Some remarks on totally imperfect sets
Authors:
Andrzej Nowik and Tomasz Weiss
Journal:
Proc. Amer. Math. Soc. 132 (2004), 231237
MSC (2000):
Primary 03E15; Secondary 03E20, 28E15
Published electronically:
May 9, 2003
MathSciNet review:
2021267
Fulltext 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
(96k:04005)
 [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
(85h:04002), http://dx.doi.org/10.1016/00018708(84)900264
 [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 (86a:00004)
 [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
(88j:54054)
 [L]
G.
G. Lorentz, On a problem of additive number
theory, Proc. Amer. Math. Soc. 5 (1954), 838–841. MR 0063389
(16,113f), http://dx.doi.org/10.1090/S00029939195400633893
 [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
(97m:54043)
 [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
(99c:54049), http://dx.doi.org/10.2307/2586602
 [NW1]
Andrzej
Nowik and Tomasz
Weiss, Not every 𝑄set is perfectly
meager in the transitive sense, Proc. Amer.
Math. Soc. 128 (2000), no. 10, 3017–3024. MR 1664434
(2000m:03116), http://dx.doi.org/10.1090/S0002993900053557
 [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
(2002j:03049), http://dx.doi.org/10.1090/S0002993901061743
 [B]
 T. Bartoszynski Remarks on small sets of reals, preprint
 [BJ]
 T. Bartoszynski, H. Judah, Borel images of sets of reals, Real Analysis Exchange 20(2) (1994/5), 536  558. MR 96k:04005
 [CS]
 T.J. Carlson, S.G. Simpson, A dual form of Ramsey's Theorem, Advances in Mathematics 53 (1984), 265  290. MR 85h:04002
 [G1]
 E. Grzegorek, Always of the first category sets, Proceedings of the 12th Winter School on Abstract Analysis Srni(Bohemian Weald), 1529 January, 1984, Section of Topology, Supplemento ai Rend. Circ. Mat. Palermo, Serie IInumero 61984, 139147. MR 86a:00004
 [G2]
 E. Grzegorek, Always of the first category sets (II), Proceedings of the 13th Winter School on Abstract Analysis Srni(Bohemian Weald), 2027 January, 1985, Section of Topology, Supplemento ai Rend. Circ. Mat. Palermo, Serie IInumero 101985, 4348. MR 88j:54054
 [L]
 G.G. Lorentz, On a problem of additive number theory, Proceedings of the American Mathematical Society 5 (1954), 838  841. MR 16:113f
 [M]
 A. W. Miller, On sets, preprint, 2003.
 [N]
 A. Nowik, Remarks about transitive version of perfectly meager sets, Real Analysis Exchange 22(1) (1996/97) 406  412. MR 97m:54043
 [NSW]
 A. Nowik, M. Scheepers, T. Weiss, The algebraic sum of sets of real numbers with strong measure zero sets. Journal of Symbolic Logic 63 (1998), 301  324. MR 99c:54049
 [NW1]
 A. Nowik, T. Weiss, Not every Qset is perfectly meager in the transitive sense, Proceedings of The American Mathematical Society 128(496), No 10 (October 2000), 3017  3024. MR 2000m:03116
 [NW2]
 A. Nowik, T. Weiss, Strongly meager sets of real numbers and tree forcing notions. Proceedings of The American Mathematical Society 110, No 4 (2002), 11831187. MR 2002j:03049
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
03E15,
03E20,
28E15
Retrieve articles in all journals
with MSC (2000):
03E15,
03E20,
28E15
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, 08110 Siedlce, Poland
Email:
weiss@wsrp.siedlce.pl
DOI:
http://dx.doi.org/10.1090/S0002993903069971
PII:
S 00029939(03)069971
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
