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• W2010069887 abstract "A cover C of a topological space X is point-countable (point-finite) if every point of X belongs to at most countably many (at most finitely many) elements of C . We say that a space X has the weak topology with respect to a cover C provided that a set F ⊆ X is closed in X if and only if its intersection F ∩ C with every C ϵ C is closed in C . A space X is an α 4 -space if for every point x ϵ X and any countable family { S n : n ϵ N } of sequences converging to x one can find a sequence S converging to x which meets infinitely many S n . The classical Birkhoff-Kakutani theorem says that a Hausdorff topological group is metrizable if (and only if) it is first countable. Quite recently Arhangel'skii generalized this theorem by showing that Hausdorff bisequential topological groups are metrizable (recall that first countable spaces are bisequential). In our paper we generalize these results by showing that a Hausdorff topological group is metrizable if it has the weak topology with respect to a point-finite cover consisting of bisequential spaces. In addition we establish the following theorem each item of which also generalizes both Birkhoff-Kakutani's and Archangel'skii's results:" @default.
• W2010069887 created "2016-06-24" @default.
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• W2010069887 date "1993-12-01" @default.
• W2010069887 modified "2023-10-17" @default.
• W2010069887 title "Metrizability of topological groups having weak topologies with respect to good covers" @default.
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• W2010069887 doi "https://doi.org/10.1016/0166-8641(93)90062-i" @default.
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