Topological Ramsey spaces and metrically Baire sets

TL;DR

This paper characterizes a class of topological Ramsey spaces where every Baire set is Ramsey, generalizing previous results and introducing new spaces related to ascending parameter words and block sequences.

Contribution

It defines a new class of topological Ramsey spaces with Baire sets being Ramsey and introduces new spaces generalizing earlier constructions.

Findings

Every Baire set in the projected spaces is Ramsey.

The class includes spaces generalizing ascending parameter words and block sequences.

Answers a question of S. Todorcevic and extends prior results.

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}, Pr\"omel-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 Pr\"omel-Voigt \cite{PromVoi} and the spaces $\FIN_m^{[\infty]}$, $0<m<\omega$, of block sequences defined by Todorcevic \cite{Todo}.