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compactness    音标拼音: [kəmp'æktnəs]
紧密度

紧密度

compactness
紧致性

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  • 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]
    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)
  • Definition of Compactness - Mathematics Stack Exchange
    To understand the definition of compactness, first you need to understand the definition of covering A covering of A in X is a set of open sets in X such that A is contained in the union of this open sets A set A is said to be compact if, for each covering of A, there exists a finite subcovering of A
  • Why is compactness so important? - Mathematics Stack Exchange
    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
  • Compactness and sequential compactness in metric spaces
    Compactness and sequential compactness are equivalent in metric space but not always in others 0 About
  • general topology - Difference between completeness and compactness . . .
    Compactness implies completeness To see that is easy Take a Cauchy sequence Since we are on a compact set, it has a convergent subsequence But a Cauchy sequence with a convergent subsequence must converge (this is a good exercise, if you don't know this fact)
  • compactness and boundedness - Mathematics Stack Exchange
    Your definition of compactness (closed and bounded) works for $\Bbb{R}$ and $\Bbb{R}^n$ (and other finite
  • compactness - Can anyone explain a compact set in a general topological . . .
    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
  • general topology - pre-compactness, total boundedness and Cauchy . . .
    Pre-compactness in the first quote is defined differently from the one in the second quote So now my question is narrowed down to whether total boundedness and Cauchy sequential compactness are equivalent in both metric spaces and uniform spaces Pete's reply says yes for metric spaces, and now what can we say about uniform spaces?





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