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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To conclude,take a look on these examples they show how worse can be lack of compactnes: R K Sinha 4 6. Consider the following Theorem:. It gives you convergent subsequences when working with arbitrary sequences that aren’t known to converge; the Arzela-Ascoli theorem is an bewitt 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.

Anyway, a topological space is finite iff it is both compact and P.

A locally compact abelian group is compact if and only if its Pontyagin dual is discrete. Essentially, compactness is “almost as good as” finiteness. Especially hewktt stating “for every” open cover makes compactness a concept that must be very difficult thing to prove in analysls – what makes it worth the effort?

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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. Here are some more useful things: Either way you look at it, though, the compactness theorem is a statement about the topological compactness of a particular space products of compact Stone spaces.

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Is there a redefinition of discrete so this principle works for all topological spaces e. And when one learns about first order logic, gets the feeling that ajalysis is, somehow, deduce information about an “infinite” object by deducing it from its “finite” or from a finite number of parts.

I can’t think of a good example hswitt make this more precise now, though. Compact spaces, being “pseudo-finite” in their nature are also well-behaved and we can prove interesting things about them.

Historically, it led to the compactness theorem for first-order logic, but that’s over my head. Mathematics Stack Exchange jn best with JavaScript enabled. The condition of having finite subcover and finite refinement are equivalent. 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.

For sequential compactness of a set, we ask: I think it’s a great example because it motivates the study of weaker notions of convergence. By the way, as always, very nice to read your te. But why finiteness is important? Evan 3, 8 FireGarden, perhaps you are reading about paracompactness? To prove your theorem without it: And yet, we work so much with these properties.


Or me ask you one thing: FireGarden 2, 2 15 This list is far from over Every compact Hausdorff space is normal.

Every universal net in a compact set converges. Clark Sep 18 ’13 at Kris 1, 8 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. One reason is that boundedness doesn’t make sense in a compzctness topological space.

general topology – Why is compactness so important? – Mathematics Stack Exchange

In probability they use the term “tightness” for measures. A very closely-related example is the compactness theorem in propositional logic: So, at least for closed sets, compactness and boundedness are the same. Thank you for the compliment.