Book contents
- Frontmatter
- Contents
- Preface
- Part I Introduction to set theory
- Appendix. An axiomatic development of set theory
- Part II Topics in combinatorial set theory
- 12 Stationary sets
- 13 Δ-systems
- 14 Ramsey's Theorem and its generalizations. Partition calculus
- 15 Inaccessible cardinals. Mahlo cardinals
- 16 Measurable cardinals
- 17 Real-valued measurable cardinals, saturated ideals
- 18 Weakly compact and Ramsey cardinals
- 19 Set mappings
- 20 The square-bracket symbol. Strengthenings of the Ramsey counterexamples
- 21 Properties of the power operation. Results on the singular cardinal problem
- 22 Powers of singular cardinals. Shelah's Theorem
- Hints for solving problems of Part II
- Bibliography
- List of symbols
- Name index
- Subject index
19 - Set mappings
Published online by Cambridge University Press: 10 May 2010
- Frontmatter
- Contents
- Preface
- Part I Introduction to set theory
- Appendix. An axiomatic development of set theory
- Part II Topics in combinatorial set theory
- 12 Stationary sets
- 13 Δ-systems
- 14 Ramsey's Theorem and its generalizations. Partition calculus
- 15 Inaccessible cardinals. Mahlo cardinals
- 16 Measurable cardinals
- 17 Real-valued measurable cardinals, saturated ideals
- 18 Weakly compact and Ramsey cardinals
- 19 Set mappings
- 20 The square-bracket symbol. Strengthenings of the Ramsey counterexamples
- 21 Properties of the power operation. Results on the singular cardinal problem
- 22 Powers of singular cardinals. Shelah's Theorem
- Hints for solving problems of Part II
- Bibliography
- List of symbols
- Name index
- Subject index
Summary
Definition 19.1. 1. The mapping F : X → P(X) on the set X is said to be a set mapping if x ∉ F(x) for each x ∈ X.
2. We say that the above set mapping has order λ if |F(x)| < λ for each x ∈ X.
3. We call the set S ⊂ X free with respect to F if x ∉ F(y) for each x,y ∈ S.
We observe that, given a set mapping F of order λ on X, if Y ⊂ X and Fy (X) = Y ∩ F(X), then the set mapping is also of order λ, and every set S ⊂ Y that is free with respect to Fy is also free with respect to F. We will frequently use this observation without calling attention to it explicitly.
It was P. Turán who first pointed out that if the order of F is small and X is large, then often there is a large free subset. He proved the existence of a free subset of cardinality 2ℵ0 in the case when λ = ω and |X| = 2ℵ0. The next lemma was established by D. Lázár in 1936.
Lemma 19.1.Assume K ≥ ω is regular, K > λ, and F is a set mapping of order λ on k. Then there is a set S ⊂ k of cardinality k that is free with respect to F.
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- Set Theory , pp. 228 - 233Publisher: Cambridge University PressPrint publication year: 1999