Countable dense homogeneous linear topological 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 conjectured that, if C_p(X) is CDH, where X is separable metrizable, then X must be discrete. We prove this conjecture. 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 linear topological 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. This is a joint work (still in progress) with Tadek Dobrowolski and Mikołaj Krupski.