Toposym 1. Edwin Hewitt. Some applications to harmonic analysis, and so clearly illustrate the importance of compactness, that they should be cited. The first. This paper traces the history of compactness from the original motivating questions E. Hewitt, The role of compactness in analysis, Amer. Compactness. The importance of compactness in analysis is well known (see Munkres, p). In real anal- ysis, compactness is a relatively easy property to.
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The only theorems I’ve seen concerning compactneess are the Heine-Borel theorem, and a proof continuous functions on R from closed subintervals of R are bounded. It seems like such a strange thing to define; why would the fact every open cover admits a finite refinement be so useful? Especially as im “for every” open cover makes compactness a concept that must be very difficult thing to prove in general – what makes it worth the anlaysis If it helps answering, I am about to enter my third year of my undergraduate degree, and came to wonder this upon preliminary reading of introductory topology, where I first found the definition of compactness.
As many have said, compactness is sort of a topological generalization rhe finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often “care about how something behaves on an open set”, and for compact spaces this means that there are only finitely many possible behaviors.
But why finiteness is important?
Well, finiteness allows us to construct things “by hand” and constructive results are a lot deeper, and to some extent useful to us. Moreover finite objects are well-behaved ones, so while compactness is not exactly finiteness, it does preserve a lot of this behavior because it behaves “like a finite set” for important topological properties and this means that we can actually work compactnss compact spaces. This is hewirt most of mathematics. And yet, we work so much with these properties.
Because those are well-behaved properties, and we can control these constructions and prove interesting things about them. Compact spaces, being “pseudo-finite” in their nature are also well-behaved anzlysis we can prove interesting things about them. So they end up being useful for that reason. 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 space looks finite on small scales. A locally compact abelian group is compact if and only if its Pontyagin dual is discrete. One reason is that boundedness doesn’t make sense in a general topological space.
So why then compactness? So, at least for closed sets, compactness and boundedness are the same. This relationship is a useful one because we now have a notion which is strongly related to boundedness which does generalise to topological spaces, unlike boundedness itself.
In addition, at least for Hausdorff topological spaces, compact sets are closed. So one way to think about compact sets in topological spaces is that they are analogous to the bounded sets in metric spaces.
I would like to give here a example showing why compactness is important. Consider the following Theorem:. In this topology we have less open sets which implies more compact sets and in particular, bounded sets are pre-compact sets. To conclude,take a look on these examples they show how worse can be lack of compactnes: 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 ocmpactness 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.
general topology – Why is compactness so important? – Mathematics Stack Exchange
It gives you the representation of regular Borel wnalysis as continuous hewiitt functionals Riesz Representation theorem. 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. A very closely-related example is the compactness theorem in propositional logic: This can be proved using topological compactness, or it can be proved using the completeness theorem: Either way you look at it, though, the compactness theorem is a statement about compactnses topological compactness of a particular space products of compact Stone spaces.
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: 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.
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.
Essentially, compactness is “almost as good as” finiteness. I can’t think of a good example to make this more precise now, though. FireGarden, perhaps you are reading about paracompactness?
It discusses the rope 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, anaalysis the problems it was invented to solve, is one of the things you should do. The condition of having finite subcover and finite refinement are equivalent.
I particularly like the phrase “finitely many possible behaviors”. In every other respect, one analysiss 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.
And when one learns about first order logic, gets the feeling that compactness is, somehow, deduce information about an “infinite” object by deducing it from its “finite” or from a finite number of parts.
By the way, as always, aalysis nice to read your answers. Thank you for the compliment. AsafKaragila This seems to be on par with what Qiaochu mentioned here: I was wondering if you had any nice examples that illustrate that first paragraph? Compactness does for continuous functions what finiteness does for functions in general.
Compactness is the next best thing to finiteness. Think about it this way: Kris 1, 8 Is there a redefinition of discrete so this principle works for all topological spaces e. Not sure what this property P analysks be called Anyway, a topological space is finite iff it is both compact and P.
Compactness is important hewift Compactness is useful even when it emerges as a property of subspaces: This list is far from over Anyone care to join in? Henrique Tyrrell 6 Historically, it led to the compactness theorem for first-order logic, but that’s over my head. Well, here are some facts that give clmpactness definitions: Every net on a compact set has a convergent subnet. Every ultrafilter on a compact set converges.
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Every filter on a compact set has a limit point. Every net in a compact set has a limit point. Every universal net in a compact set converges. Here are some more useful things: Every continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Every compact Hausdorff space is normal. The image of a compact space under a continuous function is compact. Every infinite subset of a compact space has a limit point. Consider the following Theorem: The concept of a “coercive” function was unfamiliar to me until I read your answer; I suspect the same will be true for many readers.
So I’m not sure this is a good example Clark Sep 18 ’13 at The rest of your example is very interesting and strong Thank you for your comment PeteL. Let me ask you one thing: For example, a proof which comes from my head is: Please, could you detail more your point of view to me?