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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.

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