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.