In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. It can also be shown to be equivalent that every open subset of such a space is compact, and in fact the seemingly stronger statement that every subset is compact. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

Definition

A topological space X is called noetherian if it satisfies the descending chain condition for closed subsets: for any sequence Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... A club is generally an association of people united by a common interest or goal, as opposed to any natural ties of kinship. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

of closed subsets Y_i of X, there is an integer m such that The integers are commonly denoted by the above symbol. ...

Interplay with Noetherian condition and compactness

There is a lot of interplay between the noetherian condition and compactness: This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set (possibly infinite) of first-order sentences is satisfiable, i. ...

• Every noetherian topological space is quasi-compact.
• A topological space X is Noetherian if and only if every subspace of X is compact. (i.e. X is hereditarily compact)

Note that if R is a noetherian ring, then Spec(R), the prime spectrum of R, is a noetherian topological space. Screenshot (from SSCX Star Warzone). ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

Noetherian topological spaces from algebraic geometry

Many examples of Noetherian topological spaces come from algebraic geometry, where for the Zariski topology an irreducible set has the intuitive property that any closed proper subset has smaller dimension. Since dimension can only 'jump down' a finite number of times, and algebraic sets are made up of finite unions of irreducible sets, descending chains of Zariski closed sets must eventually be constant. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition but is only weakly related to their geometric properties; it is due to Oscar Zariski and took a place of particular importance in the field around... In mathematics, an algebraic set over a field K is the set of solutions in Kn (n-tuples of elements of K, of a set of simultaneous equations P1(X1, ...,Xn) = 0 P2(X1, ...,Xn) = 0 and so on up to Pm(X1, ...,Xn) = 0 for some integer m. ...

A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chain condition. That follows because the rings of algebraic geometry, in the classical sense, are Noetherian rings. This class of examples therefore also explains the name. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...

Example

The space (affine n-space over a field k) under the Zariski topology is an example of a noetherian topological space. By properties of the ideal of a subset of , we know that if In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition but is only weakly related to their geometric properties; it is due to Oscar Zariski and took a place of particular importance in the field around... Property designates those things that are commonly recognized as being the possessions of a person or group. ...

is a descending chain of Zariski-closed subsets, then

is an ascending chain of ideals of Since is a noetherian ring, there exists an integer m such that

But because we have a one-to-one correspondence between radical ideals of and Zariski-closed sets in we have V(I(Y_i)) = Y_i for all i. Hence An injective function. ... In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. ...

as required.

See also

• compact

This article incorporates material from Noetherian topological space on PlanetMath, which is licensed under the GFDL. Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...