mathematics

Localization of a topological space

Localization of a topological space is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Localization of a topological space rather than just read about it. In short: In mathematics, well-behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in (Sullivan 2005).

Key takeaways

Localization of a 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 Localization of a topological space to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Localization of a topological space from memory before moving on to harder problems.

Reference excerpt

In mathematics, well-behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in (Sullivan 2005). The reason to do this was in line with an idea of making topology, more precisely algebraic topology, more geometric. Localization of a space X is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space X, directly, giving a second space Y.

Definitions We let A be a subring of the rational numbers, and let X be a simply connected CW complex. Then there is a simply connected CW complex Y together with a map from X to Y such that

Y is A-local; this means that all its homology groups are modules over A The map from X to Y is universal for (homotopy classes of) maps from X to A-local CW complexes. This space Y is unique up to homotopy equivalence, and is called the localization of X at A. If A is the localization of Z at a prime p, then the space Y is called the localization of X at p. The map from X to Y induces isomorphisms from the A-localizations of the homology and homotopy groups of X to the homology and homotopy groups of Y.

See also Category:Localization (mathematics)

Local analysis Localization of a category Localization of a module Localization of a ring Bousfield localization

References Adams, Frank (1978), Infinite loop spaces, Princeton, N.J.: Princeton University Press, pp. 74–95, ISBN 0-691-08206-5 Sullivan, Dennis P. (2005), Ranicki, Andrew (ed.), Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 MIT Notes (PDF), K-Monographs in Mathematics, Dordrecht: Springer, ISBN 1-4020-3511-X

Worked examples

Example 1 — a first encounter with Localization of a topological space

Start with the simplest possible case. Write down what Localization of a 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 Localization of a 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 Localization of a 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 Localization of a topological space

In research: Localization of a 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 Localization of a 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: Localization of a topological space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Commutative algebra stubs, Homotopy theory, Localization (mathematics), so understanding it makes those chapters shorter.
In everyday life: Look for Localization of a 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.

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How to study Localization of a topological space in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Localization of a 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 Localization of a topological space out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Localization of a topological space in simple terms?

Why does Localization of a 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 Localization of a 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 Localization of a topological space.