The Weihrauch lattice at the level of $\boldsymbol{\Pi}_1^1\mathsf{-CA}_0$: the Cantor-Bendixson theorem

TL;DR

This paper explores the computational complexity of the Cantor-Bendixson theorem within the Weihrauch lattice, revealing how problem strength varies with topological properties of Polish spaces at the level of -CA.

Contribution

It is the first systematic analysis of problems at the -CA level in the Weihrauch lattice, connecting reverse mathematics with computable analysis.

Findings

Problem strength depends on topological properties of spaces.

Some problems have uniform strength across sufficiently rich spaces.

This work advances understanding of the computational content of classical theorems.

Abstract

This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor-Bendixson theorem and various problems arising from them. In the framework of reverse mathematics these theorems are equivalent respectively to $\mathsf{ATR}_0$ and $\boldsymbol{\Pi}_1^1\mathsf{-CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this is the first systematic study of problems at the level of $\boldsymbol{\Pi}_1^1\mathsf{-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.