On $N^{\aleph _{1}}$ and the almost-Lindelöf property

Abstract:

In 1970, Kemperman and Maharam proved that there exists a Baire measure $\mu$ on ${N^{\mathbf {c}}}$ (where $N$ is the set of natural numbers) such that ${N^{\mathbf {c}}}$ may be covered by a famliy of elementary open $\mu$-null sets and used this to prove that ${R^{\mathbf {c}}}$ (where $R$ is the set of real numbers) does not have the “almost-Lindelöf” property. We define ${\mathbf {K}}$ to be the smallest cardinal $\kappa$ for which there exists a collection of $\kappa$ closed subsets of $R$ each of Lebesgue measure zero and which covers $R$, and we show that in the above results ${\mathbf {c}}$ can be replaced by ${\mathbf {K}}$. We then note that we have shown elsewhere that it is consistent with the negation of the continuum hypothesis that ${\mathbf {K}} = {\aleph _1}$, and this, therefore, implies that it is consistent with the negation of the continuum hypothesis that ${R^\aleph }1$ not be almost-Lindelöf.