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On dense subsets of rational numbers. (English) Zbl 1001.03045

Summary: We consider the family of dense subsets of the rational numbers as a partially ordered set. We define cardinal numbers \({\mathbf p}_{\mathbb{Q}}\) and \({\mathbf t}_{\mathbb{Q}}\) for this partial order and we prove that \({\mathbf p}_{\mathbb{Q}}={\mathbf p}\) and \({\mathbf t}_{\mathbb{Q}}={\mathbf t}\), where \({\mathbf p}\) and \({\mathbf t}\) are the classical cardinal numbers describing combinatorial properties of the family of all infinite subsets of the natural numbers. We also consider some variant of the splitting number of dense subsets of the rationals.

MSC:

03E35 Consistency and independence results
03E05 Other combinatorial set theory
06A06 Partial orders, general
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