Grey subsets of Polish spaces

TL;DR

This paper develops an analogue of descriptive set theory for functions on Polish spaces, introduces a small index property for Polish topometric groups, and extends classical theorems to this new context.

Contribution

It introduces a new descriptive set theory framework for functions on Polish spaces and extends classical theorems to Polish topometric groups.

Findings

Polish topometric groups with ample generics have the small index property

Extended Effros and Hausdorff theorems to the topometric setting

Developed foundational tools for descriptive set theory in this context

Abstract

We develop the basics of an analogue of descriptive set theory for functions on a Polish space $X$. We use this to define a version of the small index property in the context of Polish topometric groups, and show that Polish topometric groups with ample generics have this property. We also extend classical theorems of Effros and Hausdorff to the topometric context.