Classification and Non-classification of Homeomorphism Relations

TL;DR

This paper explores the complexity of homeomorphism relations on various topological spaces, extending known results to locally compact spaces and identifying limitations in non-locally compact cases, with open problems proposed.

Contribution

It extends the classification of homeomorphism relations to locally compact Polish spaces and identifies failure cases for spaces lacking local compactness.

Findings

Homeomorphism relation on locally compact Polish spaces is reducible to an orbit equivalence relation.

The reduction fails for spaces with a single non-locally compact point, such as certain subsets of C3^3.

A list of open problems in the classification of homeomorphism relations is provided.

Abstract

The paper deals with the program of determining the complexity of various homeomorphism relations. The homeomorphism relation on compact Polish spaces is known to be reducible to an orbit equivalence relation of a continuous Polish group action (Kechris-Solecki). It is shown that this result extends to locally compact Polish spaces, but does not hold for spaces in which local compactness fails at only one point. In fact it fails for those subsets of $\mathbb{R}^3$ which are unions of an open set and a point. In the end a list of open problems is given in this area of research.