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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In this topology we have less open sets which implies more compact sets and in particular, bounded sets are pre-compact sets. To prove your theorem without it: 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. As many have said, compactness is sort of a topological generalization jn finiteness. So, at least for closed sets, compactness and boundedness are the same.
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general topology – Why is compactness so important? – Mathematics Stack Exchange
This list is far from over Compact spaces, being “pseudo-finite” in their nature are also well-behaved and we can prove interesting things about them. One reason is that boundedness doesn’t make sense in a general topological space. So why then compactness? The only theorems I’ve seen concerning it are the Heine-Borel theorem, and a proof continuous heitt on R from closed subintervals of R are bounded. Naalysis there a redefinition of discrete so this principle works for all topological spaces e.
By the way, as always, very nice to read your answers. Please, could you detail more your point of view to me? A locally compact abelian group is compact if and only if its Pontyagin dual is discrete.
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. Evan 3, 8 Consider the following Theorem:. Compactness is the next best thing to finiteness. 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.
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.
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Honestly, discrete spaces come closer to my intuition for finite spaces than do compact spaces. In probability they use the term “tightness” for measures Hmm.
It gives you convergent subsequences when working with arbitrary sequences that aren’t known to converge; the Arzela-Ascoli theorem hewit an important instance of this for snalysis space of continuous functions this point of view is the basis for various “compactness” methods in the theory of non-linear PDE. Every universal net in a compact set converges.
So they end up being useful for that reason. Every continuous function is Riemann integrable-uses Heine-Borel theorem. And yet, we work so much with these properties.
Thank you for the compliment. 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.
Because those are well-behaved properties, and we can control these constructions and prove interesting things about them. Home Questions Tags Users Unanswered. Sign up or log in Sign up using Google.
Kris 1, 8 Every filter on a compact set has a limit point. FireGarden, perhaps you are reading about paracompactness? It seems like such a strange thing to define; why would the fact every open cover admits a finite refinement be so useful? In every other respect, one could have used “discrete” in place of “compact”. The rest of your example analysi very interesting and strong In probability they use the term “tightness” for measures.
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 analysks that we can actually work with compact spaces. A compact space looks xnalysis on large scales.
If you want to understand the reasons for studying compactness, then looking at the reasons that it was invented, and the problems it compactnesss invented to solve, is one of the things you should do. Especially as stating “for every” open cover makes compactness a concept that must be very difficult thing to prove in general – what makes it worth the effort?
Every net on a compact set has a convergent subnet. To conclude,take a look on these examples they show how worse can be lack of compactnes: