Interpolation sets in spaces of continuous metric-valued functions

Let X and K be a Čech-complete topological group and a compact group, 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 L‾KX is canonically homeomorphic to βω, the Stone-Čech compactifcation 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 L‾KY has countable tightness or |L‾KY|≤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 Grespects 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 kω 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.