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Provisional solution to a Comfort–van Mill problem. (English) Zbl 1017.54022
Let \(\mathcal U\) and \(\mathcal V\) be two non-empty classes of topological groups and \(X\) be a topological space. A topological group \(G=F(X, \mathcal U, \mathcal V)\) is called a free \((\mathcal U, \mathcal V)\)-group over \(X\) if \(G\in\mathcal U\), \(G\) contains \(X\) as a subspace, and every continuous map \(f\) of \(X\) to a group \(H\in\mathcal V\) uniquely extends to a continuous homomorphism \(\widetilde f: G\to H\). If such an extension \(\widetilde f: G\to H\) is not required to be unique, then the group \(G=F_w(X, \mathcal U, \mathcal V)\) is called a weakly free \((\mathcal U, \mathcal V)\)-group over \(X\). In the case \(\mathcal U=\mathcal V\) these groups are called the free \(\mathcal U\)-group \(F(X, \mathcal U)\) and the weakly free \(\mathcal U\)-group \(F_w(X, \mathcal U)\) over \(X\), respectively. W. W. Comfort and J. van Mill [Topology Appl. 29, 245-265 (1988; Zbl 0649.22001)] posed the following general problem: Does there exist a free \((\mathcal U, \mathcal V)\)-group over \(X\) for classes of topological groups \(\mathcal U\) and \(\mathcal V\) and a Tikhonov space \(X\)? The author proves that under Martin’s Axiom every Abelian group \(G\) of non-measurable cardinality is the intersection of some family of countably compact subgroups of its Bohr compactification \(bG\). Using this result he shows that weakly free countably compact topological groups do not exist. This gives an answer to the Comfort-van Mill problem. In fact, the author shows that under Martin’s Axiom a free \((\mathcal P, \mathcal C\mathcal C)\)-group over a topological space \(X\) exists if and only if \(X\) is empty, where \(\mathcal P\) and \(\mathcal C\mathcal C\) are the classes of pseudocompact and countably compact topological groups, respectively. Nevertheless, he proves the existence of a weakly free \((\mathcal P, \mathcal C\mathcal C)\)-group over a Tikhonov space \(X\) and establishes that this construction is functorial. Also the author shows that similar results remain valid in the Abelian case. The paper closes with six open problems on this topic.
MSC:
54H11 Topological groups (topological aspects)
54A35 Consistency and independence results in general topology
54B10 Product spaces in general topology
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54B05 Subspaces in general topology
22A05 Structure of general topological groups
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