P-spaces and the Whyburn property

TL;DR

This paper explores the properties of P-spaces related to the Whyburn property, constructing examples and analyzing how these properties behave under set-theoretic assumptions and products.

Contribution

It provides the first examples of non-weakly Whyburn P-spaces of size continuum and examines the influence of set-theoretic hypotheses on these spaces.

Findings

Existence of non-weakly Whyburn P-spaces of size continuum under CH.

Weak Kurepa Hypothesis implies non-weakly Whyburn P-spaces of size .

Product of Lindel4f weakly Whyburn and Whyburn P-spaces is weakly Whyburn.

Abstract

We investigate the Whyburn and weakly Whyburn property in the class of $P$-spaces, that is spaces where every countable intersection of open sets is open. We construct examples of non-weakly Whyburn $P$-spaces of size continuum, thus giving a negative answer under CH to a question of Pelant, Tkachenko, Tkachuk and Wilson. In addition, we show that the weak Kurepa Hypothesis (a set-theoretic assumption weaker than CH) implies the existence of a non-weakly Whyburn $P$-space of size $\aleph_2$. Finally, we consider the behavior of the above-mentioned properties under products; we show in particular that the product of a Lindel\"of weakly Whyburn P-space and a Lindel\"of Whyburn $P$-space is weakly Whyburn, and we give a consistent example of a non-Whyburn product of two Lindel\"of Whyburn $P$-spaces.