Topological Ramsey spaces and metrically Baire sets

We characterize a class of topological Ramsey spaces such that each element R of the class induces a collection {Rk}k<ω of projected spaces which have the property that every Baire set is Ramsey. Every projected space Rk 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 [1], Carlson-Simpson [3], Prömel-Voigt [23], and Voigt [29]. 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 [3] and Prömel-Voigt [23] and the spaces FIN[∞] m , 0 < m < ω, of block sequences defined by Todorcevic [25].