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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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.

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Some remarks on totally imperfect sets
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by Andrzej Nowik and Tomasz Weiss PDF
Proc. Amer. Math. Soc. 132 (2004), 231-237 Request permission

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 $X \subseteq 2^\omega$, if \[ \forall _{f \in \omega ^{\uparrow \omega }} \lbrace g \in \omega ^{\uparrow \omega }\colon \neg (f \prec g) \rbrace \cap X \in \mathcal {I}, \] 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
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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
  • MR Author ID: 631175
  • ORCID: 0000-0001-9201-7202
  • Email: weiss@wsrp.siedlce.pl
  • 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.
  • © Copyright 2003 American Mathematical Society
  • 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
  • MathSciNet review: 2021267