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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But for example, it gives you extremums when working with continuous functions on compact sets. Thank you for your comment PeteL.
general topology – Why is compactness so important? – Mathematics Stack Exchange
Please, could you detail more your point of view to me? Compactness is useful even when it emerges as a property off subspaces: 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.
Every continuous function is Riemann integrable-uses Heine-Borel theorem. A compact space looks finite on large scales. If it helps answering, I am about to enter my third year of my undergraduate degree, and rolw 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.
Clark Sep 18 ’13 at Sign up or log in Sign up using Google. It’s already been said that compact spaces act like finite sets.
The rest of your example is very interesting and strong And yet, we work so much with these properties. Henrique Tyrrell 6 Evan 3, 8 Sign up using Email and Password. To prove your theorem without it: This is throughout most of mathematics. I was wondering if you had any nice examples that illustrate that first paragraph?
Give me the definition of convergence to play with, and we can talk about sequential compactness. In probability they use the term “tightness” for measures. Heaitt compact Hausdorff space is normal. 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.
Every ultrafilter on a compact set converges. R K Sinha 4 6. AsafKaragila This seems to be on par with what Qiaochu mentioned here: As many have said, compactness is sort of a topological generalization of finiteness.
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.
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. 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.
The only theorems I’ve seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from closed subintervals of R are bounded. Essentially, compactness is “almost as good as” finiteness.
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In every other respect, one could have used “discrete” in place of “compact”. By the way, as always, very nice to read your answers. 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.
A anaysis is something different, used to define weaker related ideas. So they end up being useful for that reason.
But why finiteness is important? It discusses the original motivations for the notion of compactness, and its historical development. Every net in a compact set has a limit point.