On the Banach-Stone Theorem and the Manifold Topological Classification

TL;DR

This paper provides a simple proof of the Banach-Stone Theorem and applies it to classify Euclidean manifolds topologically, addressing open problems and offering a straightforward proof of the Brouwer theorem.

Contribution

It introduces a simplified set-theoretic proof of the Banach-Stone Theorem and applies it to topologically classify Euclidean manifolds, including a new proof of Brouwer's theorem.

Findings

Set-theoretic proof of Banach-Stone Theorem

Application to Euclidean manifold classification

Straightforward proof of Brouwer's theorem

Abstract

We present a simple set-theoretic proof of the Banach-Stone Theorem .We thus apply this Topological classification theorem to the still unsolved problem of topological classification of euclidean manifolds through two conjectures and additionally we give a straightforward proof of the famous Brower theorem for Manifolds topologically classified by their euclidean dimensions.