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 $D = \{ {D_i}:i \in k\}$ 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 $i \in k$.