On local quasi-convexity as a three-space property in topological abelian groups

Abstract

Let X be a topological abelian group and H a subgroup of X. We find conditions under which local quasi-convexity of both H and X/H results in the same property for X. This is true for instance if H is precompact, or if X is metrizable and H is a dually embedded subgroup which is also either discrete or bounded torsion. We also give some general principles and point out some errors we have found in the existing literature on this problem.