Zariski topology

The Zariski topology is a term from the mathematical branch of algebraic geometry. It is a natural topology on the objects of study in algebraic geometry, algebraic varieties, or more generally the schemes.

The Zariski topology in classical algebraic geometry

In classical algebraic geometry, the Zariski topology ( by Oscar Zariski ) topology that on the affine space over an algebraically closed field, by the open sets of the form

Is generated. Affine varieties carry the induced topology and the Zariski topology on general varieties is defined by affine maps.

For example, the Zariski topology on the affine line, the topology of koendlichen quantities.

On an affine variety, the Zariski topology is the coarsest topology for which the regular functions are continuous as illustrations in the affine line (with its Zariski topology).

The Zariski topology on the spectrum of a ring

Is a commutative ring with identity, then the spectrum is the set of prime ideals of the topology in which the closed sets the amounts

Are ideals.

Is a algebraically closed field then the maximal ideals of after hilbert between Nullstellensatz correspond one to one of the elements, and the topologies of these two quantities coincide.

Properties

The Zariski topology is very different from the usual, based on the real numbers topological spaces.

• The topology is not Hausdorff A.,; in fact, the space irreducible, i.e. each two non-empty open subsets intersect. Thus, irreducibility is a stronger term than context.
• Quasi- compact subsets need not necessarily be complete.

Generalizations

• The Zariski topology of a scheme is part of its structure; However, one uses the term " Zariski topology " in the context of schemes usually only to distinguish it from other Grothendieck topologies.

• Algebraic Geometry