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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Pseudocompact and countably compact abelian groups: Cartesian products and minimality

Authors: Dikran N. Dikranjan and Dmitrii B. Shakhmatov
Journal: Trans. Amer. Math. Soc. 335 (1993), 775-790
MSC: Primary 22A05; Secondary 54B10
MathSciNet review: 1085937
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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 $ \vert J\vert \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.

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Keywords: Countably compact space, pseudocompact space, $ \omega $-bounded space, topological group, minimal group, totally minimal group, Cartesian product, Tychonoff product, cardinal invariant
Article copyright: © Copyright 1993 American Mathematical Society