Cohen-stable families of subsets of integers (Q2732277)
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scientific article; zbMATH DE number 1623520
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Cohen-stable families of subsets of integers |
scientific article; zbMATH DE number 1623520 |
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Cohen-stable families of subsets of integers (English)
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11 March 2002
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Cohen forcing
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mad families
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splitting families
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maximal almost disjoint family
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Cohen extension
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0.86158055
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0.8571359
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0.8473857
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0.84496236
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0.8445449
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0.8398254
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In this paper, the author considers the (still open) problem whether there exists (in ZFC) a maximal almost disjoint family of subsets of the integers which remains maximal in any Cohen extension. Although the problem is not yet solved in this paper, some interesting information is provided. First, it is shown that for a maximal almost disjoint family in order to remain maximal in any Cohen extension, it is necessary and sufficient that every bijection from \(\omega\) to the set of rational numbers must have a somewhere dense image on some member of the family. Secondly, a characterization of the existence of such a family is given.
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