In mathematics, a Noetherian topological space, named for
Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
, is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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. These conditions played an important r ...
. 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 Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
, 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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
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
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. These conditions played an important r ...
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 cl ...
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 cal ...
:
of closed subsets
of
, there is an integer
such that
Properties
* A topological space
is Noetherian if and only if every
subspace of
is compact (i.e.,
is hereditarily compact), and if and only if every open subset of
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 Noetherian space and every compact subset is closed in a Hausdorff space; hence all subsets of X are closed and X has the discrete topology. As X is discrete and 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 of an algebraic set is an algebraic subset that is irred ...
s.
If the irreducible components are
, then
, and none of the components
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 which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, where for the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
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 set
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
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 Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
. 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-Noethe ...
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 commutative 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 ...
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, where each A_i is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noe ...
is a Noetherian topological space. The converse does not hold, since there are non-Noetherian rings with only one prime ideal, so that Spec(''R'') is not a Noetherian scheme, but consists of exactly one point and therefore is a Noetherian space.
Example
The space
(affine
-space over a
field ) under the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
is an example of a Noetherian topological space. By properties of the
ideal of a subset of
, we know that if
:
is a descending chain of Zariski-closed subsets, then
:
is an ascending chain of ideals of
Since