On countable dense homogeneous topological vector spaces

Recall that a topological space X is countable dense homogeneous (CDH) if X is separable, and given countable dense subsets D,E of X, there is an autohomeomorphism of X mapping D onto E. This is a classical notion tracing back to works of Cantor, Frechet and Brouwer. The canonical examples of CDH spaces include the Cantor set, the Hilbert cube, and all separable Banach spaces. All Borel, but not closed linear subspaces of Banach spaces are not CDH. By C_p(X) we denote the space of all continuous real-valued functions on a Tikhonov space X, endowed with the pointwise topology. V. Tkachuk asked if there exists a nondiscrete space X such that C_p(X) is CDH. Last year R. Hernandez Gutierrez gave the first consistent example of such a space X. He has asked whether a metrizable space X must be discrete, provided Cp(X) is CDH. We answer this question in the affirmative. Actually, combining our theorem with earlier results, we prove that, for a metrizable space X, C_p(X) is CDH if and only if X is discrete of cardinality less than pseudointersection number p. We also prove that every CDH topological vector space X is a Baire space. This implies that, for an infinite-dimensional Banach space E, both spaces (E,w) and (E*,w*) are not CDH. We generalize some results of Hrusak, Zamora Aviles, and Hernandez Gutierrez concerning countable dense homogeneous products. This is a joint work with Tadek Dobrowolski and Mikołaj Krupski. The preprint containing these results can be found here: https://arxiv.org/abs/2002.07423