The reverse mathematics of Cousin's lemma

TL;DR

This paper investigates the logical strength of Cousin's lemma across different classes of functions within reverse mathematics, revealing its equivalences and lower bounds relative to well-known subsystems.

Contribution

It establishes the precise axiomatic strength of Cousin's lemma for continuous, Baire 1, and Baire 2 functions in second-order arithmetic.

Findings

Cousin's lemma for continuous functions is equivalent to WKL_0.

Cousin's lemma for Baire 1 functions is at least as strong as ACA_0.

Cousin's lemma for Baire 2 functions is at least as strong as ATR_0.

Abstract

Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin's lemma for various classes of functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. We prove that, over $\mathsf{RCA}_0$: (i) Cousin's lemma for continuous functions is equivalent to the system $\mathsf{WKL}_0$; (ii) Cousin's lemma for Baire 1 functions is at least as strong as $\mathsf{ACA}_0$; (iii) Cousin's lemma for Baire 2 functions is at least as strong as $\mathsf{ATR}_0$.