A countable dense homogeneous topological vector space is a Baire space

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. Hern'andez-Guti'errez. 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 Hru\v{s}'ak, Zamora Avil'es, and Hern'andez-Guti'errez concerning countable dense homogeneous products.