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 $X$, 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.

slightly modified and expanded version

References in corpus (2)

• On the weak and pointwise topologies in function spaces II
• Countable dense homogeneity of function spaces