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.

💡 Research Summary

The paper investigates the interplay between several generalizations of P‑spaces—namely almost P‑spaces and weak P‑spaces—and the Volterra property (and its weaker version). A space is called Volterra if any two dense $G_\delta$ subsets intersect densely; it is weakly Volterra if the intersection of any two dense $G_\delta$’s is non‑empty (not necessarily dense). The authors begin by recalling that every P‑space is hereditarily Volterra, a classical result that follows from the fact that in a P‑space every $G_\delta$ set is open, so the intersection of two dense $G_\delta$’s is an open dense set.

The first major contribution is the analysis of almost P‑spaces. An almost P‑space is defined by the condition that every non‑empty $G_\delta$ set is dense. The authors prove that if $X$ is an almost P‑space, then every dense subspace of $X$ is Volterra, and every open subspace of $X$ is also Volterra. The proof relies on the observation that a dense subspace inherits the $G_\delta$ structure of the ambient space: any non‑empty $G_\delta$ in the subspace is also a non‑empty $G_\delta$ in $X$, and therefore dense in $X$, which forces it to be dense in the subspace as well. Consequently, two dense $G_\delta$’s in the subspace intersect densely. The same argument works for open subspaces because openness preserves the $G_\delta$ families.

In contrast, the authors exhibit a closed subspace of an almost P‑space that fails to be Volterra. The counterexample is constructed inside the remainder $\beta\mathbb N\setminus\mathbb N$ of the Stone–Čech compactification of the natural numbers. By selecting a suitable closed set $F$, they produce two dense $G_\delta$ subsets of $F$ whose intersection is empty, showing that the Volterra property does not descend to arbitrary closed subspaces of an almost P‑space. This highlights a sharp distinction between open/dense and closed hereditary behavior in this context.

The second line of investigation concerns weak P‑spaces, where every countable set is closed (equivalently, every countable set is a $G_\delta$). The authors construct a Tychonoff weak P‑space that is not weakly Volterra. The construction uses a product of carefully chosen spaces to create two dense $G_\delta$ sets whose intersection is empty. This demonstrates that the weak P‑space condition alone does not guarantee any form of the Volterra property, answering a natural question left open by earlier work.

A further significant result is the production of a pair of spaces $A$ and $B$, each of which is both hereditarily Volterra and hereditarily Baire, yet their product $A\times B$ fails to be weakly Volterra. The spaces are built so that each factor satisfies the strong intersection property individually, but in the product one can isolate two dense $G_\delta$ sets whose intersection is nowhere dense (or empty). This example shows that the Volterra property is not preserved under arbitrary products, even when both factors enjoy very strong regularity properties.

Finally, the authors present a Hausdorff space that simultaneously satisfies the definitions of a weak P‑space and an almost P‑space, yet contains a subspace that is not weakly Volterra. The example is obtained by taking a particular filter space that is both weakly $P$ and almost $P$, and then isolating a closed subspace that violates the weak Volterra condition. This construction underscores that the conjunction of the two “P‑type” properties does not force the Volterra property on all subspaces.

In the concluding section the authors summarize the landscape revealed by their results: while P‑spaces are robustly Volterra, their natural weakenings (almost P‑spaces and weak P‑spaces) only retain the Volterra property under restrictive hereditary conditions (dense or open subspaces) and can fail dramatically in other contexts (closed subspaces, products, or mixed P‑type spaces). The paper opens several avenues for further research, such as identifying precise combinatorial or set‑theoretic criteria that guarantee the Volterra property in products of almost P‑spaces, or exploring whether additional separation axioms can bridge the gap between weak P‑spaces and weakly Volterra spaces. Overall, the work deepens our understanding of how subtle variations in the definition of P‑type spaces affect the behavior of dense $G_\delta$ intersections, and it provides a suite of concrete counterexamples that delineate the boundaries of these interactions.