A periodic locally compact abelian group has the closed-sum property exactly when its prime components split into discrete, profinite, finite-rank, and inductively monothetic factors.
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When is the Sum of Two Closed Subgroups Closed in a Locally Compact Abelian Group
A periodic locally compact abelian group has the closed-sum property exactly when its prime components split into discrete, profinite, finite-rank, and inductively monothetic factors.