Let X and K be aČech-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 subsetis 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 itGiven 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. M X . This property, or its absence, has deep implications on the topological structure of G as a set of continuous functions on X and has found many applications in different settings (for instance, see [20,22,17,25] where there are applications to topological groups, dynamical systems, functional analysis and harmonic analysis, respectively).