Topological Ramsey spaces and metrically Baire sets

We characterize a class of topological Ramsey spaces such that each element RR of the class induces a collection {Rk}k<ω{Rk}k<ω of projected spaces which have the property that every Baire set is Ramsey. Every projected space RkRk 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 some results of Carlson, Carlson–Simpson, Prömel–Voigt, and 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 and Prömel–Voigt and the spaces FINm[∞], 0