Interpolation sets in spaces of continuous metric-valued functions

Let X and K be a Cech-complete topological group and a compact group, respectively. We prove that if G is a non-equicontinuous subset of C H o m ( X , K ) , the set of all continuous homomorphisms of X into K, then there is a countably infinite subset L ⊆ G such that L ‾ K X is canonically homeomorphic to βω, the Stone–Cech compactifcation of the natural numbers. As a consequence, if G is an infinite subset of C H o m ( X , K ) such that for every countable subset L ⊆ G and compact separable subset Y ⊆ X it holds that either L ‾ K Y has countable tightness or | L ‾ K Y | ≤ 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 ω 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.