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Zerodimensionality of free topological groups and topological groups with noncoinciding dimensions. (English) Zbl 0759.54023

Answering questions of A. V. Arkhangel’skij, the author in §2 finds a Tychonoff space \(X\) (which may be chosen to be normal or pseudocompact) such that \(\text{ind }X=0\) but \(\text{ind }F(X)\neq 0\) and \(\text{ind }A(X)\neq 0\). (Here \(F(X)\) and \(A(X)\) denote respectively the Markov free topological group over \(X\) and its Abelian analogue).
In §3 the author defines for every integer \(n\) a connected, Lindelöf, divisible Abelian group \(G=G(n)\), which is a \(\Sigma\)-space in the sense of Nagami, such that \(\dim G=n\) but \(\text{ind }G=\text{Ind }G=+\infty\); in subsequent work the author has shown that the groups \(G(n)\) may in addition be chosen to be pre-compact (=totally bounded). To the reviewer’s knowledge the groups \(G(n)\), whose construction rests in part on work of J. Keesling, are the first known examples of topological groups with non-coinciding dimensions.
Of the several problems posed but not answered by the author’s “Remark added in proof”, the following (due to Arkhangel’skij) is perhaps the most interesting: Does every Tychonoff space \(X\) with \(\dim X=0\) satisfy \(\dim F(X)=\dim A(X)=0\)?

MSC:

54H11 Topological groups (topological aspects)
54F45 Dimension theory in general topology
22A05 Structure of general topological groups
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