A new ordered compactification

D C Kent,Tom Richmond

doi:10.1155/s0161171293000146

Abstract

A new Wallman-type ordered compactificationγ∘Xis constructed using maximalCZ-filters (which have filter bases obtained from increasing and decreasing zero sets) as the underlying set. A necessary and sufficient condition is given forγ∘Xto coincide with the Nachbin compactificationβ∘X; in particularγ∘X=β∘XwheneverXhas the discrete order. The Wallman ordered compactificationω∘Xequalsγ∘XwheneverXis a subspace ofRn. It is shown thatγ∘Xis alwaysT1, but can fail to beT1-ordered orT2.

Highlights

It is well known that the Stone-(ech compactification X of a Ts.s topological space can be described as a Wallman-type compactification using maxima] filters of zero sets as the underlying set for the compactification
We have extended this construction to Ts.s-ordered spaces, and the result is a new ordered compactification which we call %X
We have not yet characterized the class of spaces for which %X woX, we have shown that this class includes all subspaces of Rn

Summary

Introduction

For an arbitrary space X we denote by CI*(X) (respectively, CD*(X)) the set of all increasing {respectively, decreasing), continuous maps PROPOSITION 1.5 Let X, Y be spaces and [ X Y an increasing, continuous map. PROPOSITION 1.6 Let X be a T.s-ordered spe. PROPOSITION 1.8 If X is a Ts.5-ordered space and z X, the CZ() is the unique maximal CZ-

Results

Conclusion