## 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 $(\ddagger )$.*

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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