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Density character of subgroups of topological groups


Authors: Arkady G. Leiderman, Sidney A. Morris and Mikhail G. Tkachenko
Journal: Trans. Amer. Math. Soc. 369 (2017), 5645-5664
MSC (2010): Primary 54D65; Secondary 22D05
DOI: https://doi.org/10.1090/tran/6832
Published electronically: December 22, 2016
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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 $ \mathfrak{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 $ \mathfrak{c}$ is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight $ \mathfrak{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.


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Additional Information

Arkady G. Leiderman
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer Sheva, Israel
Email: arkady@math.bgu.ac.il

Sidney A. Morris
Affiliation: Faculty of Science, Federation University Australia, P.O.B. 663, Ballarat, Victoria, 3353, Australia — and — Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia
Email: morris.sidney@gmail.com

Mikhail G. Tkachenko
Affiliation: Departamento de Matemáticas, Universidad Autónoma Metropolitana, Avenida San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, C.P. 09340, México, D.F., Mexico
Email: mich@xanum.uam.mx

DOI: https://doi.org/10.1090/tran/6832
Keywords: Topological group, locally compact group, pro-Lie group, separable topological space
Received by editor(s): December 29, 2014
Received by editor(s) in revised form: August 10, 2015, and September 10, 2015
Published electronically: December 22, 2016
Article copyright: © Copyright 2016 American Mathematical Society