true.

Category Theory;Rings and Algebras;Representation Theory