Cousin's lemma in second-order arithmetic

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 ${\mathsf{WKL}}_0$; (ii) Cousin's lemma for Baire class 1 functions is equivalent to ${\mathsf{ACA}}_0$; (iii) Cousin's lemma for Baire class 2 functions, or for Borel functions, are both equivalent to ${\mathsf{ATR}}_0$ (modulo some induction).

13 pages, no figures