Turning Borel sets into clopen sets effectively

TL;DR

This paper develops an effective method to transform Borel sets into clopen sets in Polish spaces, maintaining their structure and enabling hyperarithmetical parameter selection for uniformity.

Contribution

It introduces an effective version of the theorem for converting Borel sets into clopen sets, with conditions for hyperarithmetical parameter choice and a uniformity result.

Findings

Parameters can be chosen hyperarithmetically

Established a uniformity result for the transformation

Preserved Borel structure during the transformation

Abstract

We present the effective version of the theorem about turning Borel sets in Polish spaces into clopen sets while preserving the Borel structure of the underlying space. We show that under some conditions the emerging parameters can be chosen in a hyperarithmetical way and using this we prove a uniformity result.