On the topological completion

Abstract:

Let $X$ be a Tychonoff space. As is well known, the points of the Stone-Čech compactification $\beta X$ “are” the zero-set ultrafilters of $X$, and the points of the Hewitt real-compactification $\upsilon X$ are the zero-set ultrafilters which are closed under countable intersection. It is shown here that a zero-set ultrafilter is a point of the Dieudonné topological completion $\delta X$ iff the family of complementary cozero sets is $\sigma$-discretely, or locally finitely, additive. From this follows a characterization of those dense embeddings $X \subset Y$ such that each continuous metric space-valued function on $X$ extends over $Y$, and a somewhat novel proof of the Katětov-Shirota Theorem.