P-spaces and the Volterra property

We study the relationship between generalizations of $P$-spaces and Volterra (weakly Volterra) spaces, that is, spaces where every two dense $G_\delta$ have dense (non-empty) intersection. In particular, we prove that every dense and every open, but not every closed subspace of an almost $P$-space is Volterra and that there are Tychonoff non-weakly Volterra weak $P$-spaces. These results should be compared with the fact that every $P$-space is hereditarily Volterra. As a byproduct we obtain an example of a hereditarily Volterra space and a hereditarily Baire space whose product is not weakly Volterra. We also show an example of a Hausdorff space which contains a non-weakly Volterra subspace and is both a weak $P$-space and an almost $P$-space.Comment: in press on the Bulletin of the Australian Mathematical Societ

Complete nonmeasurability in regular families

We show that for a $\sigma$-ideal \ci with a Borel base of subsets of an uncountable Polish space, if \ca is (in several senses) a "regular" family of subsets from \ci then there is a subfamily of \ca whose union is completely nonmeasurable i.e. its intersection with every Borel set not in \ci does not belong to the smallest $\sigma$-algebra containing all Borel sets and \ci. Our results generalize results from \cite{fourpoles} and \cite{fivepoles}.Comment: 7 page