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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following two theorems.

**Theorem 1.** *Let be a strongly meager subset of . Then it is dual Ramsey null.*

We will say that a -ideal of subsets of satisfies the condition iff for every , if

then .

**Theorem 2.** *The -ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition .*

**[B]**T. Bartoszynski*Remarks on small sets of reals*,*preprint***[BJ]**Tomek Bartoszyński and Haim Judah,*Borel images of sets of reals*, Real Anal. Exchange**20**(1994/95), no. 2, 536–558. MR**1348078****[CS]**Timothy J. Carlson and Stephen G. Simpson,*A dual form of Ramsey’s theorem*, Adv. in Math.**53**(1984), no. 3, 265–290. MR**753869**, 10.1016/0001-8708(84)90026-4**[G1]***Proceedings of the 12th winter school on abstract analysis*, Circolo Matematico di Palermo, Palermo, 1984. Section of topology; Held at Srní, January 15–29, 1984; Rend. Circ. Mat. Palermo (2) 1984, Suppl. No. 6. MR**782700****[G2]**E. Grzegorek,*Always of the first category sets. II*, Proceedings of the 13th winter school on abstract analysis (Srní, 1985), 1985, pp. 43–48 (1986). MR**894270****[L]**G. G. Lorentz,*On a problem of additive number theory*, Proc. Amer. Math. Soc.**5**(1954), 838–841. MR**0063389**, 10.1090/S0002-9939-1954-0063389-3**[M]**A. W. Miller,*On -sets, preprint*, 2003.**[N]**Andrzej Nowik,*Remarks about a transitive version of perfectly meager sets*, Real Anal. Exchange**22**(1996/97), no. 1, 406–412. MR**1433627****[NSW]**Andrej Nowik, Marion Scheepers, and Tomasz Weiss,*The algebraic sum of sets of real numbers with strong measure zero sets*, J. Symbolic Logic**63**(1998), no. 1, 301–324. MR**1610427**, 10.2307/2586602**[NW1]**Andrzej Nowik and Tomasz Weiss,*Not every 𝑄-set is perfectly meager in the transitive sense*, Proc. Amer. Math. Soc.**128**(2000), no. 10, 3017–3024. MR**1664434**, 10.1090/S0002-9939-00-05355-7**[NW2]**Andrzej Nowik and Tomasz Weiss,*Strongly meager sets of real numbers and tree forcing notions*, Proc. Amer. Math. Soc.**130**(2002), no. 4, 1183–1187 (electronic). MR**1873795**, 10.1090/S0002-9939-01-06174-3

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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

Email:
weiss@wsrp.siedlce.pl

DOI:
http://dx.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