Topological Ramsey spaces and metrically Baire sets

Abstract

We characterize a class of topological Ramsey spaces such that each element $\mathcal{R}$ of the class induces a collection $\{{\mathcal{R}}_k{\}}_{k<\omega }$ of projected spaces which have the property that every Baire set is Ramsey. Every projected space ${\mathcal{R}}_k$ is a subspace of the corresponding space of length-$k$ approximation sequences with the Tychonoff, equivalently metric, topology. This answers a question of S. Todorcevic and generalizes the results of Carlson \cite{Carlson}, Carlson-Simpson \cite{CarSim2}, Prömel-Voigt \cite{PromVoi}, and Voigt \cite{Voigt}. We also present a new family of topological Ramsey spaces contained in the aforementioned class which generalize the spaces of ascending parameter words of Carlson-Simpson \cite{CarSim2} and Prömel-Voigt \cite{PromVoi} and the spaces $\FIN_m^{[\infty]}$, $0<m<\omega$, of block sequences defined by Todorcevic \cite{Todo}.

17 pp