κ-Borel sets, κ-Baire spaces, and filtrations between topologies

Filtrations are certain transfinite sequences of topologies increasing in strength and interpolating between two given topologies σ and τ, with τ being stronger than σ. We prove general results on stabilization at τ of filtrations interpolating between σ and τ. These topological results involve an interplay between κ-Borel sets with respect to the topology σ and κ-Baireness of the topology τ.