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.