HOME

TheInfoList



OR:

In mathematics, a Noetherian topological space, named for
Emmy Noether Amalie Emmy NoetherEmmy 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 Noethe ...
, 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 points ...
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 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 ...
, 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 clo ...
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 calle ...
: 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 to ...
. : 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 (for ...
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 In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
an
irreducible set In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
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 set Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a dat ...
s 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 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 ...
. 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-Noether ...
s. This class of examples therefore also explains the name. If ''R'' is a commutative Noetherian ring, then Spec(''R''), the
prime spectrum In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
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. Thus ...
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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
k) under the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
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(Y_1) \subseteq I(Y_2) \subseteq I(Y_3) \subseteq \cdots is an ascending chain of ideals of k _1,\ldots,x_n Since k _1,\ldots,x_n/math> is a Noetherian ring, there exists an integer m such that :I(Y_m)=I(Y_)=I(Y_)=\cdots. Since V(I(Y)) is the closure of ''Y'' for all ''Y'', V(I(Y_i))=Y_i for all i. Hence :Y_m=Y_=Y_=\cdots as required.


Notes


References

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