Zariski topologies on groups

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.

6 pages

References in corpus (2)

• Reflection principle characterizing groups in which unconditionally closed sets are algebraic
• The number of non-solutions to an equation in a group and non-topologizable torsion-free groups

Cited by in corpus (2)

• Algebraically determined topologies on permutation groups
• Hindman's finite sums theorem and its application to topologizations of algebras