Noetherian topological space

The Noetherian topological space, named after Emmy Noether, is a mathematical term from the branch of topology. He is motivated by the algebraic concept of a Noetherian ring and found mainly in algebraic geometry application.

Definition

If we consider open sets of a topological space in analogy to the ideals of a ring, the following definition is natural to look at the concept of a Noetherian ring:

• A topological space is called noetherian if every ascending chain of open sets is stationary, ie is a family of open sets, so there is a with for all.

As in algebra shows a simple argument:

• A topological space is Noetherian if and only if a maximum condition for open sets is considered, that is: Every non-empty family of open sets contains a maximal element.

Since the closed sets are exactly the complements of open sets, we have:

• A topological space is Noetherian if and only if every descending chain of closed sets is stationary, ie is a family of closed sets, so there is a with for all.

• A topological space is Noetherian if and only if a minimum condition for closed sets is considered, that is: Every non-empty family of closed sets contains a minimal element.

Importance

On the spectrum of a ring usually one considers the Zariski topology. Easy, one can show that the spectrum of a commutative Noetherian ring is a Noetherian topological space. Since affine varieties correspond to radical ideals in the ring of polynomials in finitely many variables over the coordinates body ( Hilbert Nullstellensatz ), and this ring Noetherian is ( Hilbert basis theorem ), we obtain that affine varieties with the Zariski topology is Noetherian. Therefore, this term has a role in the algebraic geometry such varieties are tested in the.

Application

• A Noetherian topological space has only finitely many irreducible components.

In particular, there is an affine variety of finitely many irreducible components.

Since the simple proof illustrates the typical Noetherian final way it should be played here briefly: Let the set of all closed subsets which are not finite union of irreducible sets. If we assume that this set is not empty, then it contains a minimal element because of the minimal condition for closed sets. This can not be irreducible as an element of, so is the union of two proper closed sets and. Since is minimal, and are not, and therefore a finite union of irreducible sets. But then also a finite union of irreducible quantities, leading to a contradiction. Therefore, it is empty, in particular, the space itself finite union of irreducible quantities, which was to be shown.

Compactness

If we define compactness by the coverage property and waives the Hausdorf fig stem, then some authors speak of quasi- compact spaces, then:

• Each Noetherian space is quasi- compact.
• A topological space is Noetherian if and only if every subset is quasi- compact with the relative topology.

Other properties

• Every subspace of a Noetherian space is Noetherian again.
• If the topological space union of subspaces and each is noetherian, so is noetherian.