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

##### 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
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