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 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.