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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.

We define the g-extension of a topological Abelian group G as the set of all characters on G such that the restriction to every equicontinuous subset of G is continuous with respect to the pointwise convergence topology. A g-group is a topological Abelian group (G,τ) such that its g-extension coincides with its completion. The Bohr topology of a topological group (G,τ) is the topology that the group inherits as a subset of its Bohr compactification. A topological group (G,τ) respects a property P if the subsets A of G that satisfy the property P are exactly the same for the Bohr topology and for the original topology of the group [Trigos-Arrieta, J. Pure Appl. Algebra 70 (1991) 199]. All groups here are assumed to be Abelian. We prove that every complete g-group when endowed with its Bohr topology is a μ -space. As a consequence, we obtain that for a complete g-group the properties of respecting functionally boundedness, pseudocompactness, countable compactness and compactness are all equivalent and a characterization of this property is also provided. Finally, we extend a theorem of Rosenthal about the existence of sequences equivalent to the l 1 -basis. We prove that for a Cech-complete g-group the property of respecting compactness is equivalent to the existence of conveniently placed sequences equivalent to the l 1 -basis.

Let G be a MAPA group that is metrizable and satisfies Pontryagin duality; that is, it coincides with its topological bidual. We prove that the Bohr topology of G respects compactness if and only if every non-totally bounded subset contains an infinite discrete subset which is C ∗ -embedded in the Bohr compactification of G. This result is used to characterize the Banach spaces which respect compactness, or, with a different terminology, have the Schur property (defined below). Among other equivalent properties, we prove that a Banach space E has the Schur property if and only if every bounded basic sequence contains an infinite subsequence equivalent to a l 1 -basis.