A countable dense homogeneous topological vector space is a Baire space

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 X , the function space C p ( X ) 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 E (dual Banach space E ∗ E^\ast ), the space E E equipped with the weak topology ( E ∗ 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.