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)

 
 

 

Some remarks on totally imperfect sets


Authors: Andrzej Nowik and Tomasz Weiss
Journal: Proc. Amer. Math. Soc. 132 (2004), 231-237
MSC (2000): Primary 03E15; Secondary 03E20, 28E15
DOI: https://doi.org/10.1090/S0002-9939-03-06997-1
Published electronically: May 9, 2003
MathSciNet review: 2021267
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following two theorems.


Theorem 1. Let $X$ be a strongly meager subset of $2^{\omega\times\omega}$. Then it is dual Ramsey null.

We will say that a $\sigma$-ideal $\mathcal{I}$of subsets of $2^{\omega}$ satisfies the condition $(\ddagger)$ iff for every $X \subseteq2^\omega$, if

\begin{displaymath}\forall_{f \in \omega^{\uparrow \omega }} \lbrace g \in \omeg... ...\omega }\colon \neg(f \prec g) \rbrace \cap X \in \mathcal{I}, \end{displaymath}

then $X \in \mathcal{I}$.


Theorem 2. The $\sigma$-ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition $(\ddagger)$.


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

  • [B] T. Bartoszynski Remarks on small sets of reals, preprint
  • [BJ] T. Bartoszynski, H. Judah, Borel images of sets of reals, Real Analysis Exchange 20(2) (1994/5), 536 - 558. MR 96k:04005
  • [CS] T.J. Carlson, S.G. Simpson, A dual form of Ramsey's Theorem, Advances in Mathematics 53 (1984), 265 - 290. MR 85h:04002
  • [G1] E. Grzegorek, Always of the first category sets, Proceedings of the 12th Winter School on Abstract Analysis Srni(Bohemian Weald), 15-29 January, 1984, Section of Topology, Supplemento ai Rend. Circ. Mat. Palermo, Serie II-numero 6-1984, 139-147. MR 86a:00004
  • [G2] E. Grzegorek, Always of the first category sets (II), Proceedings of the 13th Winter School on Abstract Analysis Srni(Bohemian Weald), 20-27 January, 1985, Section of Topology, Supplemento ai Rend. Circ. Mat. Palermo, Serie II-numero 10-1985, 43-48. MR 88j:54054
  • [L] G.G. Lorentz, On a problem of additive number theory, Proceedings of the American Mathematical Society 5 (1954), 838 - 841. MR 16:113f
  • [M] A. W. Miller, On $\lambda'$-sets, preprint, 2003.
  • [N] A. Nowik, Remarks about transitive version of perfectly meager sets, Real Analysis Exchange 22(1) (1996/97) 406 - 412. MR 97m:54043
  • [NSW] A. Nowik, M. Scheepers, T. Weiss, The algebraic sum of sets of real numbers with strong measure zero sets. Journal of Symbolic Logic 63 (1998), 301 - 324. MR 99c:54049
  • [NW1] A. Nowik, T. Weiss, Not every Q-set is perfectly meager in the transitive sense, Proceedings of The American Mathematical Society 128(496), No 10 (October 2000), 3017 - 3024. MR 2000m:03116
  • [NW2] A. Nowik, T. Weiss, Strongly meager sets of real numbers and tree forcing notions. Proceedings of The American Mathematical Society 110, No 4 (2002), 1183-1187. MR 2002j:03049

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E15, 03E20, 28E15

Retrieve articles in all journals with MSC (2000): 03E15, 03E20, 28E15


Additional Information

Andrzej Nowik
Affiliation: Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80 – 952 Gdańsk, Poland
Address at time of publication: Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, 81–825 Sopot, Poland
Email: matan@julia.univ.gda.pl, nowik@impan.gda.pl

Tomasz Weiss
Affiliation: Institute of Mathematics, WSRP, 08-110 Siedlce, Poland
Email: weiss@wsrp.siedlce.pl

DOI: https://doi.org/10.1090/S0002-9939-03-06997-1
Keywords: Strongly meager sets, dual Ramsey null sets, partitions
Received by editor(s): March 14, 2002
Received by editor(s) in revised form: August 19, 2002
Published electronically: May 9, 2003
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society