On the covering and the additivity number of the real line

Abstract

We show that the real line R cannot be covered by k many nowhere dense sets iff whenever D = { D i : i ∈ k } 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 ∖ D i | > ω |G\backslash {D_i}| > \omega for all i ∈ k 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 ∈ k } 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 ⊆ G Q \subseteq G such that | Q ∖ D i | > ω |Q\backslash {D_{i}}| > \omega for all i ∈ k i \in k .