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Bourbaki spaces of topological groups

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Abstract

A study is presented of the relationship between the topological and uniformity properties of a group G and the spaces ℐ(G), ℒ(G) of all nonempty closed subsets and closed subgroups of G. A base for the neighborhood system of a closed subset X of G is formed by the sets S(X, U)={Y ∶ Y ⊒ XU, X ⊒ YU}, where U ranges over all neighborhoods of the identity in G. Criteria are obtained for the space ℐ(G) and some of its subspaces to be totally bounded and locally totally bounded. Some classes of groups with compact spaces ℒ(G) are described. It is proved that the spaces ℐ(G), ℒ(G) are complete in the case of projective metrizable groups G.

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Authors and Affiliations

  1. Kiev University, USSR

    I. V. Protasov & A. Charyev

Authors
  1. I. V. Protasov
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  2. A. Charyev
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Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 542–549, April, 1990.

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Protasov, I.V., Charyev, A. Bourbaki spaces of topological groups. Ukr Math J 42, 480–486 (1990). https://doi.org/10.1007/BF01071339

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  • DOI: https://doi.org/10.1007/BF01071339

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