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The Birkhoff–Kakutani Theorem asserts that a topological group is metrizable if, and only if, it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups.We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin–van Kampen duals...