The reverse mathematics of Cousin's lemma

Jordan Mitchell Barrett

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arXiv · 2011.13060

The reverse mathematics of Cousin's lemma

Jordan Mitchell Barrett

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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 $\mathsf{ATR}_0$.

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Jordan Mitchell Barrett. 2020-11-25. The reverse mathematics of Cousin's lemma. https://arxiv.org/abs/2011.13060

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