In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $\alpha \le {2}^{}$, a topological group G such that ${G}^{\gamma}$ is countably compact for all cardinals γ < α, but ${G}^{\alpha}$ is not countably compact?
Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M{A}_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M{A}_{countable}$. However, the question has remained...

We will show that under ${M\phantom{\rule{-1.8pt}{0ex}}A\phantom{\rule{0.2pt}{0ex}}}_{countable}$ for each $k\in \mathbb{N}$ there exists a group whose $k$-th power is countably compact but whose ${2}^{k}$-th power is not countably compact. In particular, for each $k\in \mathbb{N}$ there exists $l\in [k,{2}^{k})$ and a group whose $l$-th power is countably compact but the $l+1$-st power is not countably compact.

It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size $\u212d$ admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its $\omega $-th power countably compact. In particular, a free Abelian group does not admit a Hausdorff $p$-compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff...

Under 𝔭 = 𝔠, we prove that it is possible to endow the free abelian group of cardinality 𝔠 with a group topology that makes its square countably compact. This answers a question posed by Madariaga-Garcia and Tomita and by Tkachenko. We also prove that there exists a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group) whose square is countably compact. This answers a question posed by Grant.

For $\varnothing \ne M\subseteq {\omega}^{*}$, we say that $X$ is quasi $M$-compact, if for every $f:\omega \to X$ there is $p\in M$ such that $\overline{f}\left(p\right)\in X$, where $\overline{f}$ is the Stone-Čech extension of $f$. In this context, a space $X$ is countably compact iff $X$ is quasi ${\omega}^{*}$-compact. If $X$ is quasi $M$-compact and $M$ is either finite or countable discrete in ${\omega}^{*}$, then all powers of $X$ are countably compact. Assuming $CH$, we give an example of a countable subset $M\subseteq {\omega}^{*}$ and a quasi $M$-compact space $X$ whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi...

We show that if $\mathcal{A}$ is an uncountable AD (almost disjoint) family of subsets of $\omega $ then the space $\Psi \left(\mathcal{A}\right)$ does not admit a continuous selection; moreover, if $\mathcal{A}$ is maximal then $\Psi \left(\mathcal{A}\right)$ does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.

Following Malykhin, we say that a space $X$ is if $X$ contains a family $\mathcal{D}$ of dense subsets such that $\left|\mathcal{D}\right|>\Delta \left(X\right)$ and the intersection of every two elements of $\mathcal{D}$ is nowhere dense, where $\Delta \left(X\right)=min\left\{\right|U|:U$ is a nonempty open subset of $X\}$ is the of $X$. We show that, for every cardinal $\kappa $, there is a compact extraresolvable space of size and dispersion character ${2}^{\kappa}$. In connection with some cardinal inequalities, we prove the equivalence of the following statements: 1) ${2}^{\kappa}<{2}^{{\kappa}^{+}}$, 2) ${\left({\kappa}^{+}\right)}^{\kappa}$ is extraresolvable and 3) $A{\left({\kappa}^{+}\right)}^{\kappa}$ is extraresolvable, where $A\left({\kappa}^{+}\right)$...

In this paper, we deal with the product of spaces which are either $\mathcal{G}$-spaces or ${\mathcal{G}}_{p}$-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 $\mathcal{G}$-spaces, and every ${\mathcal{G}}_{p}$-space is a $\mathcal{G}$-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 $\mathcal{G}$-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 $\mathcal{G}$-space iff ${X}^{{\omega}_{1}}$ is countably compact, and if ${X}^{{2}^{\U0001d520}}$ is a $\mathcal{G}$-space, then all...

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