Convergent sequences in iterated ultrapowers as \(p\)-compact groups. (English) Zbl 07854787
Summary: We prove, in \(\mathsf{ZFC} \), that if \(\mathbb{G}\) is an infinite countable Abelian group, then there is an ultrafilter \(p \in \omega^\ast\) such that \(( \mathsf{Ult}_p(\mathbb{G}), \tau_{\overline{\text{Bohr}}})\) has non-trivial convergent sequences, consequently \(( \mathsf{Ult}_p^{\omega_1}(\mathbb{G}), \tau_{\overline{\text{Bohr}}})\) has non-trivial convergent sequences, extending Theorem 3.9 from [13]. In addition, we prove that the Remark 3.8 from [13] is false; so, the proof of the Corollary 3.11 is false too.
MSC:
| 22A05 | Structure of general topological groups |
| 03C20 | Ultraproducts and related constructions |
| 03E05 | Other combinatorial set theory |
| 54H11 | Topological groups (topological aspects) |
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