The Mackey problem for free abelian topological groups

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The classical Mackey–Arens theorem states that every locally convex space has a Mackey space topology. However, in the wider class of locally quasi-convex (lqc) groups an analogous result does not hold. Indeed, Außenhofer and the author showed independently that the free abelian topological group A(s) over a convergent sequence s does not admit a Mackey group topology. We essentially extend this example by showing that the free abelian topological group A(X) over a non-discrete zero-dimensional (for example, countable) metrizable space X does not have a Mackey group topology.

Original languageAmerican English
Pages (from-to)2073-2079
Number of pages7
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Issue number3
StatePublished - 1 Jul 2019


  • Free abelian topological group
  • Mackey group topology
  • Metrizable space
  • Zero-dimensional space

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Computational Mathematics
  • Applied Mathematics


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