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 Free Access

Abstract | References | Similar Articles | Additional Information

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

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