On the (in)equivalence of Brouwer's fixed point theorem and Sperner's lemma

TL;DR

This paper explores the relationship between Brouwer's fixed point theorem and Sperner's lemma in one dimension, demonstrating their interdependence and fundamental differences related to the properties of the underlying number fields.

Contribution

It provides a proof of Brouwer's fixed point theorem using Sperner's lemma and clarifies that they are not equivalent due to the need for completeness in Brouwer's theorem.

Findings

Sperner's lemma holds in the rational numbers

Brouwer's fixed point theorem requires completeness

The two theorems are not equivalent in general

Abstract

We consider Brouwer's fixed point theorem and Sperner's lemma in one dimension. We present a proof of the Brouwer theorem using the Sperner lemma, and vice versa. However, we also show that they are not equivalent, because the Sperner lemma holds in the ordered field of rational numbers, whereas proving the Brouwer theorem requires the property of completeness.