Answer by Kapil for Why is compactness so important?
We can think of a closed set in $X$ as being like the locus of $x\in X$ where a continuous function $f(x)$ is $0$.Now suppose there is a continuous function $f(x,p)$ which depends on a parameter $p$....
View ArticleAnswer by R K Sinha for Why is compactness so important?
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...
View ArticleAnswer by John Gowers for Why is compactness so important?
If you have some object, then compactness allows you to extend results that you know are true for all finite sub-objects to the object itself. The main result used to prove this kind of thing is the...
View ArticleAnswer by Evan for Why is compactness so important?
I want to elaborate Sargera's and Tomás' theme. Topological considerations are great, but to me the examples are not as concrete as for when we speak of "sequential compactness" (which unfortunately in...
View ArticleAnswer by Chris Culter for Why is compactness so important?
It's already been said that compact spaces act like finite sets. A variation on that theme is to contrast compact spaces with discrete spaces.A compact space looks finite on large scales. A discrete...
View ArticleAnswer by Tomás for Why is compactness so important?
I would like to give here a example showing why compactness is important. Consider the following Theorem:Theorem: Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous coercive function. Then, there exist...
View ArticleAnswer by Sargera for Why is compactness so important?
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...
View ArticleAnswer by Michael Hardy for Why is compactness so important?
Compactness does for continuous functions what finiteness does for functions in general.If a set $A$ is finite, then every function $f:A\to \mathbb R$ has a max and a min, and every function...
View ArticleAnswer by Asaf Karagila for Why is compactness so important?
As many have said, compactness is sort of a topological generalization of finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often "care about...
View ArticleAnswer by Michael Albanese for Why is compactness so important?
One reason is that boundedness doesn't make sense in a general topological space. For example $(-1, 1) \subset \mathbb{R}$ is bounded when viewing $\mathbb{R}$ as a metric space with the usual...
View ArticleAnswer by Henrique Tyrrell for Why is compactness so important?
Compactness is important because:1)It behaves greatly when using topological operationsa)It's a condition that is carried on by continuous functions on any topological space, that is, if $C$ is compact...
View ArticleAnswer by dfeuer for Why is compactness so important?
Well, here are some facts that give equivalent definitions:Every net on a compact set has a convergent subnet.Every ultrafilter on a compact set converges.Every filter on a compact set has a limit...
View ArticleAnswer by Kris for Why is compactness so important?
Compactness is the next best thing to finiteness.Think about it this way: Let $A$ be a finite set, let $f: A \to \mathbb{R}$ be a function. Then $f$ is trivially bounded. Now let $X$ be a compact set,...
View ArticleWhy is compactness so important?
I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel...
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