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Equivalence of certain free topological groups. (English) Zbl 0759.54022
In the earlier paper of the author and J. de Groot [Comment. Math. Univ. Carol. 29, 577-595 (1988; Zbl 0684.54011)], a complete linear homeomorphic classification of function spaces \(C_ p(X)\) of locally compact zero-dimensional separable metric spaces \(X\) was given. In the paper under review the author presents a clear proof that the above classification is completely analogous to an isomorphic classification of free topological groups \(FM(X)\) of the same spaces \(X\). This implies, in particular, that the groups \(FM(X)\) and \(FM(Y)\) are isomorphic for any uncountable zero-dimensional compact metric spaces \(X\) and \(Y\). The zero- dimensionality assumption is essential, because the free topological groups of the unit interval and the “letter” \(T\) are not isomorphic by the well-known result of M. I. Graev.

MSC:
54H11 Topological groups (topological aspects)
54C35 Function spaces in general topology
22A05 Structure of general topological groups
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