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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:
 3e+35 Consistency and independence results 3e+40 Other aspects of forcing and Boolean-valued models
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