Dual properties in totally bounded Abelian groups

Abstract

Let \( \mathcal{T}_A \) denote the category of totally bounded Abelian groups and their continuous group homomorphisms. Each object \( (G, \tau) \) in \( \mathcal{T}_A \) has associated a dual group \( (G', \tau') \) also in \( \mathcal{T}_A \) such that \( (G'', \tau'') \) is canonically isomorphic to \( (G, \tau) \). Two (topological) properties \( \{\mathcal{P}, \mathcal{Q} \} \) are in duality when for each \( (G, \tau) \in \mathcal{T}_A \) it holds that \( (G, \tau) \) satisfies \( \mathcal{P} \) if and only if \( (G', \tau') \) satisfies \( \mathcal{Q} \). For instance, the pair of properties {compactness, largest totally bounded group topology} and {metrizability, countable cardinal} are both in duality. In the first part of this paper we find the dual properties of realcompactness, hereditarily realcompactness and pseudocompactness.¶ A topological space is called countably pseudocompact when for each countable subset B of X there is a countable subset A of X such that \( B \subseteq cl_{X}A \) and \( cl_{X}A \) is pseudocompact. In the last part of this paper we prove that if X is a countably pseudocompact space and Y is metrizable then \( C_{p}(X, Y) \) is a \( \mu \)-space. As a consequence, it follows that if \( (G, \tau) \) is a countably pseudocompact group then \( (G', \tau') \) is a \( \mu \)-space.