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Proceedings of the American Mathematical Society

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

 
 

 

A note on $ \mathcal{Z}$-realcompactifications


Author: Anthony J. D’Aristotle
Journal: Proc. Amer. Math. Soc. 32 (1972), 615-618
MSC: Primary 54.53
DOI: https://doi.org/10.1090/S0002-9939-1972-0288730-9
MathSciNet review: 0288730
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Abstract: Orrin Frink showed that the real-valued functions over a Tychonoff space X which may be continuously extended to $ \omega (\mathcal{Z})$, the Wallman-type compactification associated with a normal base $ \mathcal{Z}$ for X, are those which are $ \mathcal{Z}$-uniformly continuous

Let $ \mathcal{Z}$ be a delta normal base on a Tychonoff space X, and let $ \eta (\mathcal{Z})$ be the corresponding $ \mathcal{Z}$-realcompactification of X. In this note we show that countable $ \mathcal{Z}$-uniform continuity is a sufficient but not a necessary condition for continuously extending real-valued functions over X to $ \eta (\mathcal{Z})$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0288730-9
Keywords: Normal base, delta normal base, $ \mathcal{Z}$-uniform continuity, countable $ \mathcal{Z}$-uniform continuity
Article copyright: © Copyright 1972 American Mathematical Society

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