We present a complete
description of the closed sets in the coset ring ℛ(G) of an abelian topological group
G. Using this result we show that every such set in a separable, metrizable, locally
compact, abelian group Γ is a strong Ditkin set in the sense of Wik, yielding the
converse of a theorem of Rosenthal and thus completing the characterization of the
strong Ditkin sets with void interior for certain choices of Γ. These two results were
first obtained by J. E. Gilbert. Our development of the former rests on the following
theorem, which seems to be of independent interest: If φ : G → G∗ is a
homomorphism and A ∈ℛ(G), then φ(A) ∈ℛ(G∗).
