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 | 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 is a completely regular Hausdorff space which is locally compact and -compact, then each closed Baire set in is a zero-set; the same conclusion is known to hold in case is Lindelöf and a in . 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 is Baire in and is a closed Baire set in , then is a zero-set in . Finally, we indicate how our theorem, and hence those of Ross and Stromberg, can be derived from early and forthcoming work of Frolík.

**[F]**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]**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]**Stelios Negrepontis,*Absolute Baire sets*, Proc. Amer. Math. Soc.**18**(1967), 691-694. MR**35**#4883. MR**0214031 (35:4883)****[N]**-,*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)**

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Additional Information

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