Pathological Topologies and Random Walks on Abelian Groups

Abstract

It is shown in this paper that a part of the structure theory of locally compact abelian groups, which does not explicitly involve compactness, still fails to apply to complete separable metric abelian groups. Specifically, Theorem 2 asserts that any countable abelian group may be discretely imbedded in a metrizable abelian group with a dense cyclic subgroup. This result is independent of the rest of the paper.

Received by the editors August 21, 1962 and, in revised form, January 11, 1963

Some of the work on this paper was done while the author held a National Science Foundation Cooperative Fellowship in 1961–1962.