Moscow spaces, Pestov-Tkačenko Problem, and \(C\)-embeddings. (English) Zbl 1038.54013

Arhangel’skii, A.

Commentat. Math. Univ. Carol. 41, No. 3, 585-595 (2000).

An example of an Abelian topological group \(H\) is given such that \(H\) is not the topological subgroup of a topological group defined on the Dieudonné completion of \(H\). The example is the product of a zero-dimensional group \(G\), e.g. \(G=\{0,1\}^{\omega _1}\), and of a particular topological group defined on clopen subsets of \(G\). This solves a problem posed by V. G. Pestov and M. G. Tkachenko [Unsolved Problems of Topological Algebra, Academy of Sciences, Moldova, Kishinev, p. 18 (1985)]. The notions of a \(C\)-embedding and Rajkov completion are used essentially to verify the properties of the example.
On the other hand, new classes of topological groups \(G\) are shown to admit an extension of the operations to its Dieudonné completion \(\mu G\) so that \(G\) becomes a topological subgroup of \(\mu G\). Here the notion of Moscow space and some special types of tightness play an important role.
New results on products of Hewitt-Nachbin completions of topological groups are obtained.

Reviewer: Petr Holický (Praha)

MSC:

54H11	Topological groups (topological aspects)
54E15	Uniform structures and generalizations
54C35	Function spaces in general topology
54C45	\(C\)- and \(C^*\)-embedding
54G20	Counterexamples in general topology
22A05	Structure of general topological groups

Keywords:

Dieudonné completion; Rajkov completion; Hewitt-Nachbin completion; \(C\)-embedding; topological group