On countable dense homogeneous topological vector spaces

Friday, October 1, 2021 - 10:00


Speaker Information
Mikolaj Krupski
University of Warsaw

Abstract or Additional Information

A topological space X is countable dense homogeneous (CDH) if X is separable, and given any two countable dense subsets D, E of X, there is an autohomeomorphism of X that maps D onto E. Canonical examples of CDH spaces include the Cantor set, the Hilbert cube, and all separable Banach spaces. 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. In 2020, R. Hernandez-Gutierrez gave the first consistent example of such a space X. He asked whether a metrizable space X must be discrete, provided C_p(X) is CDH. We answer this question in the affirmative by proving that every CDH topological vector space is a Baire space. This is a joint work with T. Dobrowolski and W. Marciszewski.