End compactifications and general compactifications

Matt Insall,Malgorzata Marciniak,Peter Loeb

doi:10.4115/jla.2014.6.7

Abstract

We use the insights of Robinson's nonstandard analysis as a powerful tool to extend and simplify the construction of compactifications of regular spaces. In particular, we deal with the Stone-Čech compactification and compactifications formed from topological ends. For the nonstandard extension of a metric space, the monad of a standard point x is the set of all points infinitesimally close to x. Monads of standard points can also be defined for non-metric spaces. The new points of a compactification are formed from equivalence classes of points that are not in the monad of any standard point. Adjoining such points to the original point set allows a better understanding of the relationship between the original space and the set of compactifying points. Our results for end compactifications are established for regular, connected and locally connected spaces. Simple examples of end compactifications are the two point compactification of the real line and the one point compactification of the complex plane. In general, one or more ends form the "far" termination of a non-compact space, while any "hole" in the space corresponds to an end that is "near". Our results on ends extend previous work initiated by Hans Freudenthal on ends understood as equivalence classes of nested sequences of nonempty open sets with compact boundaries. We show, for example, that a product of spaces with at least two non-compact factors has only one end. A brief overview and introduction to nonstandard analysis begins the discussion.

Highlights

We use the insights of Robinson’s nonstandard analysis ([12], [10], see section) to extend and simplify previous works in the literature on compactifications
The new points of a compactification are formed from equivalence classes of remote points
The resulting compactification is a compact space containing the original point set as a dense subset

Summary

Introduction

We use the insights of Robinson’s nonstandard analysis ([12], [10], see section) to extend and simplify previous works in the literature on compactifications. Given a family of bounded real-valued functions on the original space, one can call two remote points equivalent if the nonstandard extension of each of the functions in the family has infinitesimal variation on the two point set. This leads to such compactifications as the Stone-Cech compactification, but constructed here for spaces that need only be regular. Our approach to compactifications of topological spaces, extending work of Salbany and Todorov ([14], [15]), will utilize equivalence relations on the nonstandard extension of a given topological space, independent of any algebraic structure, thereby producing quite general compactifications. We do not assume here that our underlying space is locally compact

Background for Nonstandard Methods

General Compactifications

End Compactifications

Product Spaces