Interpolation sets in spaces of continuous metric-valued functions

Abstract

Let X and K be a Cech-complete topological group and a compact group, ˇ respectively. We prove that if G is a non-equicontinuous subset of CHom(X; K), the set of all continuous homomorphisms of X into K, then there is a countably infinite subset L ⊆ G such that LKX is canonically homeomorphic to β!, the Stone-Cech compactifca- ˇ tion of the natural numbers. As a consequence, if G is an infinite subset of CHom(X; K) such that for every countable subset L ⊆ G and compact separable subset Y ⊆ X it holds that either LKY has countable tightness or jLKY j ≤ c, then G is equicontinuous. Given a topological group G, denote by G+ the (algebraic) group G equipped with the Bohr topology. It is said that G respects a topological property P when G and G+ have the same subsets satisfying P. As an application of our main result, we prove that if G is an abelian, locally quasiconvex, locally k! group, then the following holds: (i) G respects any compact-like property P stronger than or equal to functional boundedness; (ii) G strongly respects compactness.