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UNIVERSAL SUBGROUPS OF POLISH GROUPS

Published online by Cambridge University Press:  12 December 2014

KONSTANTINOS A. BEROS
KONSTANTINOS A. BEROS*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS GENERAL ACADEMICS BUILDING 435 1155 UNION CIRCLE DENTON, TX 76203-5017, USA E-mail:beros@unt.edu
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Abstract

Given a class ${\cal C}$ of subgroups of a topological group G, we saythat a subgroup $H \in {\cal C}$ is a universal${\cal C}$subgroup of G if every subgroup $K \in {\cal C}$ is a continuous homomorphic preimage of H.Such subgroups may be regarded as complete members of ${\cal C}$ with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable power Gω has a universal Kσ subgroup and a universal compactly generated subgroup. We prove a weakerversion of this in the nonlocally compact case and provide an example showingthat this result cannot readily be improved. Additionally, we show that manystandard Banach spaces (viewed as additive topological groups) have universalKσ and compactly generated subgroups. As an aside, we explore therelationship between the classes of Kσ and compactly generated subgroups and give conditions under which the twocoincide.

Keywords

03E1554H1122A0520B3020B35

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 4 , December 2014 , pp. 1148 - 1183
Copyright
Copyright © The Association for Symbolic Logic 2014 

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