Constructive equivalence between Brouwer's fixed-point theorem and weak K\"onig's lemma

Tatsuji Kawai

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arXiv · 2609.04644

Constructive equivalence between Brouwer's fixed-point theorem and weak K\"onig's lemma

Tatsuji Kawai

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Abstract

In the context of constructive reverse mathematics, we show that Brouwer's fixed-point theorem and weak K\"onig's lemma (WKL) are equivalent. To derive WKL from Brouwer's fixed-point theorem, the construction of a continuous function on the unit square without fixed points due to Orevkov [Soviet Math. Doklady (1963), 1253--1256] is generalised to yield a uniformly continuous function on the unit square whose fixed points encode information about infinite paths of a given infinite tree.

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Tatsuji Kawai. 2026-09-04. Constructive equivalence between Brouwer's fixed-point theorem and weak K\"onig's lemma. https://arxiv.org/abs/2609.04644

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