End compactifications and general compactifications

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.