Cardinal invariants and independence results in the poset of precompact group topologies
type: Item
instance of: scholarly article
title: Cardinal invariants and independence results in the poset of precompact group topologies
zbMATH Open document ID: 0909.22003
DOI: 10.1016/S0022-4049(96)00149-1
author: Alessandro Berarducci
author: Dikran Dikranjan
author: Marco Forti
author: Stephen Watson
published in: Journal of Pure and Applied Algebra
publication date: 1998-02-25T00:00:00Z
review text: In this nice paper the poset \({\mathcal B} (G)\) of all Hausdorff precompact group topologies on an infinite group \(G\) and its subposet \({\mathcal B}_\sigma (G)\) of topologies of weight \(\sigma\) are studied. If \({\mathcal B}_\sigma (G)\neq \emptyset\) and \(2^{| G/G'|} = 2^{| G |}\) -- \(G'\) is the commutator subgroup of \(G\) -- then the poset of all subsets of \(2^{| G |}\) of size \(\sigma\) can be embedded into \({\mathcal B}_\sigma (G)\) and vice versa (cf. Section 7). (Example: \(G\) is Abelian, or \(G\) is free.) This extends and improves earlier results of W. W. Comfort and the reviewer. For a poset \(P\) let \[ \begin{aligned} Ded (P) & =\min \{\kappa: \text{ there is no totally ordered and bounded } A\subseteq P\text{ with } | A|= \kappa\}; \\ Ded^e(P) & =\min \{\kappa: \text{ there is no totally ordered } A\subseteq P\text{ with } | A |= \kappa\}. \end{aligned} \] In Section 4 it is proved that \(Ded^e(\sigma) =Ded (\sigma)\) if and only if \(\text{cf} (Ded (\sigma)) \neq \sigma^+\) holds. Then in Section 5 earlier results of Baumgartner and Mitchell are used to show that \(Ded^e(\aleph_1) =Ded (\aleph_1)\) is independent of (ZFC). Finally this result is used to prove that it is independent of (ZFC) whether \({\mathcal B}_{\aleph_1} (\mathbb{Z})\) has chains of bigger size than those of the bounded chains (Theorem 8.1). The paper which contains many other interesting results should be read by everyone who is interested in the structure of \({\mathcal B} (G)\).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 03G10
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05C35
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 05C40
zbMATH DE Number: 1121690
zbMATH Keywords: cardinal invariant
zbMATH Keywords: independence result
zbMATH Keywords: poset
zbMATH Keywords: precompact group topologies
reviewed by: Dieter Remus
MaRDI profile type: Publication
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full work available at URL: s0022-4049(96)00149-1
OpenAlex ID: W1978435415
description: scientific article; zbMATH DE number 1121690