Injective tests of low complexity in the plane

TL;DR

This paper explores injective tests for classifying Borel sets and oriented graphs in the plane, focusing on low complexity and continuous homomorphisms to understand their structure and classification.

Contribution

It introduces injective characterization methods for Borel sets and graphs, extending previous non-injective approaches to injective versions with low complexity.

Findings

Injective tests effectively characterize Borel sets in specific classes.

Continuous homomorphisms relate to the classification of oriented graphs.

New methods for injective classification of sets and graphs in the plane.

Abstract

We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.