In this paper, we study acc
(ascending chain condition) and dcc (descending chain condition) on different types
of subgroups of LCA (locally compact abelian) groups, such as open subgroups,
compact subgroups, discrete subgroups, metrizable subgroups, closed divisible
subgroups, proper dense subgroups. We characterize compactly generated LCA
groups as the class of LCA groups whose open subgroups satisfy acc; the compactly
cogenerated groups as the class of LCA groups whose discrete subgroups satisfy dcc.
We also show that acc and dcc on the following classes of subgroups are pairwise
equivalent: (a) closed subgroups (b) closed totally disconnected subgroups
(c) closed σ-compact subgroups (d) closed metrizable subgroups. We also
obtain a characterization of those LCA groups which contain no proper dense
subgroups.
