On Mackey Topologies in Topological Abelian

Let C be a full subcategory of the category of topological abelian groups and SPC denote the full subcategory of subobjects of products of objects of C. We say that SPC has Mackey coreeections if there is a functor that assigns to each object A of SPC an object A that has the same group of characters as A and is the nest topology with that property. We show that the existence of Mackey coreeections in SPC is equivalent to the injectivity of the circle with respect to topological subgroups of groups in C. 1. Introduction Consider a full subcategory C of the category of topological abelian groups that includes the circle group T = R=Z. If A is a topological group, then a character on A is a continuous homomorphism : A ?! T. We let A b denote the discrete group of all characters of A. Let SPC denote the closure of C with respect to arbitrary products and subobjects. If A is an object of SPC, we say that A is a Mackey group in SPC if, whenever e