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)



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
MathSciNet review: 0259053
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

  • [F$ _{1}$] Z. Frolík, A contribution to the descriptive theory of sets and spaces, Proc. Sympos. General Topology and its Relations to Modern Analysis and Algebra (Prague, 1961), Academic Press, New York, 1962, pp. 157-173. MR 26 #3002. MR 0145471 (26:3002)
  • [F$ _{2}$] Zdenek Frolík, A survey of separable descriptive theory of sets and spaces, Czechoslovak Math. J. (to appear). MR 0266757 (42:1660)
  • [GJ] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [H] Paul R. Halmos, Measure theory, Van Nostrand, Princeton, N. J., 1950. MR 11, 504. MR 0033869 (11:504d)
  • [N$ _{1}$] Stelios Negrepontis, Absolute Baire sets, Proc. Amer. Math. Soc. 18 (1967), 691-694. MR 35 #4883. MR 0214031 (35:4883)
  • [N$ _{2}$] -, Baire sets in topological spaces, Arch. Math. (Basel) 18 (1967), 603-608. MR 36 #3314. MR 0220248 (36:3314)
  • [RS] Kenneth A. Ross and Karl R. Stromberg, Baire sets and Baire measures, Ark. Mat. 6 (1965), 151-160. MR 33 #4224. MR 0196029 (33:4224)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28.10, 54.00

Retrieve articles in all journals with MSC: 28.10, 54.00

Additional Information

Keywords: Baire set, zero-set, Stone-Čech compactification
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society