  For Pontryagin's group duality in the setting of locally compact topological
Abelian groups, the topology on the character group is the compact open
topology. There exist at present two extensions of this theory to topological
groups which are not necessarily locally compact. The first, called the
Pontryagin dual, retains the compact-open topology. The second, the continuous
dual, uses the continuous convergence structure. Both coincide on locally
compact topological groups but differ dramatically otherwise. The Pontryagin
dual is a topological group while the continuous dual is usually not. On the
other hand, the continuous dual is a left adjoint and enjoys many categorical
properties which fail for the Pontryagin dual. An examination and comparison of
these dualities was initiated in \cite{CMP1}. In this paper we extend this
comparison considerably.
