Zariski topology: Difference between revisions
No edit summary |
No edit summary |
||
| (One intermediate revision by the same user not shown) | |||
| Line 3: | Line 3: | ||
==Definition== | ==Definition== | ||
Suppose <math>G</math> is a [[group]] (viewed ''purely'' as an abstract group). The '''Zariski topology''' of <math>G</math> is defined as the topology where the closed subsets are precisely the [[defining ingredient::algebraic subset]]s of <math>G</math> (here, the term ''algebraic'' is being used in a purely group-theoretic sense, ''not'' in relation to algebraic structure over a field). | Suppose <math>G</math> is a [[group]] (viewed ''purely'' as an abstract group). The '''Zariski topology''' or '''verbal topology''' of <math>G</math> is defined as the topology where the closed subsets are precisely the [[defining ingredient::algebraic subset]]s of <math>G</math> (here, the term ''algebraic'' is being used in a purely group-theoretic sense, ''not'' in relation to algebraic structure over a field). | ||
Note that although all subsets in the Zariski topology are closed subsets for any [[T0 topological group]] structure of <math>G</math> (since they are [[unconditionally closed subset]]s of <math>G</math>), it is '''not necessary''' that <math>G</math> under the Zariski topology would itself be a topological group. In fact, it is often ''not'' a topological group. For instance, the Zariski 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 Zariski topology are closed subsets for any [[T0 topological group]] structure of <math>G</math> (since they are [[unconditionally closed subset]]s of <math>G</math>), it is '''not necessary''' that <math>G</math> under the Zariski topology would itself be a topological group. In fact, it is often ''not'' a topological group. However, it ''is'' a [[quasitopological group]] (see [[Zariski topology defines a quasitopological group]]). | ||
For instance, the Zariski 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]]). | |||
===Confusion with algebraic group definition=== | ===Confusion with algebraic group definition=== | ||
| Line 15: | Line 17: | ||
* In general, the [[Markov topology]], where the closed subsets are precisely ''all'' the [[unconditionally closed subset]]s, is a [[topospaces:finer topology|finer topology]]. | * In general, the [[Markov topology]], where the closed subsets are precisely ''all'' the [[unconditionally closed subset]]s, is a [[topospaces:finer topology|finer topology]]. | ||
* For an [[abelian group]], this coincides with the [[Markov topology]] (the topology where the closed subsets are precisely the [[unconditionally closed subset]]s), and is called the [[Markov-Zariski topology of abelian group]]. | * For an [[abelian group]], this coincides with the [[Markov topology]] (the topology where the closed subsets are precisely the [[unconditionally closed subset]]s), and is called the [[Markov-Zariski topology of abelian group]]. | ||
==References== | |||
===Journal references=== | |||
* {{paperlink|Bryantverbal}} defines it using the term "verbal topology." | |||
Latest revision as of 20:01, 27 July 2013
WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with Zariski topology in the context of an algebraic group over a field
Definition
Suppose is a group (viewed purely as an abstract group). The Zariski topology or verbal topology of is defined as the topology where the closed subsets are precisely the algebraic subsets of (here, the term algebraic is being used in a purely group-theoretic sense, not in relation to algebraic structure over a field).
Note that although all subsets in the Zariski topology are closed subsets for any T0 topological group structure of (since they are unconditionally closed subsets of ), it is not necessary that under the Zariski topology would itself be a topological group. In fact, it is often not a topological group. However, it is a quasitopological group (see Zariski topology defines a quasitopological group).
For instance, the Zariski 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).
Confusion with algebraic group definition
Note that the term Zariski topology, when used in the context of algebraic groups over fields, refers to the Zariski topology of the underlying algebraic variety.
Relation with other topologies
- In general, the Markov topology, where the closed subsets are precisely all the unconditionally closed subsets, is a finer topology.
- For an abelian group, this coincides with the Markov topology (the topology where the closed subsets are precisely the unconditionally closed subsets), and is called the Markov-Zariski topology of abelian group.
References
Journal references
- The verbal topology of a group by Roger M. Bryant, Journal of Algebra, ISSN 00218693, Volume 48,Number 2, Page 340 - 346(October 1977): PDF (gated)More info defines it using the term "verbal topology."