Abstract
We prove that groups for which every countable subgroup is free (ℵ1-free groups) are n-slender, cm-slender, and lcH-slender. In particular, every homomorphism from a completely metrizable group to an ℵ1-free group has an open kernel. We also show that ℵ1-free abelian groups are lcHslender, which is especially interesting in light of the fact that some ℵ1-free abelian groups are neither n-slender nor cm-slender. The strongly ℵ1-free abelian groups are shown to be n-slender, cm-slender, and lcH-slender. We also give a characterization of the cm-slender and lcH-slender abelian groups.
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The author thanks the anonymous referee for timely helpful feedback to improve the paper.
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The author is supported by European Research Council grant PCG-336983.
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Corson, S.M. Automatic continuity of ℵ1-free groups. Isr. J. Math. 237, 267–285 (2020). https://doi.org/10.1007/s11856-020-2006-z
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DOI: https://doi.org/10.1007/s11856-020-2006-z