Local operations and eventually open actions

TL;DR

This paper explores continuous actions of Polish groups on Polish spaces, introducing the concept of eventually open actions and linking them to invariant structures through Scott analysis.

Contribution

It develops Scott analysis for orbit equivalence relations and characterizes eventually open actions via hereditarily countable invariant structures.

Findings

Introduces the concept of eventually open actions.

Characterizes actions with complete systems of invariant structures.

Extends Scott analysis to new classes of group actions.

Abstract

We study continuous actions of Polish groups on Polish spaces. We develop Scott analysis introduced by Hjorth for studying orbit equivalence relations. We define eventually open actions and prove that this property characterizes the actions endowed with a complete system of hereditarily countable invariant structures.