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Closures of weakened analytic groups


Author: T. Christine Stevens
Journal: Proc. Amer. Math. Soc. 119 (1993), 291-297
MSC: Primary 22A05; Secondary 22E15
DOI: https://doi.org/10.1090/S0002-9939-1993-1123667-8
MathSciNet review: 1123667
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Abstract: Let $ (G,\mathcal{G})$ be a topological group with dense subgroup $ L$, and suppose that $ L$ is an analytic group in a topology $ \tau $ that is stronger than the topology that $ L$ inherits from $ \mathcal{G}$. It is known that $ L$ contains a $ \tau $-closed abelian subgroup $ H$ that completely determines the topology of $ L$. We now prove that the $ \mathcal{G}$-closure $ \overline H $ of $ H$ similarly determines the topology of $ G$. $ (G,\mathcal{G})$ always has a left-completion in the category of topological groups, and the properties of $ \overline H $ determine whether $ (G,\mathcal{G})$ is locally compact, analytic, metrizable, left-complete, or finite dimensional. We discuss the relationship between these results and recent work of Goto.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1123667-8
Keywords: Analytic group, weakened analytic group, completion of a group
Article copyright: © Copyright 1993 American Mathematical Society

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