Topology Atlas Document # vaaa-74

## Dense subgroups may or may not produce the same character group

by

### W. W. Comfort, S. U. Raczkowski and F. Javier Trigos-Arrieta

#### Topology Atlas Preprint # 410

A character of an Abelian topological group G is a continuous homomorphism from G into the usual torus T. Denote by [^G] the character group of G, i.e., the group of characters of G with operation defined pointwise and equipped with the compact-open topology. We say that an Abelian group G is determined if for every dense subgroup D of G the restriction map \phi: [^G] --> [^D], defined by \phi(\lambda) : = \lambda\upharpoonright D, is a topological isomorphism between [^G] and [^D]. It is a theorem of Chasco that every Abelian metric group G is determined. Examples and characterizations of certain classes of non-determined groups are presented. We also show that a continuous homomorphic image of a determined group need not be determined, and that the property of being determined is not \mathfrak c-productive.

Mathematics Subject Classification: 22A10, 22B99, 22C05, 43A40, 54H11 (03E35, 03E50, 54D30, 54E35)
Keywords: Bohr compactification, Bohr topology, character (group), compact-open topology, dense subgroups, determined groups, duality