Compactness and its Generalizations in Topological Groups

Abstract

This chapter introduces the reader to the Pontryagin–van Kampen duality theory, the Bohr topology on Abelian groups, and the study of algebraic and topological structure of compact, countably compact, and pseudocompact Abelian groups. A special emphasis is given to the study of Abelian groups in each of the directions just mentioned. The main objective in Sections 9.1–9.4 is to prepare all necessary material for the proof of an important corollary to Peter–Weyl’s theorem on irreducible representations of compact topological groups. We refer to Theorem 9.4.11 saying that the family of continuous homomorphisms of a compact Abelian group to the circle group separates points of the group.