mathematics

Locally discrete collection

Locally discrete collection 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 Locally discrete collection rather than just read about it. In short: In mathematics, particularly topology, collections of subsets are said to be locally discrete if they look like they have precisely one element from a local point of view. The study of locally discrete collections is worthwhile as Bing's metrization theorem shows.

Key takeaways

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

Reference excerpt

In mathematics, particularly topology, collections of subsets are said to be locally discrete if they look like they have precisely one element from a local point of view. The study of locally discrete collections is worthwhile as Bing's metrization theorem shows.

Formal definition Let X be a topological space. A collection {Ga} of subsets of X is said to be locally discrete, if each point of the space has a neighbourhood intersecting at most one element of the collection. A collection of subsets of X is said to be countably locally discrete, if it is the countable union of locally discrete collections.

Properties and examples Locally discrete collections are by definition locally finite. If a collection of subsets of a topological space X is locally discrete, it must satisfy the property that each point of the space belongs to at most one element of the collection. This means that only collections of pairwise disjoint sets can be locally discrete. A Hausdorff space cannot have a locally discrete basis unless it is itself discrete. The same property holds for a T1 space. The following is known as Bing's metrization theorem: A space X is metrizable iff it is regular and has a basis that is countably locally discrete. A countable collection of sets is necessarily countably locally discrete. Therefore, if X is a metrizable space with a countable basis, one implication of Bing's metrization theorem holds. In fact, Bing's metrization theorem is almost a corollary of the Nagata–Smirnov theorem.

References James Munkres (1999). Topology, 2nd edition, Prentice Hall. ISBN 0-13-181629-2.

Worked examples

Example 1 — a first encounter with Locally discrete collection

Start with the simplest possible case. Write down what Locally discrete collection 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 Locally discrete collection 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 Locally discrete collection 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 Locally discrete collection

In research: Locally discrete collection 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 Locally discrete collection 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: Locally discrete collection is common in secondary-school and first-year university syllabi. It links to neighbouring topics Families of sets, General topology, so understanding it makes those chapters shorter.
In everyday life: Look for Locally discrete collection 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 Locally discrete collection in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Locally discrete collection 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 Locally discrete collection out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Locally discrete collection in simple terms?

Why does Locally discrete collection 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 Locally discrete collection?

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 Locally discrete collection.