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 Gδ 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.