A countable dense homogeneous topological vector space is a Baire space

Dobrowolski, Tadeusz; Krupski, Mikołaj; Marciszewski, Witold

doi:10.1090/proc/15271

A countable dense homogeneous topological vector space is a Baire space
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by Tadeusz Dobrowolski, Mikołaj Krupski and Witold Marciszewski PDF

Proc. Amer. Math. Soc. 149 (2021), 1773-1789 Request permission

Abstract:

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hernández-Gutiérrez. We also conclude that, for any infinite-dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hrušák, Zamora Avilés, and Hernández-Gutiérrez concerning countable dense homogeneous products.

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