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Proceedings of the American Mathematical Society
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
Published electronically: May 9, 2003
MathSciNet review: 2021267
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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 $(\ddagger)$.

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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

Tomasz Weiss
Affiliation: Institute of Mathematics, WSRP, 08-110 Siedlce, Poland

PII: S 0002-9939(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

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