Skip to Main Content

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 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Extending Baire Property by countably many sets
HTML articles powered by AMS MathViewer

by Piotr Zakrzewski PDF
Proc. Amer. Math. Soc. 129 (2001), 271-278 Request permission

Abstract:

We prove that if ZFC is consistent so is ZFC + “for any sequence $(A_{n})$ of subsets of a Polish space $\langle X,\tau \rangle$ there exists a separable metrizable topology $\tau ’$ on $X$ with $\mathbf {B}(X,\tau )\subseteq \mathbf {B}(X,\tau ’)$, $\operatorname {MGR}(X,\tau ’)\cap \mathbf {B}(X,\tau )=\operatorname {MGR} (X,\tau )\cap \mathbf {B}(X,\tau )$ and $A_{n}$ Borel in $\tau ’$ for all $n$.” This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets. A uniform argument is presented, which gives a new proof of the latter as well. Some consequences of these extension properties are also studied.
References
  • S. Banach and K. Kuratowski, Sur une gĂ©nĂ©ralization du problème de la mesure, Fund. Math. 14 (1929), 127–131.
  • Tomek BartoszyĹ„ski and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295
  • J. B. Brown and G. V. Cox, Classical theory of totally imperfect spaces, Real Anal. Exchange 7 (1981/82), no. 2, 185–232. MR 657320
  • Tim Carlson, Extending Lebesgue measure by infinitely many sets, Pacific J. Math. 115 (1984), no. 1, 33–45. MR 762199
  • D. H. Fremlin, Measure algebras, in Handbook of Boolean algebras, North-Holland, 1989, 876–980.
  • E. Grzegorek Always of the first category sets, Rend. Circ. Mat. Palermo, II. Ser. Suppl. 6 (1984), 139–147.
  • E. Grzegorek, Always of the first category sets. II, Proceedings of the 13th winter school on abstract analysis (SrnĂ­, 1985), 1985, pp. 43–48 (1986). MR 894270
  • Anastasis Kamburelis, A new proof of the Gitik-Shelah theorem, Israel J. Math. 72 (1990), no. 3, 373–380 (1991). MR 1120228, DOI 10.1007/BF02773791
  • A. Kamburelis, On cardinal numbers related to Baire property, preprint, WrocĹ‚aw 1989.
  • Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
  • Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
  • Arnold W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), no. 3, 575–584. MR 716618, DOI 10.2307/2273449
  • I. RecĹ‚aw and P. Zakrzewski, Strong Fubini properties of ideals, Fund. Math. 159 (1999), 135–152.
  • Robert M. Solovay, Real-valued measurable cardinals, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428. MR 0290961
  • Piotr Zakrzewski, Strong Fubini axioms from measure extension axioms, Comment. Math. Univ. Carolin. 33 (1992), no. 2, 291–297. MR 1189659
  • P. Zakrzewski, Universally meager sets, Proc. Amer. Math. Soc., to appear.
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E35, 54E52, 28A05
  • Retrieve articles in all journals with MSC (2000): 03E35, 54E52, 28A05
Additional Information
  • Piotr Zakrzewski
  • Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
  • MR Author ID: 239503
  • Email: piotrzak@mimuw.edu.pl
  • Received by editor(s): July 10, 1998
  • Received by editor(s) in revised form: March 16, 1999
  • Published electronically: June 14, 2000
  • Additional Notes: The author was partially supported by KBN grant 2 P03A 047 09 and by the Alexander von Humboldt Foundation
  • Communicated by: Carl G. Jockusch, Jr.
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 271-278
  • MSC (2000): Primary 03E35, 54E52; Secondary 28A05
  • DOI: https://doi.org/10.1090/S0002-9939-00-05505-2
  • MathSciNet review: 1695095