Abstract
We prove a result on perfect cliques with respect to countably many \(G_{\delta }\) relations on a complete metric space. As an application, we show that a Polish group contains a free subgroup generated by a perfect set as long as it contains any uncountable free subgroup. This answers a recent question of Gła̧b and Strobin.
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Research of Martin Doležal was supported by RVO: 67985840. Research of Wiesław Kubiś was supported by GAČR grant P201 14 07880S and RVO: 67985840.
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Doležal, M., Kubiś, W. Perfect independent sets with respect to infinitely many relations. Arch. Math. Logic 55, 847–856 (2016). https://doi.org/10.1007/s00153-016-0498-3
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DOI: https://doi.org/10.1007/s00153-016-0498-3