  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}.
