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ON VARIETIES OF ABELIAN TOPOLOGICAL GROUPS WITH COPRODUCTS

Part of: Peculiar spaces General theory of categories and functors Topological and differentiable algebraic systems Structure and classification of infinite or finite groups Connections with other structures, applications Abelian groups

Published online by Cambridge University Press:  19 October 2016

SAAK S. GABRIYELYAN  and
SIDNEY A. MORRIS
SAAK S. GABRIYELYAN
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva P.O. 653, Israel email saak@math.bgu.ac.il
SIDNEY A. MORRIS*
Affiliation:
Faculty of Science and Technology, Federation University Australia, PO Box 663, Ballarat, Victoria, 3353, Australia email morris.sidney@gmail.com Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, 3086, Australia
*
✉ morris.sidney@gmail.com
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Abstract

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A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

Keywords

varieties of abelian topological groupscoproducts

MSC classification

Primary: 22A05: Structure of general topological groups
Secondary: 18A30: Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 20E10: Quasivarieties and varieties of groups 20K25: Direct sums, direct products, etc. 54G10: $P$-spaces 54H11: Topological groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 54 - 65
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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