# Markov topology

## Definition

Suppose is a group (viewed *purely* as an abstract group). The **Marjov topology** of is defined as the topology where the closed subsets are precisely the unconditionally closed subsets of (this means that the subset is closed for *any* topology on ).

Note that although all subsets in the Markov topology are closed subsets for any T0 topological group structure of , it is **not necessary** that 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.