Pseudocompact and countably compact abelian groups: Cartesian products and minimality

Abstract:

Denote by $\mathcal {G}$ the class of all Abelian Hausdorff topological groups. A group $G \in \mathcal {G}$ is minimal (totally minimal) if every continuous group isomorphism (homomorphism) $i:G \to H$ of $G$ onto $H \in \mathcal {G}$ is open. For $G \in \mathcal {G}$ let $\kappa (G)$ be the smallest cardinal $\tau \geq 1$ such that the minimality of ${G^\tau }$ implies the minimality of all powers of $G$. For $\mathcal {Q} \subset \mathcal {G}$, $\mathcal {Q} \ne \emptyset$, we set $\kappa (\mathcal {Q}) = \sup \{ \kappa (G):G \in \mathcal {G}\}$ and denote by $\alpha (\mathcal {Q})$ the smallest cardinal $\tau \geq 1$ having the following property: If $\{ {G_i}:i \in I\} \subset \mathcal {Q}$, $I \ne \emptyset$, and each subproduct $\prod {\{ {G_i}:i \in J\} }$, with $J \subset I$, $J \ne \emptyset$, and $|J| \leq \tau$, is minimal, then the whole product $\prod {\{ {G_i}:i \in I\} }$ is minimal. These definitions are correct, and $\kappa (G) \leq {2^\omega }$ and $\kappa (\mathcal {Q}) \leq \alpha (\mathcal {Q}) \leq {2^\omega }$ for all $G \in \mathcal {G}$ and any $\mathcal {Q} \subset \mathcal {G}$, $\mathcal {Q} \ne \emptyset$, while it can happen that $\kappa (\mathcal {Q}) < \alpha (\mathcal {Q})$ for some $\mathcal {Q} \subset \mathcal {G}$. Let $\mathcal {C} = \{ G \in \mathcal {G}:G\;{\text {is}}\;{\text {countably}}\;{\text {compact}}\}$ and $\mathcal {P} = \{ G \in \mathcal {G}:G\;{\text {is}}\;{\text {pseudocompact}}\}$. If $G \in \mathcal {C}$ is minimal, then $G \times H$ is minimal for each minimal (not necessarily Abelian) group $H$; in particular, ${G^n}$ is minimal for every natural number $n$. We show that $\alpha (\mathcal {C}) = \omega$, and so either $\kappa (\mathcal {C}) = 1$ or $\kappa (\mathcal {C}) = \omega$. Under Lusin’s Hypothesis ${2^{{\omega _1}}} = {2^\omega }$ we construct $\{ {G_n}:n \in \mathbb {N}\} \subset \mathcal {P}$ and $H \in \mathcal {P}$ such that: (i) whenever $n \in \mathbb {N}$, $G_n^n$ is totally minimal, but $G_n^{n + 1}$ is not even minimal, so $\kappa ({G_n}) = n + 1$; and (ii) ${H^n}$ is totally minimal for each natural number $n$, but ${H^\omega }$ is not even minimal, so $\kappa (H) = \omega$. Under ${\text {MA}} + \neg {\text {CH}}$, conjunction of Martin’s Axiom with the negation of the Continuum Hypothesis, we construct $G \in \mathcal {P}$ such that ${G^\tau }$ is totally minimal for each $\tau < {2^\omega }$, while ${G^{{2^\omega }}}$ is not minimal, so $\kappa (G) = {2^\omega }$. This yields $\alpha (\mathcal {P}) = \kappa (\mathcal {P}) = {2^\omega }$ under ${\text {MA}} + \neg {\text {CH}}$. We also present an example of a noncompact minimal group $G \in \mathcal {C}$, which should be compared with the following result obtained by the authors quite recently: Totally minimal groups $G \in \mathcal {C}$ are compact.