Construction of nearly pseudocompactification

TL;DR

This paper presents a novel construction of nearly pseudocompact extensions of a space using $z$-ultrafilters, avoiding reliance on properties of the Stone-Čech compactification, and proves the uniqueness of this extension.

Contribution

It introduces an independent method to construct nearly pseudocompact extensions via $z$-ultrafilters, differing from traditional approaches based on $eta X$ properties.

Findings

Constructed $eta X$ using $z$-ultrafilters.

Developed a new construction of $ ext{nearly pseudocompact}$ extension $ ext{} ext{} ext{delta}X$.

Proved the uniqueness of the constructed extension.

Abstract

A space is nearly pseudocompact if and only if $\upsilon X\backslash X$ is dense in $\beta X\backslash X$. If we denote $K=cl_{\beta X}(\upsilon X\backslash X)$, then $\delta X=X\cup(\beta X\backslash K)$ is referred by Henriksen and Rayburn \cite{hr80} as nearly pseudocompact extension of $X$. Henriksen and Rayburn studied the nearly pseudocompact extension using different properties of $\beta X$. In this paper our main motivation is to construct nearly pseudocompact extension of $X$ independently and not using any kind of extension property of $\beta X$. An alternative construction of $\beta X$ is made by taking the family of all $z$-ultrafilters on $X$ and then topologized in a suitable way. In this paper we also adopted the similar idea of constructing the $\delta X$ from the scratch, taking the collection of all $z$-ultrafilters on $X$ of some kind, called $hz$-ultrafilters, together with fixed $z$-ultrafilter and then be topologized in the similar way what we do in the construction of $\beta X.$ We have further shown that the extension $\delta X$ is unique with respect to certain properties.