Dual properties in totally bounded Abelian groups

摘要

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.