How to understand compactness? - Mathematics Stack Exchange
Compactness extends local stuff to global stuff because it's easy to make something satisfy finitely many restraints- this is good for bounds. Connectedness relies on the fact that ``clopen'' properties should be global properties, and usually the closed' part is easy, whereas the open' part is the local thing we're used to checking. $\endgroup$
What is Compactness and why is it useful? [closed]
2021年10月3日 · The wiki definiton defines a compactness of an interval as closed and bounded. In mathematics, specifically general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (containing all its limit points) and bounded (having all its points lie within some fixed distance of each other).
Understanding compactness and how it relates to finiteness
2017年12月21日 · One can use compactness to prove the intermediate value property of continuous functions as well and also the fact that continuous functions attain their maximum/minimum values on a closed interval. There is another result about uniform continuity of continuous functions on a closed interval which does not have a proof without the idea of ...
Discreteness"+" Compactness= Finiteness: What does this really …
2019年1月30日 · What are some ways in which compactness doesn't include discreteness? We can approach this by considering specific compact non-discrete spaces, like $[0,1]$ , but a better approach is to realize that any topology coarser than a compact topology is compact.
compactness - Can anyone explain a compact set in a general …
2020年2月16日 · Hence a key observation concerning compactness is that it is a non-trivial generalisation of finiteness. In Edwin Hewitt's Essay, "The rôle of compactness in analysis" he says that: "The thesis of this essay is that a great many propositions of analysis are: trivial for finite sets. true and reasonably simple for infinite compact sets.
Why is compactness so important? - Mathematics Stack Exchange
2013年9月7日 · As many have said, compactness is sort of a topological generalization of finiteness. 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.
What's going on with "compact implies sequentially compact"?
Here we have that sequential closure always equals closure, and for Hausdorff sequential spaces we can prove (e.g. see this blog post) quite easily that sequential compactness is equivalent to countable compactness. The latter is always implied by compactness, so that for sequential spaces we have compact implies sequentially compact (but not ...
general topology - compactness / sequentially compact
a problem about compactness and sequential compactness in metric space. 1. Question on countably compact ...
compactness - Every co-finite topology is compact? - Mathematics …
2023年9月15日 · Has trouble understanding finite set while I was solving theorem on topological space which is Every co-finite topology is compact. Let (X,T) be topological space. In co-finite topology X is infin...
compactness and boundedness - Mathematics Stack Exchange
Your definition of compactness (closed and bounded) works for $\Bbb{R}$ and $\Bbb{R}^n$ (and other finite ...