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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the topological completion

Authors: Howard Curzer and Anthony W. Hager
Journal: Proc. Amer. Math. Soc. 56 (1976), 365-370
MSC: Primary 54D60
MathSciNet review: 0415573
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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.

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Keywords: Topological completion, zero-set ultrafilter, <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\sigma$">-discrete, locally finite, extension of functions, Kat&#283;tov-Shirota Theorem
Article copyright: © Copyright 1976 American Mathematical Society