mathematics

Infinite broom

Infinite broom 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 Infinite broom rather than just read about it. In short: In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also referred to as the broom space.

Key takeaways

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

Reference excerpt

In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also referred to as the broom space.

Definition The infinite broom is the subset of the Euclidean plane that consists of all closed line segments joining the origin to the point (1, 1/n) as n varies over all positive integers, together with the interval (½, 1] on the x-axis. The closed infinite broom is then the infinite broom together with the interval (0, ½] on the x-axis. In other words, it consists of all closed line segments joining the origin to the point (1, 1/n) or to the point (1, 0).

Properties Both the infinite broom and its closure are connected, as every open set in the plane which contains the segment on the x-axis must intersect slanted segments. Neither is locally connected. Despite the closed infinite broom being arc connected, the standard infinite broom is not path connected. The interval [0,1] on the x-axis is a deformation retract of the closed infinite broom, but it is not a strong deformation retract.

See also Comb space Integer broom topology List of topologies

References

Illustrations

Worked examples

Example 1 — a first encounter with Infinite broom

Start with the simplest possible case. Write down what Infinite broom 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 Infinite broom 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 Infinite broom 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 Infinite broom

In research: Infinite broom 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 Infinite broom 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: Infinite broom is common in secondary-school and first-year university syllabi. It links to neighbouring topics Topological spaces, Trees (topology), so understanding it makes those chapters shorter.
In everyday life: Look for Infinite broom 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Infinite broom in 20 minutes

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

Frequently asked questions

What is Infinite broom in simple terms?

Why does Infinite broom 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 Infinite broom?

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 Infinite broom.