Markov topology

Definition
Suppose $$G$$ is a group (viewed purely as an abstract group). The Marjov topology of $$G$$ is defined as the topology where the closed subsets are precisely the defining ingredient::unconditionally closed subsets of $$G$$ (this means that the subset is closed for any $$T_0$$ topology on $$G$$).

Note that although all subsets in the Markov topology are closed subsets for any T0 topological group structure of $$G$$, it is not necessary that $$G$$ under the Markov topology would itself be a topological group. In fact, it is often not a topological group. However, it is a quasitopological group (see Markov topology defines a quasitopological group).

For instance, the Markov topology on the group of integers is the cofinite topology, which makes it a quasitopological group (see infinite group with cofinite topology is a quasitopological group) but not a topological group (see infinite group with cofinite topology is not a topological group).

Relation with Zariski topology

 * In general, the Zariski topology, defined as the topology where the closed subsets are precisely the algebraic subsets, is a coarser topology.
 * The Markov topology and Zariski topology coincide for an abelian group, and the topology is called the Markov-Zariski topology of abelian group.