mathematics

Relatively compact subspace

Relatively compact subspace 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 Relatively compact subspace rather than just read about it. In short: In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact).

Key takeaways

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

Reference excerpt

In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact.

Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). In an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is not necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. Some major theorems characterize relatively compact subsets, in particular in function spaces. An example is the Arzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis. Mahler's compactness theorem in the geometry of numbers characterizes relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices).

Counterexample As a counterexample take any finite neighbourhood of the particular point of an infinite particular point space. The neighbourhood itself is compact but is not relatively compact because its closure is the whole non-compact space.

Almost periodic functions

The definition of an almost periodic function F at a conceptual level has to do with the translates of F being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.

See also Compactly embedded Totally bounded space

References

page 12 of V. Khatskevich, D.Shoikhet, Differentiable Operators and Nonlinear Equations, Birkhäuser Verlag AG, Basel, 1993, 270 pp. at google books

Worked examples

Example 1 — a first encounter with Relatively compact subspace

Start with the simplest possible case. Write down what Relatively compact subspace 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 Relatively compact subspace 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 Relatively compact subspace 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 Relatively compact subspace

In research: Relatively compact subspace 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 Relatively compact subspace 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: Relatively compact subspace is common in secondary-school and first-year university syllabi. It links to neighbouring topics Compactness (mathematics), Properties of topological spaces, so understanding it makes those chapters shorter.
In everyday life: Look for Relatively compact subspace 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 Relatively compact subspace in 20 minutes

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

Frequently asked questions

What is Relatively compact subspace in simple terms?

In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact).

Why does Relatively compact subspace 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 Relatively compact subspace?

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 Relatively compact subspace.