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Extensions of topological and semitopological groups and the product operation. (English) Zbl 1053.54043
Summary: The main results concern commutativity of the Hewitt-Nachbin realcompactification or Dieudonné completion with products of topological groups. It is shown that for every topological group $$G$$ that is not Dieudonné complete one can find a Dieudonné complete group $$H$$ such that the Dieudonné completion of $$G\times H$$ is not a topological group containing $$G\times H$$ as a subgroup. Using Korovin’s construction of $$G_\delta$$-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological groups.

##### MSC:
 54H11 Topological groups (topological aspects) 22A05 Structure of general topological groups 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D60 Realcompactness and realcompactification