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Maximal resolvability of bounded groups. (English) Zbl 0866.54032

A topological group \(G=\langle G,{\mathcal T}\rangle\) is said to be \(m\)-bounded if for \(\emptyset\neq U\in{\mathcal T}\) there exists \(K\subseteq G\) such that \(|K|<m\) and \(G=KU\). Using this terminology, the \(\omega\)-bounded groups are exactly the totally bounded groups. The dispersion character of a space \(X=\langle X,{\mathcal T}\rangle\) is by definition the cardinal number \(\Delta(X)= \min\{|V|:\emptyset\neq V\in{\mathcal T}\}\). A space \(X\) is said to be \(\kappa\)-resolvable [resp., maximally resolvable] if \(X\) admits a family of \(\kappa\)-many [resp., \(\Delta(X)\)-many] pairwise disjoint dense subsets; a 2-resolvable space is said to be resolvable.
In the present paper, which supersedes and generalizes results obtained over several years by the reviewer, H. Gladdines, O. Masaveu, J. van Mill, L. Villegas-Silva and H. Zhou, and by the authors themselves, the first-listed author exploits a new method discovered by the second-listed author to prove that many topological groups are maximally resolvable. The crucial lemma is the assertion that for every infinite group \(G\) there is a pairwise disjoint family \({\mathcal A}\) of subsets of \(G\) such that (1) \(|{\mathcal A}|=|G|\) and (2) \(AK\neq G\) and \((G\smallsetminus K)\cdot K\neq G\) for each \(A\in{\mathcal A}\) and each finite \(K\subseteq G\); indeed when \(|G|\) is regular one may arrange that \(AK\neq G\) and \((G\smallsetminus A)\cdot K\neq G\) for each \(K\subseteq G\) such that \(|K|<|G|\). Since each \(A\in{\mathcal A}\) has \(\text{int}_{\mathcal T} A=\text{int}_{\mathcal T}(G\smallsetminus A)=\emptyset\) with respect to every totally bounded group topology \({\mathcal T}\) on \(G\), each such topology \({\mathcal T}\) has the property that \(\langle G,{\mathcal T}\rangle\) is \(|G|\) resolvable, i.e., is maximally resolvable. Similarly, each \(m\)-bounded group is maximally resolvable for \(m\) regular, \(m\leq|G|\).
In Corollary 11, the authors deduce that an uncountable group is maximally resolvable in each of these cases: (a) \(G\) is \(\aleph_1\)-bounded; (b) \(G\) is a subgroup of a product of separable, metrizable groups; (c) \(G\) satisfies the countable chain condition; (d) \(G\) is algebraically generated by a Lindelöf subspace; (e) \(G\) is the free topological group over a Lindelöf space.

MSC:

54H11 Topological groups (topological aspects)
20K45 Topological methods for abelian groups
54A35 Consistency and independence results in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
22A30 Other topological algebraic systems and their representations
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References:

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