Interval of group topologies satisfying Pontryagin duality

Abstract

Hewitt [2] proved that the only locally compact group topology on the additive group 1R of real numbers strictly finer than the usual topology is the discrete topology. Suppose instead we ask the question: how many group topologies satisfying Pontryagin duality exist on IR in the interval of topologies between the usual topology and the discrete topology? The answer is (Corollary3.4 below) that it is at least 22~! Our approach is to find a method of obtaining all group topologies ~ on an abelian group G such that (G, r) satisfies Pontryagin duality. This is done in Section 3 and 4 of this paper. We call a group topology "c on G which is the finest group topology on G coinciding on the compact sets of ~, a kg-topology. If ~1, % are kg-topologies on G such that "c2>z 1 and (G, zl), i = 1, 2 satisfies Pontryagin duality with Xi as character group, then the interval of all kg-topologies ~ [ r l , z2] on G such that (G, z) satisfies Pontryagin duality is order isomorphic with the lattice of subgroups of the abstract group X 2 / X 1 where X~ is identified as a subgroup of the abstract group X 2 in an obvious manner (Theorem 3.3 below). Hewitt [2], Rajagopalan [8], Janakiraman and Rajagopalan [4], Rickert [9] and Miller and Rajagopalan [6] studied the above problem with the added restriction that z be locally compact.