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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Home Questions Tags Hhe Unanswered. Not sure what this property P should be called 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.
Thank you for the compliment. R K Sinha 4 6. Every continuous function is Riemann integrable-uses Heine-Borel theorem.
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A variation on that theme is to contrast compact spaces with discrete spaces. Every net in a compact set has a limit point.
It discusses the original motivations for the notion of compactness, and its historical development. Sign up using Email and Password. It gives you the representation of regular Borel measures as continuous linear functionals Riesz Representation theorem.
Anyone care to join in? For example, a proof which comes from my head is: 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.
Sign up using Facebook. Honestly, discrete spaces come closer to my intuition for finite spaces than do compact spaces. Essentially, compactness is “almost as good as” finiteness. So why then compactness?
Every continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Is there a redefinition of discrete so this principle works for all topological spaces e. I can’t think of a good example to make this more precise now, though.
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. Anyway, a topological space is finite iff it is both compact and P. I’ve read many times that ‘compactness’ is such an extremely important and useful concept, though it’s still not very apparent why.
general topology – Why is compactness so important? – Mathematics Stack Exchange
Compactness is important because: A locally clmpactness abelian group is compact if and only if its Pontyagin dual is discrete. Well, finiteness allows us to construct things “by hand” and constructive results are a lot deeper, and to some extent useful to us.
In addition, at least for Hausdorff topological spaces, compact sets are closed. Every universal net in a compact set converges. The only theorems I’ve seen concerning it are the Rolle theorem, and a proof continuous functions on R from closed subintervals of R are bounded. Consider the following Theorem: In every other respect, one could have used “discrete” in place of “compact”. To conclude,take a look on these examples they show how worse can be lack of compactnes: 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.
Moreover finite objects are well-behaved ones, so while compactness hewott not exactly finiteness, rolle 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 with compact spaces.
Compact spaces, being “pseudo-finite” in their nature are also well-behaved and we can prove interesting things about them. As many have said, compactness is sort of a topological generalization of finiteness. In this topology we have less open sets which implies more compact sets and in particular, bounded sets are pre-compact sets. This is throughout most of mathematics.
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. Simply put, compactness gives you something to work with, this “something” depending on the particular context. Post analysos a guest Name. I particularly like the phrase “finitely many possible behaviors”.