A countable dense homogeneous topological vector space is a Baire space

We prove that every homogeneous countable dense topological space containing a copy of the Cantor set is Baire space. In particular, vector It follows that, for any nondiscrete metrizable X X , function C Subscript p Baseline left-parenthesis upper X right-parenthesis"> C p ( stretchy="false">) encoding="application/x-tex">C_p(X) not homogeneous. This answers question posed recently by R. Hernández-Gutiérrez. also conclude infinite-dimensional Banach E"> E encoding="application/x-tex">E (dual E Superscript asterisk"> ∗ encoding="application/x-tex">E^\ast ), equipped with weak topology (^\ast topology) generalize some results Hrušák, Zamora Avilés, and Hernández-Gutiérrez concerning products.