Nonstandard Analysis in Topology

TL;DR

This paper introduces a nonstandard analysis framework with three axioms and applies it to various topics in point-set topology, providing new insights and simplifications for existing concepts.

Contribution

It presents a novel axiomatic approach to nonstandard analysis and applies it to topology, including new characterizations of realcompactification and sober spaces.

Findings

New nonstandard characterization of Hewitt realcompactification

Simplified proofs of existing topological results

Introduction of nonstandard methods to analyze sober spaces

Abstract

We present Nonstandard Analysis by three axioms: the {\em Extension, Transfer and Saturation Principles} in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactification and the nonstandard characterization of the sober spaces seem to be new in the literature on nonstandard analysis. Others have already close counterparts but they are presented here with essential simplifications.