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Limits in function spaces and compact groups. (English) Zbl 1074.54013

Given an abelian discrete group \(G\), let \(\hat G\) denote the (compact) Pontryagin dual group of \(G\) (that is, the group of all homomorphisms of \(G\) into the circle group \(\mathbb T\) with pointwise product as composition law). For any subgroup \(H\) of \(\hat G\), let \((G,t_p(H))\) denote the group \(G\) equipped with the topology of pointwise convergence on the elements of \(H\). In response to a question of S. V. Raczkowski [Topology Appl. 121, No. 1-2, 63–74 (2002; Zbl 1007.22003)], G. Barbieri, D. Dikranjan, C. Milan and H. Weber [Topology Appl. 132, No. 1, 89–101 (2003; Zbl 1022.22001)] showed (under the assumption of Martin’s axiom) that there exists a measure-zero subgroup \(H\) of \(\mathbb T\) such that \((\mathbb Z, t_p(H))\) has no nontrivial convergent sequences. In the same paper the authors asked whether it was possible to accomplish a similar construction in the axiom system \(ZFC\).
The main result of the paper under review is a solution to the latter question. This is accomplished as follows. For \(B\) an infinite subset of \(\omega\) and \(X\) a topological group, let \(\mathcal C^X_B\) be the set of all \(x\in X\) such that \(\{x_n:n\in B\}\) converges to \(1\). The set \(\mathcal C^\mathbb T_B\) always has measure \(0\) in the circle group \(\mathbb T\). If \(\mathcal F\) is a filter of infinite subsets, let \(\mathcal D^X_\mathcal F=\bigcup \{\mathcal C^X_B: B\in \mathcal F\}\). Then \(\mathcal C^X_B\) and \(\mathcal D^X_\mathcal F\) are subgroups of \(X\) when \(X\) is abelian. Theorem: Let \(\mathcal F\subset [\omega]^\omega\) be the filter generated by all sets of the form \(\{k!+1:k\in D\}\), where \(D\subset \omega\) has asymptotic density 1; then: (1) \(\mathcal F\) is a Borel subset of \(\mathcal P(\omega)\cong2^\omega\); (2) Whenever \(X\) is an infinite compact group, the subgroup \(\mathcal D^X_\mathcal F\) has zero measure and, if \(X\) is not totally disconnected, then \(\mathcal D^X_\mathcal F\) is not a subset of \(\mathcal C^X_B\) for any infinite \(B\). Taking \(X\) to be \(\mathbb T\), the theorem above gives a solution to Raczkowski’s question. The paper also contains other interesting results and clarifying examples.

MSC:

54H11 Topological groups (topological aspects)
22C05 Compact groups
46E25 Rings and algebras of continuous, differentiable or analytic functions
54C35 Function spaces in general topology
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References:

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