Locally compact topologies for groups

Abstract

The investigations leading to this paper were suggested by the papers of Hewitt [3] and Ross [5]. In [3] Hewitt was interested in proving that if an abelian group is locally compact in two topologies, one strictly stronger than the other, there is a character continuous in one topology and discontinuous in the other (actually a special case of a theorem of Kaplansky-see Theorem 1.1 of [2]). Actually Hewitt proved a stronger result. His arguments were based on the fact that both the additive group of reals, and the multiplicative group of complex numbers of absolute value 1 have the property that every stronger locally compact group topology is discrete. A natural question to ask is what other groups have this property. The answer is very simple (2.1 of this paper). We consider in the second section of this paper the obvious generalization. Namely which groups have the property that there are only finitely many stronger locally compact group topologies. The investigations in the first section of this paper were suggested by the paper of Ross [5]. Ross was considering the same question as Hewitt, and was led to consider the relationship between two locally compact group topologies on a group G such that G has the same closed subgroups in the two topologies. We investigate this further in the first section of this paper, and are able to say that many of the properties of G as a topological group can be recovered once we know the closed subgroups.