Determining a Weakly Locally Compact Group Topology by Its System of Closed Subgroups

ABSTRACT: Definition: A topological group G is weakly locally compact if (i) G is locally bounded and (ii) for every closed, Gδ subgroup H of G the quotient space G/H is locally compact. The authors show: Theorem: A Hausdorff topological group G is weakly locally compact if and only if some pseudocompact subset of G has nonempty interior (i.e., G is locally pseudocompact). This characterization generates the following result, which is motivated by a question posed by K. A. Ross [Fundamenta Mathematicae, 1965, 56: 241–244]. Theorem: If T1 and T2 are weakly locally compact group topologies on an Abelian group G such that 〈G,T1〉 and 〈G,T2〉 have the same closed subgroups, and if T1, T2, then T1=T2.