The reverse mathematics of Cousin's lemma

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 ${\mathsf{ATR}}_0$.

Honours thesis, Victoria University of Wellington, 30 Oct 2020. Supervised by Rod Downey and Noam Greenberg. 6+51 pages, 10 figures