Nonclosed pure subgroups of locally compact abelian groups

Abstract:

We study the existence of many nonclosed pure subgroups of nondiscrete locally compact abelian groups. It is shown that every nondiscrete locally compact abelian group has uncountably many nonclosed pure subgroups. This in particular solves a question of Armacost. It is also shown that, if $G$ is a nondiscrete locally compact abelian group and if either $G$ is a compact group or the torsion part $t\left ( G \right )$ of $G$ is nonopen, then $G$ has ${2^c}$ proper dense pure subgroups, where $c$ denotes the power of the continuum. This in particular gives a partial answer to another question of Armacost.