Noetherian topological space

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In mathematics, a Noetherian topological space, named for
Emmy Noether Amalie Emmy Noether Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
, is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
in which closed subsets satisfy the
descending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong
compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that ''every'' subset is compact.

# Definition

A topological space $X$ is called Noetherian if it satisfies the
descending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
for
closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s: for any
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
:$Y_1 \supseteq Y_2 \supseteq \cdots$ of closed subsets $Y_i$ of $X$, there is an integer $m$ such that $Y_m=Y_=\cdots.$

# Properties

* A topological space $X$ is Noetherian if and only if every subspace of $X$ is compact (i.e., $X$ is hereditarily compact), and if and only if every open subset of $X$ is compact. * Every subspace of a Noetherian space is Noetherian. * The continuous image of a Noetherian space is Noetherian. * A finite union of Noetherian subspaces of a topological space is Noetherian. * Every Hausdorff Noetherian space is finite with the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest t ...
. : Proof: ''Every subset of X is compact in a Hausdorff space, hence closed. So X has the discrete topology, and being compact, it must be finite.'' * Every Noetherian space ''X'' has a finite number of
irreducible component In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (fo ...
s. If the irreducible components are $X_1,...,X_n$, then $X=X_1\cup\cdots\cup X_n$, and none of the components $X_i$ is contained in the union of the other components.

# From algebraic geometry

Many examples of Noetherian topological spaces come from
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, 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. 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 ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
s. This class of examples therefore also explains the name. If ''R'' is a commutative Noetherian ring, then Spec(''R''), the prime spectrum of ''R'', is a Noetherian topological space. More generally, a
Noetherian scheme In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Th ...
is a Noetherian topological space. The converse does not hold, since Spec(''R'') of a one-dimensional valuation domain ''R'' consists of exactly two points and therefore is Noetherian, but there are examples of such rings which are not Noetherian.

# Example

The space $\mathbb^n_k$ (affine $n$-space over a field $k$) under the Zariski topology is an example of a Noetherian topological space. By properties of the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of a subset of $\mathbb^n_k$, we know that if :$Y_1 \supseteq Y_2 \supseteq Y_3 \supseteq \cdots$ is a descending chain of Zariski-closed subsets, then :$I\left(Y_1\right) \subseteq I\left(Y_2\right) \subseteq I\left(Y_3\right) \subseteq \cdots$ is an ascending chain of ideals of Since

# References

* {{PlanetMath attribution, id=3465, title=Noetherian topological space Algebraic geometry Properties of topological spaces Scheme theory Wellfoundedness