$k$-groups and duality

Recall that a function is Ac-continuous if its restriction to each compact subset of its domain is continuous.We call a topological group G a Ac-group if each ¿-continuous homomorphism on G is continuous.As we will see in §1, where elementary properties of Ac-groups are studied, Ac-groups are the appropriate topological group analogue to Ac-spaces.As one would expect, they provide a useful tool for the study of dual groups topologized with the compact-open topology.In §2 we show that the duality map on an abelian Ac-group is always continuous, and in §3 we give some extensions of the Pontrjagin-Van Kampen duality theorem.As a corollary we show that each closed subgroup of a countable product of locally compact abelian groups satisfies duality, extending the principal result of [8]-that the inverse limit of a sequence of locally compact abelian groups satisfies duality.Proof.Since the supremum of any collection of group topologies is a group topology, kg(t) is a group topology ; since t S kg(t) ^ k(t), kg(t) coincides on compact sets with / and by definition kg(t) is the largest group topology with this property.If (G, t) is a Ac-group, then the identity map from (C7, t) to (G, kg(t)), which is a Ac-continuous homomorphism, is continuous, so t=kg(t).Conversely, suppose t=kg(t) and let/be a ¿-continuous homomorphism on G. Let t' be the smallest