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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
DOI: https://doi.org/10.1090/S0002-9939-1976-0415573-9
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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DOI: https://doi.org/10.1090/S0002-9939-1976-0415573-9
Keywords: Topological completion, zero-set ultrafilter, $ \sigma $-discrete, locally finite, extension of functions, Katětov-Shirota Theorem
Article copyright: © Copyright 1976 American Mathematical Society

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