Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54D60

Retrieve articles in all journals with MSC: 54D60

Additional Information

Keywords: Topological completion, zero-set ultrafilter, $ \sigma $-discrete, locally finite, extension of functions, Katětov-Shirota Theorem
Article copyright: © Copyright 1976 American Mathematical Society