Polish group actions and computability

TL;DR

This paper explores the computability aspects of Polish group actions, focusing on how elements of a Polish G-space can be characterized by computable functions based on a countable basis.

Contribution

It introduces a framework for analyzing the computability of characteristic functions associated with elements of Polish G-spaces under group actions.

Findings

Characterization of computable functions for elements in Polish G-spaces

Analysis of the complexity of these functions in the context of group actions

Foundations for further research on computability in topological group actions

Abstract

Let G be a closed subgroup of the group of all permutations of a countably infinite set. Let X be a Polish G-space with a countable basis A of clopen sets. Each x from X defines a characteristic function f on A by f(U)=1 iff x belongs to U (where U is from A). We consider computable complexity of f and some related questions.