For a locally compact (LC) group G, denote by G + its underlying group equipped with the topology inherited from its Bohr compactification. G is maximally almost periodic (MAP) if and only if G + is Hausdorff. If P denotes a topological property, then we say that a MAP group G respects P if G and G + have the same subspaces with P . In 1962 I. Glicksberg proved that LC Abelian groups respect compactness. We extend this result by showing that LC groups such that all their irreducible unitary representations are finite-dimensional, i.e., [MOORE] groups, do so as well. Moreover, we prove that G equipped with the topology induced by its topological dual is equal to G + if and only if G belongs to the class [MOORE]. If this is indeed the case, then (a) G additionally respects pseudocompactness, (relative) functional boundedness, and the Lindelöf property, (b) G is connected (respectively zero-dimensional, respectively realcompact) if and only if G + is connected (respectively zero-dimensional, respectively realcompact), and (c) G is σ-compact if and only if G + normal. We end the paper by showing the existence of a discrete group that is not [MOORE] and which still respects compactness.