P-spaces and the Volterra property

TL;DR

This paper explores the relationships between various generalizations of P-spaces and Volterra properties, providing new examples and counterexamples that clarify their interactions and differences.

Contribution

It establishes new results about which subspaces of almost P-spaces are Volterra and provides examples of spaces with specific Volterra properties and their products.

Findings

Not all subspaces of almost P-spaces are Volterra.

Existence of hereditarily Volterra spaces with non-Volterra products.

Examples of spaces with mixed P-space and Volterra properties.

Abstract

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.