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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Closed Baire sets are (sometimes) zero-sets
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by W. W. Comfort
Proc. Amer. Math. Soc. 25 (1970), 870-875
DOI: https://doi.org/10.1090/S0002-9939-1970-0259053-7

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.
References
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Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • 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