Abstract

We find a characterization of the covering number $cov({\mathbb R})$ , of the real line in terms of trees. We also show that the cofinality of $cov({\mathbb R})$ is greater than or equal to ${\mathfrak n}_\lambda$ for every $\lambda \in cov({\mathbb R}),$ where $\mathfrak n_\lambda \geq add({\mathcal L})$ ( $add( {\mathcal L})$ is the additivity number of the ideal of all Lebesgue measure zero sets) is the least cardinal number k for which the statement: $(\exists{\mathcal G}\in [^\omega \omega ]^{\leq \lambda })(\forall{\mathcal F}\in [^\omega \omega ]^{\leq k})(\exists g\in{\mathcal G})(\exists h\in ^\omega \omega )(\forall f\in{\mathcal F})(\forall ^\infty n)(\exists u\in [g(n),g(n+1))(f(u)=h(u))$ fails