Identifying and distinguishing various varieties of abelian topological groups

Volume 458 / 2008

Carolyn E. McPhail, Sidney A. Morris Dissertationes Mathematicae 458 (2008), 1-45 MSC: 22A05, 20E10, 54D35, 54D50. DOI: 10.4064/dm458-0-1


A variety of topological groups is a class of (not necessarily Hausdorff) topological groups closed under the operations of forming subgroups, quotient groups and arbitrary products. The variety of topological groups generated by a class of topological groups is the smallest variety containing the class. In this paper methods are presented to distinguish a number of significant varieties of abelian topological groups, including the varieties generated by (i) the class of all locally compact abelian groups; (ii) the class of all $k_\omega$-groups; (iii) the class of all $\sigma$-compact groups; and (iv) the free abelian topological group on $[0,1]$. In all cases, hierarchical containments are determined.


  • Carolyn E. McPhailSchool of Mathematics and Applied Statistics
    University of Wollongong
    Wollongong, NSW 2522
  • Sidney A. MorrisSchool of Information Technology and Mathematical Sciences
    University of Ballarat
    PO Box 663
    Ballarat, VIC 3353

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