The Shape of Infinity

TL;DR

This paper introduces a novel approach to understanding the Stone-Cech compactification by developing its theory without relying on traditional topological concepts, using gauge spaces and categorical ideas instead.

Contribution

It presents a new method to develop the Stone-Cech compactification theory without open or closed sets, using gauge spaces and categorical completeness.

Findings

Provides a self-contained construction of the Stone-Cech compactification

Defines compactness via total boundedness and completeness in gauge spaces

Uses Lawvere's categorical approach to define completeness

Abstract

In these expository notes, intended for students without background in point-set topology, we develop the basic theory of the Stone-Cech compactification without reference to open sets, closed sets, filters, or nets. In particular, this means we cannot use any of the usual definitions of topological space. This may seem like proposing to run a marathon while hopping on one foot, but it is easier than it may appear, and not devoid of interest. We use gauge spaces (uniform spaces presented by a family of pseudometrics); we define compactness as total boundedness plus completeness; and we define completeness using a variation on Lawvere's categorical characterization of completeness for metric spaces.