We consider properties of Tanaka spaces (introduced in Mynard F., More on strongly sequential spaces, Comment. Math. Univ. Carolin. 43 (2002), 525–530), strongly sequential spaces, and weakly sequential spaces. Applications include product theorems for these types of spaces.

In this paper the following two propositions are proved: (a) If $X_{\alpha }$, $\alpha \in A$, is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product ${\square }_{\alpha \in A}{X}_{\alpha }$ can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If $X_{\alpha }$, $\alpha \in A$, is an infinite system of Brown Hausdorff topological spaces then the box product ${\square }_{\alpha \in A}{X}_{\alpha }$ is also Brown Hausdorff, and hence, it is connected. A space is Brown if...

We establish a relation between covering properties (e.gĿindelöf degree) of two standard topological spaces (Lemmas 4 and 5). Some cardinal inequalities follow as easy corollaries.

We introduce and study, following Z. Frol’ık, the class $ℬ\left(𝒫\right)$ of regular $P$-spaces $X$ such that the product $X\times Y$ is pseudo-${\aleph }_1$-compact, for every regular pseudo-${\aleph }_1$-compact $P$-space $Y$. We show that every pseudo-${\aleph }_1$-compact space which is locally $ℬ\left(𝒫\right)$ is in $ℬ\left(𝒫\right)$ and that every regular Lindelöf $P$-space belongs to $ℬ\left(𝒫\right)$. It is also proved that all pseudo-${\aleph }_1$-compact $P$-groups are in $ℬ\left(𝒫\right)$. The problem of characterization of subgroups of $ℝ$-factorizable (equivalently, pseudo-${\aleph }_1$-compact) $P$-groups is considered as well. We give some necessary...

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...

A refined common generalization of known theorems (Arhangel’skii, Michael, Popov and Rančin) on the Fréchetness of products is proved. A new characterization, in terms of products, of strongly Fréchet topologies is provided.