Universally meager sets

Author:
Piotr Zakrzewski

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

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

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

Published electronically:
November 2, 2000

MathSciNet review:
1814112

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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.**E. Grzegorek*Solution to a problem of Banach on -fields without continuous measures*, Bull. Acad. Pol. Sci.**28**(1980), 7-10. MR**82h:04005****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*, Rend. Circ. Mat. Palermo, II. Ser. Suppl.**10**(1985), 43-48. MR**88j:54054****4.**W. Just, A.W. Miller, M. Scheepers, P.J. Szeptycki,*The combinatorics of open covers (II)*, Topology Appl.**73**(1996), 241-266. MR**98g:03115a****5.**A. S. Kechris,*Classical descriptive set theory*, Graduate Texts in Math. 156, Springer-Verlag 1995. MR**96e:03057****6.**A.W. Miller,*Mapping a set of reals onto the reals*, J. Symb. Logic**48**(1983), 575-584. MR**84k:03125****7.**A.W. Miller,*Special subsets of the real line*in*Handbook of set-theoretic topology*, North-Holland 1984, 201-233. MR**86i:54037****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.**S. Plewik,*Towers are universally measure zero and always of first category*, Proc. Amer. Math. Soc.**119(3)**(1993), 865-868. MR**93m:04003****10.**I. Recaw,*Products of perfectly meager sets*, Proc. Amer. Math. Soc.**112(4)**(1991), 1029-1031. MR**91j:28001****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