The dual space of precompact groups. (Q2856929)

The main result: Let \(G\) be a precompact group (i.e., a subgroup of a compact group) which is almost metrizable (i.e., there is a compact subgroup \(K\subset G\) such that the quotient space \(G/K\) is metrizable). Let \(\hat {G}\) be the dual object of \(G\) (i.e., the set of equivalence classes of irreducible unitary representations of \(G\)) equipped with the Fell topology. Then \(\hat {G}\) is a discrete space.