zbMATH — the first resource for mathematics

On the covering and the additivity number of the real line. (English) Zbl 0823.03026
Summary: We show that the real line \(R\) cannot be covered by \(k\) many nowhere dense sets iff whenever \(D= \{D_ i: i\in k\}\) is a family of dense open sets of \(R\) there exists a countable dense set \(G\) of \(R\) such that \(| G\backslash D_ i|< \omega\) for all \(i\in k\). We also show that the union of \(k\) meagre sets of the real line is a meagre set iff for every family \(D= \{D_ i: i\in k\}\) of dense open sets of \(R\) and for every countable dense set \(G\) of \(R\) there exists a dense set \(Q\subseteq G\) such that \(| Q\backslash D_ i|< \omega\) for all \(i\in k\).

03E35 Consistency and independence results
03E40 Other aspects of forcing and Boolean-valued models
Full Text: DOI
[1] Tomek Bartoszyński, Additivity of measure implies additivity of category, Trans. Amer. Math. Soc. 281 (1984), no. 1, 209 – 213. · Zbl 0538.03042
[2] Tomek Bartoszyński, Combinatorial aspects of measure and category, Fund. Math. 127 (1987), no. 3, 225 – 239. · Zbl 0635.04001
[3] Tomek Bartoszyński and Jaime I. Ihoda, On the cofinality of the smallest covering of the real line by meager sets, J. Symbolic Logic 54 (1989), no. 3, 828 – 832. · Zbl 0686.03023
[4] Tomek Bartoszyński, Jaime I. Ihoda, and Saharon Shelah, The cofinality of cardinal invariants related to measure and category, J. Symbolic Logic 54 (1989), no. 3, 719 – 726. · Zbl 0686.03022
[5] Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111 – 167. · Zbl 0561.54004
[6] D. H. Fremlin, The partially ordered sets of measure and category and Tukey’s ordering, preprint. · Zbl 0799.06004
[7] -, Cichon’s diagram, Seminaire d’Initiation a l’Analyse , 23eme anne: 1983-1984, Université Pierre et Marie Curie, (Paris-VI), Paris, expose 5. · Zbl 1270.03087
[8] Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. · Zbl 0419.03028
[9] Kyriakos Keremedis, Partition reals and the consistency of \?&lt;\?\?\?(&real;), Math. Logic Quart. 39 (1993), no. 4, 545 – 550. · Zbl 0799.03057
[10] Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. · Zbl 0534.03026
[11] Kenneth Kunen, Random and Cohen reals, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 887 – 911. · Zbl 0588.03035
[12] Arnold W. Miller, A characterization of the least cardinal for which the Baire category theorem fails, Proc. Amer. Math. Soc. 86 (1982), no. 3, 498 – 502. · Zbl 0506.03012
[13] Arnold W. Miller, The Baire category theorem and cardinals of countable cofinality, J. Symbolic Logic 47 (1982), no. 2, 275 – 288. · Zbl 0487.03026
[14] Arnold W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), no. 1, 93 – 114. , https://doi.org/10.1090/S0002-9947-1981-0613787-2 Arnold W. Miller, Corrections and additions to: ”Some properties of measure and category”, Trans. Amer. Math. Soc. 271 (1982), no. 1, 347 – 348. · Zbl 0472.03040
[15] Jerry E. Vaughan, Small uncountable cardinals and topology, Open problems in topology, North-Holland, Amsterdam, 1990, pp. 195 – 218. With an appendix by S. Shelah.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.