$\kappa$-Borel sets, $\kappa$-Baire spaces, and filtrations between topologies

TL;DR

This paper investigates the properties of filtrations of topologies between two given topologies, focusing on stabilization and the interplay between $orel$ sets and $aireness$ in the context of $orel$ hierarchies and transfinite sequences.

Contribution

It establishes general results on the stabilization of filtrations at a stronger topology, linking $orel$ sets and $aireness$ properties in a transfinite setting.

Findings

Filtrations stabilize at the stronger topology under certain conditions.

The interplay between $orel$ sets and $aireness$ influences the behavior of filtrations.

Results connect transfinite topological sequences with descriptive set theory concepts.

Abstract

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