Transfinite sequences of topologies, descriptive complexity, and approximating equivalence relations

TL;DR

This paper introduces the concept of filtration between topologies, explores its stabilization, and uses descriptive set theory to analyze transfinite sequences that approximate equivalence relations.

Contribution

It presents a novel notion of filtration between topologies and studies its properties, linking topological and descriptive set theoretic concepts to approximate equivalence relations.

Findings

Filtrations can be stabilized through transfinite sequences.

Descriptive complexity influences the behavior of these filtrations.

The approach provides a new framework for analyzing equivalence relations.

Abstract

We introduce the notion of filtration between topologies and study its stabilization properties. Descriptive set theoretic complexity plays a role in this study. Filtrations lead to natural transfinite sequences approximating a given equivalence relation. We investigate those.