Order and minimality of some topological groups

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Abstract

A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group H+(X) of order-preserving homeomorphisms of a compact linearly ordered connected space X. We provide a sufficient condition on X under which the topological group H+(X) is minimal. This condition is satisfied, for example, by: the unit interval, the ordered square, the extended long line and the circle (endowed with its cyclic order). In fact, these groups are even a-minimal, meaning, in this setting, that the compact-open topology on G is the smallest Hausdorff group topology on G. One of the key ideas is to verify that for such X the Zariski and the Markov topologies on the group H+(X) coincide with the compact-open topology. The technique in this article is mainly based on a work of Gartside and Glyn [21].

Original languageEnglish
Pages (from-to)131-144
Number of pages14
JournalTopology and its Applications
Volume201
DOIs
StatePublished - 15 Mar 2016

Bibliographical note

Publisher Copyright:
© 2015 Elsevier B.V.

Keywords

  • A-Minimal group
  • Compact LOTS
  • Markov's topology
  • Minimal groups
  • Order-preserving homeomorphisms
  • Zariski's topology

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