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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Stone-Čech remainders which make continuous images normal


Authors: William Fleissner and Ronnie Levy
Journal: Proc. Amer. Math. Soc. 106 (1989), 839-842
MSC: Primary 54D40; Secondary 54C05, 54D15
MathSciNet review: 963571
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Abstract: If $ f$ is a continuous surjection from a normal space $ X$ onto a regular space $ Y$, then there are a space $ Z$ and a perfect map $ bf:Z \to Y$ extending $ f$ such that $ X \subset Z \subset \beta X$. If $ f$ is a continuous surjection from normal $ X$ onto Tychonov $ Y$ and $ \beta X\backslash X$ is sequential, then $ Y$ is normal. More generally, if $ f$ is a continuous surjection from normal $ X$ onto regular $ Y$ and $ \beta X\backslash X$ has the property that countably compact subsets are closed (this property is called $ C$-closed), then $ Y$ is normal. There is an example of a normal space $ X$ such that $ \beta X\backslash X$ is $ C$-closed but not sequential. If $ X$ is normal and $ \beta X\backslash X$ is first countable, then $ \beta X\backslash X$ is locally compact.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0963571-8
PII: S 0002-9939(1989)0963571-8
Keywords: Stone-Cech remainder, continuous image, normal, ACRIN, $ C$-closed, normality inducing
Article copyright: © Copyright 1989 American Mathematical Society