Topology Proceedings 37 (2011) pp. 315-330: Continuous and Pontryagin duality of topological groups

For Pontryagin’s group duality in the setting of locally compact topological Abelian groups, the topology on the character group is the compact open topology. There exist at present two extensions of this theory to topological groups which are not necessarily locally compact. The first, called the Pontryagin dual, retains the compact-open topology. The second, the continuous dual, uses the continuous convergence structure. Both coincide on locally compact topological groups but differ dramatically otherwise. The Pontryagin dual is a topological group while the continuous dual is usually not. On the other hand, the continuous dual is a left adjoint and enjoys many categorical properties which fail for the Pontryagin dual. An examination and comparison of these dualities was initiated in [19]. In this paper we extend this comparison considerably. 1. Preliminaries Definition 1.1. Let X be a set and, for each x ∈ X, let λ(x) be a collection of filters satisfying: (i) The ultrafilter ẋ := {A ⊆ X : x ∈ A} ∈ λ(x), (ii) If F ∈ λ(x) and G ∈ λ(x), then F ∩ G ∈ λ(x), (iii) If F ∈ λ(x), then G ∈ λ(x) for all filters G ⊇ F . 2010 Mathematics Subject Classification. Primary 54A20, 54H11; Secondary 22D35, 18A30.