Zariski topologies on groups

TL;DR

This paper investigates the properties of Zariski topologies on groups, showing that the second Zariski topology is never discrete, providing examples with specific topological characteristics, and analyzing a special non-topologizable group.

Contribution

It establishes fundamental properties of the 2-nd Zariski topology on groups and provides explicit examples illustrating diverse topological behaviors.

Findings

The 2-nd Zariski topology on any group is never discrete.

Existence of a group with continuum cardinality whose 2-nd Zariski topology has countable pseudocharacter.

A non-topologizable group has a discrete 665-th Zariski topology.

Abstract

The $n$-th Zariski topology on a group $G$ is generated by the sub-base consiting of the cozero sets of monomials of degree $\le n$ on $G$. We prove that for each group $G$ the 2-nd Zariski topology is not discrete and present an example of a group $G$ of cardinality continuum whose 2-nd Zariski topology has countable pseudocharacter. On the other hand, the non-topologizable group $G$ constructed by Ol'shanskii has discrete 665-th Zariski topology.