Topology Atlas Document # ideb-23

## Duality of totally bounded Abelian groups

by

### S. U. Raczkowski and F. Javier Trigos-Arrieta

#### Topology Atlas Preprint # 396

Let G be a totally bounded Abelian (Hausdorff) group and denote by [G\tilde] the group of characters, i.e. continuous homomorphisms from G into the usual Torus T, equipped with operation defined pointwise, and endowed with the topology of pointwise convergence on G. Define the evaluation mapping \phi: G --> [[G\tilde]\tilde] by the relation \phi(x)(\lambda): = \lambda(x) for x \in G, \lambda \in [G\tilde]. We show that \phi is a topological isomorphism of G onto [[G\tilde]\tilde]. We compare this with the usual duality on locally compact Abelian (LCA) groups. As an application, a new proof is presented of the fact that LCA groups respect compactness when equipped with their Bohr topology.

Mathematics Subject Classification: 03E50, 22A05, 22A10, 22B05, 54D30, 54D65, 54H11
Keywords: Ascoli's Theorem, Baire space, Bohr topology, character, compact-open topology, equicontinuity, finite-open topology, pointwise convergence, Pontryagin-van Kampen duality, totally bounded group