The paper constructs a countable abelian group whose bounded Zariski topologies are all distinct, shows that groups with no algebraicity have hyperconnected semigroup Zariski topology, and proves that on symmetric groups the semigroup Hausdorff-Markov topology equals the pointwise convergence…
Zariski topologies on groups
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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.
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A note on intrinsic topologies of groups
The paper constructs a countable abelian group whose bounded Zariski topologies are all distinct, shows that groups with no algebraicity have hyperconnected semigroup Zariski topology, and proves that on symmetric groups the semigroup Hausdorff-Markov topology equals the pointwise convergence…