On the Mackey problem for free abelian topological groups

2018/04/01 by Saak Gabriyelyan, Gabriyelyan, Saak
Mathematics · #22A10 #54H11 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #math.GN #math.GR #msc:22A10 #msc:54H11

paper · pdf · doi:10.48550/arxiv.1804.00303

arxiv created 2018/04/01 · arxiv updated 2018/04/03

Abstract

Recently Außenhofer and the author independently have shown that the free abelian topological group A(s) over a convergent sequence s does not admit the strongest compatible locally quasi-convex group topology that gives the first example of a locally quasi-convex abelian group without a Mackey group topology. In this note we considerably extend this example by showing that the free abelian topological group A(X) over a non-discrete zero-dimensional metrizable space X does not have a Mackey group topology. In particular, for every countable non-discrete metrizable space X, the group A(X) does not have a Mackey group topology.

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