Cardinal invariants and independence results in the poset of precompact group topologies
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 \({\mathcal B} (G)\).
scientific article; zbMATH DE number 1121690
scientific article