Markov topology

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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