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 | References | Similar Articles | Additional Information

Abstract: We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an -narrow topological group : (i) is homeomorphic to a subspace of a separable regular space; (ii) is topologically isomorphic to a subgroup of a separable topological group; (iii) 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 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 which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that 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 is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight . 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