Countably continuous group homomorphisms and the property of realcompactness
by
Sergio Macario
Universitat Jaume I
Coauthors: Salvador Hernández (Universitat Jaume I)

Let G be an Abelian topological group. Let (P) be the property:

'Every homomorphism from G to the unit circle T, continuous over countable subsets of G is continuous'

We prove that a group G has the property (P) if, and only if, ([^G], \sigma(G)) is realcompact. As a consequence we identify new classes of topological groups that are realcompact; in particular, if G is a complete g-group and contains a \sigma-compact dense subset, then (G, \sigma([^G])) is realcompact. This means, for exemple, that every countably weakly compactly generated locally convex space is w-realcompact.

A topological group is said to be Mazur if every sequentially continuous homomorphism is continuous. Some applications are also provided to the study of topological groups that enjoy this property.

Date received: June 1, 1999