Noetherian topological space

Key takeaways

Noetherian topological space belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Noetherian topological space to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Noetherian topological space from memory before moving on to harder problems.

Reference excerpt

In mathematics, a Noetherian topological space, named for Emmy Noether, 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. The Noetherian property of a topological space can also be seen as a strong compactness 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 {\displaystyle X} is called Noetherian if it satisfies the descending chain condition for closed subsets: for any sequence

Y 1 ⊇ Y 2 ⊇ ⋯ {\displaystyle Y_{1}\supseteq Y_{2}\supseteq \cdots }

of closed subsets Y i {\displaystyle Y_{i}} of X {\displaystyle X} , there is an integer m {\displaystyle m} such that Y m = Y m + 1 = ⋯ . {\displaystyle Y_{m}=Y_{m+1}=\cdots .}

Properties A topological space X {\displaystyle X} is Noetherian if and only if every subspace of X {\displaystyle X} is compact (i.e., X {\displaystyle X} is hereditarily compact), and if and only if every open subset of X {\displaystyle 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. 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 components. If the irreducible components are X 1 , . . . , X n {\displaystyle X_{1},...,X_{n}} , then X = X 1 ∪ ⋯ ∪ X n {\displaystyle X=X_{1}\cup \cdots \cup X_{n}} , and none of the components X i {\displaystyle X_{i}} is contained in the union of the other components.

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. 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. 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 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 A k n {\displaystyle \mathbb {A} _{k}^{n}} (affine n {\displaystyle n} -space over a field k {\displaystyle k} ) under the Zariski topology is an example of a Noetherian topological space. By properties of the ideal of a subset of A k n {\displaystyle \mathbb {A} _{k}^{n}} , we know that if

Y 1 ⊇ Y 2 ⊇ Y 3 ⊇ ⋯ {\displaystyle Y_{1}\supseteq Y_{2}\supseteq Y_{3}\supseteq \cdots }

is a descending chain of Zariski-closed subsets, then

I ( Y 1 ) ⊆ I ( Y 2 ) ⊆ I ( Y 3 ) ⊆ ⋯ {\displaystyle I(Y_{1})\subseteq I(Y_{2})\subseteq I(Y_{3})\subseteq \cdots }

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Noetherian topological space

Start with the simplest possible case. Write down what Noetherian topological space claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Noetherian topological space before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Noetherian topological space ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Noetherian topological space

In research: Noetherian topological space appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Noetherian topological space in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Noetherian topological space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Properties of topological spaces, Scheme theory, so understanding it makes those chapters shorter.
In everyday life: Look for Noetherian topological space outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Noetherian topological space in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Noetherian topological space means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Noetherian topological space out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Noetherian topological space in simple terms?

In mathematics, a Noetherian topological space, named for Emmy Noether, 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.

Why does Noetherian topological space matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Noetherian topological space?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Noetherian topological space.