On a characterization of path connected topological fields

TL;DR

This paper characterizes path connected topological fields by identifying the key property that enables a Gelfand-type correspondence with compact Hausdorff spaces.

Contribution

It establishes that being a path connected topological field is both necessary and sufficient for a Gelfand correspondence to hold for all compact Hausdorff spaces.

Findings

Path connectedness is the key property for the correspondence.

Real numbers are a prototypical example of such a field.

The characterization applies to all compact Hausdorff spaces.

Abstract

The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space $X$ and the space of maximal ideals of the ring of real valued continuous functions $C(X,\mathbb{R})$. More explicitly, our motivation is the following question: What is the essential property of the topological field $F=\mathbb{R}$ that makes such a correspondence valid for all compact Hausdorff spaces? It turns out that such a perfect correspondence exists if and only if $F$ is a path connected topological field.