Regular rigid Korovin orbits

Abstract An example of an infinite regular feebly compact quasitopological group is presented such that all continuous real-valued functions on the group are constant. The example delineates the boundaries for possible generalizations of Banakh–Ravsky’s theorem concerning the complete regularity of regular paratopological groups. Our construction is based on the use of Korovin orbits in $$X^G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>G</mml:mi> </mml:msup> </mml:math> , where X is a special regular countably compact space constructed by Bardyla and Zdomskyy and G is an abstract Abelian group of an appropriate cardinality. Also, we study the interplay between the separation properties of the space X and Korovin orbits in $$X^G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>G</mml:mi> </mml:msup> </mml:math> . We show in particular that if X contains two nonempty disjoint open subsets, then every Korovin orbit in $$X^G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>G</mml:mi> </mml:msup> </mml:math> is Hausdorff. Several open problems in the field are presented.