Vol. 86, No. 2, 1980

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Chain conditions on subgroups of LCA groups

Mohammad Ahmad Khan

Vol. 86 (1980), No. 2, 517–534

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.

Mathematical Subject Classification 2000
Primary: 22B05
Received: 24 April 1978
Revised: 16 March 1979
Published: 1 February 1980
Mohammad Ahmad Khan