Construction of nearly pseudocompactification

Abstract

A space is nearly pseudocompact if and only if $\upsilon X\backslash X$ is dense in $\beta X\backslash X$. If we denote $K=c{l}_{\beta X}(\upsilon X\backslash X)$, 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.