- Title
- Density character of subgroups of topological groups
- Creator
- Leiderman, Arkady; Morris, Sidney; Tkachenko, Mikhail
- Date
- 2017
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/159323
- Identifier
- vital:11955
- Identifier
-
https://doi.org/10.1090/tran/6832
- Identifier
- ISSN:0002-9947
- Abstract
- We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.
- Publisher
- American Mathematical Society
- Relation
- Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664
- Rights
- Copyright © Copyright 2016 American Mathematical Society
- Rights
- Open Access
- Rights
- This metadata is freely available under a CCO license
- Subject
- 0101 Pure Mathematics; 0102 Applied Mathematics; Topological group; Locally compact group; pro-Lie group; Separable topological space
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