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We answer the question, raised more than thirty years ago by Dikranjan and Shakhmatov [Comp. Rend. Acad. Sci. Bulg., 41 (1990), pp. 13-15] and Dikranjan and Shakhmatov [Trans. Amer. Math. Soc. 335 (1993), pp. 775-790], on whether the power G ω G^\omega of a countably compact minimal Abelian group G G is minimal, by showing that the negative answer is equivalent to the existence of measurable cardinals. The proof is carried out in the larger class of sequentially complete groups. We characterize the sequentially complete minimal Abelian groups G G such that G ω G^\omega is minimal – these are exactly those G G that contain the connected component of their completion. This naturally leads to the next step, namely, a better understanding of the structure of the sequentially complete minimal Abelian groups, and in particular, their connected components which turns out to depend on the existence of Ulam measurable cardinals. More specifically, all connected sequentially complete minimal Abelian groups are compact if Ulam measurable cardinals do not exist. On the other hand, for every Ulam measurable cardinal σ \sigma we build a non-compact torsion-free connected minimal ω \omega -bounded Abelian group of weight σ \sigma , thereby showing that the Ulam measurable cardinals are precisely the weights of non-compact sequentially complete connected minimal Abelian groups.
We answer the question, raised more than thirty years ago by Dikranjan and Shakhmatov [Comp. Rend. Acad. Sci. Bulg., 41 (1990), pp. 13-15] and Dikranjan and Shakhmatov [Trans. Amer. Math. Soc. 335 (1993), pp. 775-790], on whether the power G ω G^\omega of a countably compact minimal Abelian group G G is minimal, by showing that the negative answer is equivalent to the existence of measurable cardinals. The proof is carried out in the larger class of sequentially complete groups. We characterize the sequentially complete minimal Abelian groups G G such that G ω G^\omega is minimal – these are exactly those G G that contain the connected component of their completion. This naturally leads to the next step, namely, a better understanding of the structure of the sequentially complete minimal Abelian groups, and in particular, their connected components which turns out to depend on the existence of Ulam measurable cardinals. More specifically, all connected sequentially complete minimal Abelian groups are compact if Ulam measurable cardinals do not exist. On the other hand, for every Ulam measurable cardinal σ \sigma we build a non-compact torsion-free connected minimal ω \omega -bounded Abelian group of weight σ \sigma , thereby showing that the Ulam measurable cardinals are precisely the weights of non-compact sequentially complete connected minimal Abelian groups.
Abstract. It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group K admits a proper totally dense pseudocompact subgroup, then K cannot have a torsion closed G δ -subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup H of K that contains an ω-bounded dense subgroup of K (such an H is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc.
Abstract. We prove that hereditarily disconnected countably compact groups are zero-dimensional. This gives a strongly positive answer to a question of Shakhmatov. We show that hereditary or total disconnectedness yields zerodimensionality in various classes of pseudocompact groups.
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