# Dual properties in totally bounded Abelian groups

Research paper by S. Hernández, S. Macario

Indexed on: 01 May '03Published on: 01 May '03Published in: Archiv der Mathematik

#### 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.