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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 \(i\in k\).

MSC:
03E35 Consistency and independence results
03E40 Other aspects of forcing and Boolean-valued models
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