For brevity’s sake, a subgroup
of finite index is called “large.” A discontinuous torsion character is (clearly) one
whose kernel is a large nonopen subgroup. Compact Abelian groups (and certain
other LCA groups) have the same number (either none or infinitely many) of large
nonopen subgroups and discontinuous torsion characters. Locally compact
Abelian groups of which all large subgroups are open include connected,
locally connected, and monothetic (possibly totally disconnected) groups.
Contrariwise, there are locally compact groups G which have as many as
2|𝜃| large nonopen subgroups. These include nondiscrete torsion Abelian
groups of bounded order and all totally disconnected, nonmetrizable, compact
groups.
