The pseudocompact extension $\alpha X$

Abstract:

For any Tychonoff space we define $\alpha X = (\beta X - vX) \cup X = \beta X - (vX - X)$. We show that $\alpha X$ is the smallest pseudocompactification $Y$ of $X$ contained is $\beta X$ such that every free hyperreal $z$-ultrafilter on $X$ converges in $Y$ and is the largest pseudocompactification $Y$ of $X$ contained in $\beta X$ such that every point in $Y - X$ is contained in a zero set of $Y$ which does not intersect $X$. A space $S$ is defined to be $\alpha$-embedded in a space $X$ if $\alpha S \subset \beta X$. Properties of $\alpha$-embeddings and its relation to $v$-embeddings of Blair ${C^*}$-embeddings, $C$-embeddings, and well-embeddings are investigated. For instance, if $S$ is $\alpha$-embedded and dense in $X,S$ is fully well-embedded (for $P,R \subset X$, where $S \subset P \subset R \subset X$, $P$ is well-embedded in $R$) in $X$ iff $\alpha X - \alpha S = X - S$.