An Expository on Some Nonstandard Compactifications

Alagu Somasundaram,Rukhmoni Kala

doi:10.4236/apm.2021.1110054

Abstract

Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement *X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group Γ of the locally compact Abelian group G is extended to the set of the above mentioned equivalence classes. A compact topology on is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in .

Highlights

The general procedure is as follows: A suitable equivalence relation is defined on an enlargement ∗X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group
The foundation stone for Nonstandard analysis was laid by Abraham Robinson in 1966 with the publication of his book “Nonstandard analysis” [1]
A topological space X is said to be a compactification of X if X can be embedded in X as a dense subspace

Summary

Introduction

The foundation stone for Nonstandard analysis was laid by Abraham Robinson in 1966 with the publication of his book “Nonstandard analysis” [1]. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement ∗X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. With compact support) or the dual group Γ of the locally compact Abelian group G is extended to the set X of the above mentioned equivalence classes. Non-Standard, Compactification, Completely Regular Space, Locally Compact Hausdorff Space, Locally Compact Abelian Group

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Conclusion