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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
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