Teoremas de Jordan y Brouwer: demostraciones elementales

Abstract

[en] In mathematics, an elementary proof is one that uses only basic techniques. In this work we provide the elementary proof of the Jordan curve at $\mathbb{R}^2$ and the Brouwer fixed point theorems. The Jordan curve theorem on $\mathbb{R}^2$ tells us that every simple closed curve separates the plane into two connected components. The elementary proof uses four lemmas, whose proofs we also carry out, and consists of approximating the curve by means of polygons.

Let $\mathbb{B}^n \subset \mathbb{R}^n$ be the n-dimensional unit ball. Brouwer's fixed point theorem tells us that every continuous function $f: \mathbb{B}^n \rightarrow \mathbb{B}^n$ has a fixed point. The elementary proof is based on the fact that the compact, convex and non-empty subsets of $\mathbb{R}^n$ are homeomorphic and on Sperner's Lemma, which we also state and prove. Sperner's Lemma is a combinatorial result of coloring n-dimensional simplices.