Sobre algunos espacios de funciones continuas en el círculo unidad

Abstract

This paper studies the topological group structure of C(X,T), the group of continuous functions on the topological space X with values in the circle group T, with the topology of uniform convergence on compact subsets of X. For the main part, attention is restricted to the case X=Q, the rational numbers with either the Euclidean or Bohr topologies. The style of the paper is largely expository, though some new results are proved. It is shown for instance that while the homomorphism group Hom(Q,T) (also known as Qˆ) is topologically isomorphic to Hom(R,T) (and, thus, to R), the group C(Q,T) is not even first countable. The group C(Q,T) is next realized as the completion of C(Qb,T), where Qb stands for the group Q equipped with its Bohr topology, the one induced by all continuous characters (homomorphisms into T) of Q. Another set of results concerns the duality properties of these groups. Here the authors represent C(Qb,T) as the character group of the free abelian topological group A(Qb,T) and exploit the duality properties of the latter to show that C(Qb,T) is a reflexive topological group.