Regular rigid Korovin orbits

TL;DR

This paper constructs an example of a regular feebly compact quasitopological group with all continuous real-valued functions constant, using Korovin orbits in a specially constructed space, and explores their separation properties.

Contribution

It introduces a new example of a regular feebly compact quasitopological group based on Korovin orbits and analyzes their separation properties in relation to the space's topology.

Findings

All continuous real-valued functions on the constructed group are constant.

If the space contains two disjoint open sets, Korovin orbits are Hausdorff.

The example demonstrates specific topological properties 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.