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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Well, here are some facts that give equivalent definitions: A very closely-related th is the compactness theorem in propositional logic: I can’t think of a good example to make this more precise now, though.
I’ve read many times that ‘compactness’ is such an extremely important and useful concept, though it’s still not very apparent why.
Is there a redefinition of discrete so this principle works for all topological spaces e. R K Sinha 4 6. Every compact Hausdorff space is normal. Thank you for the compliment.
Let me ask you one thing: 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. So they end up being useful for that reason.
general topology – Why is compactness so important? – Mathematics Stack Exchange
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. By the way, as always, very nice to read your answers. To conclude,take a look on these examples they show how worse can be lack of compactnes: In probability they use the term “tightness” for measures Hmm. FireGarden, perhaps you are reading about paracompactness?
So, at least for closed sets, compactness and boundedness are the same.
It seems like such a strange thing to define; why would the fact every open cover admits a finite refinement be so useful? 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 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 with compact spaces. I think it’s a great example because it motivates the study of weaker notions of convergence.
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But for example, it gives you extremums when working with continuous functions on compact sets. FireGarden 2, 2 15 Think about it this way: Consider the following Theorem:. Here are some more useful things: This can be proved using topological compactness, or it can be hewitt using the completeness theorem: I was wondering if you had any nice examples that illustrate that first paragraph?
One reason is that boundedness doesn’t make sense in a general topological space. Compact spaces, being “pseudo-finite” in their nature are also well-behaved and we can prove interesting things about them.
Why is compactness so important? 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.
To prove your theorem without it: Evan 3, 8 Honestly, discrete spaces come closer to my intuition for finite spaces than do compact spaces. Historically, it led to the compactness theorem for first-order logic, but that’s over my head. Simply put, compactness gives you something to work with, this “something” depending on the particular context. Because those are well-behaved properties, and we can control these constructions and prove interesting things about them.
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. A variation on that theme is to contrast compact spaces with discrete spaces.