2014/08/09 by Andrea Medini, Medini, Andrea · 1 citation
Mathematics · #54E52 #54G20 #54H05 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1408.2137
openalex publication_date 2014/08/09 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We show that, for a coanalytic subspace X of 2^\ω, the countable\ndense homogeneity of X^\ω is equivalent to X being Polish. This\nstrengthens a result of Hru vs 'ak and Zamora Avil 'es. Then, inspired by\nresults of Hern 'andez-Guti 'errez, Hru vs 'ak and van Mill, using a\ntechnique of Medvedev, we construct a non-Polish subspace X of 2^\ω\nsuch that X^\ω is countable dense homogeneous. This gives the first\n\ZFC answer to a question of Hru vs 'ak and Zamora Avil 'es.\nFurthermore, since our example is consistently analytic, the equivalence result\nmentioned above is sharp. Our results also answer a question of Medini and\nMilovich. Finally, we show that if every countable subset of a zero-dimensional\nseparable metrizable space X is included in a Polish subspace of X then\nX^\ω is countable dense homogeneous.\n