Locally compact Abelian groups with totally suitable sets

We study the class Y-t of topological groups G that admit a totally suitabl e set, i.e., a relatively discrete subset S subset of or equal to G\ {1} su ch that SU {1} is closed in G and N boolean AND [S] is dense in N for each closed normal subgroup N of G. We resolve an open question from [10] by giv ing a complete description of the locally compact Abelian groups in Y-l . W e also give a characterization of the groups of p-adic numbers in terms of totally suitable sets.