Markov topology: Difference between revisions

Latest revision as of 20:01, 27 July 2013

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.

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 Suppose <math>G</math> is a [[group]] (viewed ''purely'' as an abstract group). The '''Marjov topology''' of <math>G</math> is defined as the topology where the closed subsets are precisely the [[defining ingredient::unconditionally closed subset]]s of <math>G</math> (this means that the subset is closed for ''any'' <math>T_0</math> topology on <math>G</math>).
-Note that although all subsets in the Markov topology are closed subsets for any [[T0 topological group]] structure of <math>G</math>, it is '''not necessary''' that <math>G</math> under the Markov topology would itself be a topological group. In fact, it is often ''not'' a topological 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]]).
+Note that although all subsets in the Markov topology are closed subsets for any [[T0 topological group]] structure of <math>G</math>, it is '''not necessary''' that <math>G</math> 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==