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Proceedings of the American Mathematical Society

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

 

 

Closed Baire sets are (sometimes) zero-sets


Author: W. W. Comfort
Journal: Proc. Amer. Math. Soc. 25 (1970), 870-875
MSC: Primary 28.10; Secondary 54.00
DOI: https://doi.org/10.1090/S0002-9939-1970-0259053-7
MathSciNet review: 0259053
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Abstract: It is a theorem essentially due to Paul Halmos [H, 51.D] that each compact Baire set is a zero-set. Kenneth A. Ross and Karl Stromberg [RS] have shown (a bit more than the fact that) if $ X$ is a completely regular Hausdorff space which is locally compact and $ \sigma $-compact, then each closed Baire set in $ X$ is a zero-set; the same conclusion is known to hold in case $ X$ is Lindelöf and a $ {G_\delta }$ in $ \beta X$. In the present paper we prove the following theorem, and we show how the ``closed Baire set'' theorems of Ross and Stromberg emerge as corollaries: If $ X$ is Baire in $ \beta X$ and $ A$ is a closed Baire set in $ X$, then $ A$ is a zero-set in $ X$. Finally, we indicate how our theorem, and hence those of Ross and Stromberg, can be derived from early and forthcoming work of Frolík.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0259053-7
Keywords: Baire set, zero-set, Stone-Čech compactification
Article copyright: © Copyright 1970 American Mathematical Society