Universally meager sets

Author:
Piotr Zakrzewski

Journal:
Proc. Amer. Math. Soc. **129** (2001), 1793-1798

MSC (1991):
Primary 03E20, 54E52; Secondary 54G99, 28A05

Published electronically:
November 2, 2000

MathSciNet review:
1814112

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We study category counterparts of the notion of a universal measure zero set of reals.

We say that a set is universally meager if every Borel isomorphic image of is meager in . We give various equivalent definitions emphasizing analogies with the universally null sets of reals.

In particular, two problems emerging from an earlier work of Grzegorek are solved.

**1.**Edward Grzegorek,*Solution of a problem of Banach on 𝜎-fields without continuous measures*, Bull. Acad. Polon. Sci. Sér. Sci. Math.**28**(1980), no. 1-2, 7–10 (1981) (English, with Russian summary). MR**616191****2.**E. Grzegorek*Always of the first category sets*, Rend. Circ. Mat. Palermo, II. Ser. Suppl.**6**(1984), 139-147. CMP**17:10****3.**E. Grzegorek,*Always of the first category sets. II*, Proceedings of the 13th winter school on abstract analysis (Srní, 1985), 1985, pp. 43–48 (1986). MR**894270****4.**Winfried Just, Arnold W. Miller, Marion Scheepers, and Paul J. Szeptycki,*The combinatorics of open covers. II*, Topology Appl.**73**(1996), no. 3, 241–266. MR**1419798**, 10.1016/S0166-8641(96)00075-2**5.**Alexander S. Kechris,*Classical descriptive set theory*, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR**1321597****6.**Arnold W. Miller,*Mapping a set of reals onto the reals*, J. Symbolic Logic**48**(1983), no. 3, 575–584. MR**716618**, 10.2307/2273449**7.**Arnold W. Miller,*Special subsets of the real line*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233. MR**776624****8.**A. Nowik, T. Weiss,*Not every -set is perfectly meager in the transitive sense*, to appear in Proc. Amer. Math. Soc. CMP**99:06****9.**Szymon Plewik,*Towers are universally measure zero and always of first category*, Proc. Amer. Math. Soc.**119**(1993), no. 3, 865–868. MR**1152287**, 10.1090/S0002-9939-1993-1152287-4**10.**Ireneusz Recław,*Products of perfectly meagre sets*, Proc. Amer. Math. Soc.**112**(1991), no. 4, 1029–1031. MR**1059635**, 10.1090/S0002-9939-1991-1059635-2**11.**I. Recaw and P. Zakrzewski,*Strong Fubini properties of ideals*, Fund. Math.**159**(1999), 135-152. CMP**99:08****12.**P. Zakrzewski,*Extending Baire property by countably many sets*, to appear in Proc. Amer. Math. Soc. CMP**99:14**

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Additional Information

**Piotr Zakrzewski**

Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

Email:
piotrzak@mimuw.edu.pl

DOI:
https://doi.org/10.1090/S0002-9939-00-05726-9

Keywords:
Measure and category,
Borel sets,
Baire property,
$\sigma$-algebra,
$\sigma$-ideal

Received by editor(s):
March 23, 1999

Received by editor(s) in revised form:
September 7, 1999

Published electronically:
November 2, 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.

Article copyright:
© Copyright 2000
American Mathematical Society