In Dually discrete spaces, Topology Appl. 155 (2008), 1420–1425, Alas et. al. proved that ordinals are hereditarily dually discrete and asked whether the product of two ordinals has the same property. In Products of certain dually discrete spaces, Topology Appl. 156 (2009), 2832–2837, Peng proved a number of partial results and left open the question of whether the product of two stationary subsets of ${\omega }_1$ is dually discrete. We answer the first question affirmatively and as a consequence also give...

Let $W$ be the subspace of ${\mathbb{N}}^{*}$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p\in {\mathbb{N}}^{*}$.

In this paper, we deal with the product of spaces which are either $𝒢$-spaces or ${𝒢}_p$-spaces, for some $p\in {\omega }^{*}$. These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are $𝒢$-spaces, and every ${𝒢}_p$-space is a $𝒢$-space, for every $p\in {\omega }^{*}$. We prove that if $\{X_{\mu }:\mu <{\omega }_1\}$ is a set of spaces whose product $X={\prod }_{\mu <{\omega }_1}{X}_{\mu }$ is a $𝒢$-space, then there is $A\in {\left[{\omega }_1\right]}^{\le \omega }$ such that $X_{\mu }$ is countably compact for every $\mu \in {\omega }_1\setminus A$. As a consequence, $X^{{\omega }_1}$ is a $𝒢$-space iff $X^{{\omega }_1}$ is countably compact, and if $X^{2^{𝔠}}$ is a $𝒢$-space, then all...