Regular rigid Korovin orbits
Evgenii Reznichenko, Mikhail Tkachenko
TL;DR
The article investigates the limits of regularity in semitopological and quasitopological group contexts via Korovin orbits. It constructs an infinite regular feebly compact quasitopological group $H$ that is $\mathbb{R}$-rigid, realized as a Korovin orbit in $X^G$, to show that regularity does not imply complete regularity or functional Hausdorffness. A cohesive framework links the separation properties of product spaces, subspaces that fill subproducts, and Korovin orbits, yielding transfer criteria for $T_3$ and $T_{3.5}$ properties and for $C$-embedding to factors. The paper also provides an explicit high-cardinality example using a regular separable ${\mathbb R}$-rigid space $X$ and a Boolean group $G$, proving feebly compactness and rigidity, and concludes with open questions on normality, paracompactness, and separability of 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 is based on the use of Korovin orbits in $X^G$, where $X$ is a special regular countably compact space constructed by S.Bardyla and L.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 $X^G$. We show in particular that if $X$ contains two nonempty disjoint open subsets, then every Korovin orbit in $X^G$ is Hausdorff.