Construction of nearly pseudocompactification

Abstract

A space is nearly pseudocompact if and only if $\upsilon X\backslash X$ is dense in $βX\backslash X$. If we denote $K=cl_{βX}(\upsilon X\backslash X)$, then $δX=X\cup(βX\backslash K)$ 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 $βX$. In this paper our main motivation is to construct nearly pseudocompact extension of $X$ independently and not using any kind of extension property of $βX$. An alternative construction of $βX$ 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 $δX$ 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 $βX.$ We have further shown that the extension $δX$ is unique with respect to certain properties.