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topological group; Dieudonné completion; PT-group; realcompactness; \newline Moscow space; $C$-embedding; product
The main results concern commutativity of 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.
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