Why is compactness important

Classifying non-compact surfaces is more difficult and less satisfying. Compact surfaces are more constrained. Non-compact ones can squirm out of your hands like blobs of rice pudding.

Compact ones are more like jello: they might wobble a bit, but you can hold on to them if you don't mind getting your hands a little dirty. The post-rigorous understanding of compactness allows the word "compact" to circle around from something that feels like robot speak to something that aligns very closely with an English meaning of the word. I like to think of it as a delightful accident of mathematical-linguistic convergence.

The views expressed are those of the author s and are not necessarily those of Scientific American. Follow Evelyn Lamb on Twitter. Already a subscriber?

Sign in. Thanks for reading Scientific American. Create your free account or Sign in to continue. See Subscription Options.

Discover World-Changing Science. A collection of many open intervals on the real number line. Credit: Evelyn Lamb. Get smart. Sign up for our email newsletter. Sign Up. Read More Previous. There are many other theorems for abstract topological spaces where compactness is important.

You might want to look at the answers for this question: Applications of compactness. This indicates that one might be able to formulate the extreme value theorem in more general situations, and it might be fruitful to study the notion of compactness further. My guess is that there is no really simple way to motivate the modern definition compactness. Does anyone know when the modern definition appeared?

Im guessing that the definition of sequential compactness appeared before the definition of compactness. Sign up to join this community. The best answers are voted up and rise to the top. Asked 11 years, 4 months ago. Active 11 years, 4 months ago. Viewed 2k times. Improve this question.

Community Bot 1 2 2 silver badges 3 3 bronze badges. Simply put, compactness gives you something to work with, this "something" depending on the particular context. But for example, it gives you extremums when working with continuous functions on compact sets. It gives you convergent subsequences when working with arbitrary sequences that aren't known to converge; the Arzela-Ascoli theorem is an important instance of this for the space of continuous functions this point of view is the basis for various "compactness" methods in the theory of non-linear PDE.

It gives you the representation of regular Borel measures as continuous linear functionals Riesz Representation theorem. In this situation, for practical purposes, all I want to know about topologically for a given setting is, given a sequence of points in my space, define a notion of convergence. Give me the definition of convergence to play with, and we can talk about sequential compactness. For sequential compactness of a set, we ask: "Given an arbitrary sequence in the set, does there exist a convergent subsequence?

In probability they use the term "tightness" for measures. I think it's a great example because it motivates the study of weaker notions of convergence.

Put differently, we know that a union of closed sets need not be closed. Under what condition is the union of a parametric family of closed sets closed? With appropriate definitions, if it is a "continuous family" where the parameter space is compact. This can be useful in many contexts. Every continuous function is Riemann integrable-uses Heine-Borel theorem.

Since there are a lot of theorems in real and complex analysis that uses Heine-Borel theorem, so the idea of compactness is too important. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn more. Why is compactness so important? Ask Question. Asked 8 years, 2 months ago. Active 1 year, 4 months ago. Viewed 22k times. FireGarden FireGarden 5, 4 4 gold badges 20 20 silver badges 31 31 bronze badges.

A refinement is something different, used to define weaker related ideas. I can't think of a good example to make this more precise now, though. It discusses the original motivations for the notion of compactness, and its historical development.

If you want to understand the reasons for studying compactness, then looking at the reasons that it was invented, and the problems it was invented to solve, is one of the things you should do.

Show 5 more comments. Active Oldest Votes. I particularly like the phrase "finitely many possible behaviors". In every other respect, one could have used "discrete" in place of "compact".

Honestly, discrete spaces come closer to my intuition for finite spaces than do compact spaces. However, as you pointed out, compactness is deep; in contrast, discreteness is the ultimate separation axiom while most spaces we're interested in are comparatively low on the separation hierarchy.

By the way, as always, very nice to read your answers. I was wondering if you had any nice examples that illustrate that first paragraph? Add a comment. Michael Hardy Michael Hardy 1. Kris Kris 1, 10 10 silver badges 12 12 bronze badges.