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What would be importance of cover in topological space?

And how is the concept of cover being used in other areas, more specifically set theory?

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What do you mean by 'cover?' A covering space? The idea that a collection of open balls might cover a set? A covering map? A universal cover? –  mixedmath Jun 1 '12 at 8:02
    
One of the most useful thing about cover is that it shows whether a space is compact or not . and indeed compactness is of great importance, not very easy to summerize. –  Theorem Jun 1 '12 at 8:02
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2 Answers

It would take a whole book really to explain the importance of covers, and especially of open covers, in topology. For starters, compactness is a topological property of central importance, and although this fact wasn’t immediately apparent in the early years of topology, it is best defined in terms of open covers of spaces. Metric spaces are clearly important, and not just in topology proper; one of the less obvious reasons for their importance is the fundamental fact that they are paracompact: this property, which is also defined in terms of open covers, is the reason for some of their nicer properties. Study of metrization theory, i.e., of when a space is metrizable, has led naturally to the study of a large variety of so-called covering properties, properties defined in terms of covers of various sorts.

In purely set-theoretic terms a cover of a set $X$ is simply a family $\mathscr{C}\subseteq\wp(X)$ such that $X=\bigcup\mathscr{C}$. Covers aren’t generally interesting or useful unless they have additional properties, however. One might, for instance, require that the members of a cover be pairwise disjoint and non-empty, in which case one has a very familiar object: a partition of a set $X$.

If $X$ is partially ordered, one might be interested in covering it with chains (linearly ordered subsets); the Dilworth decomposition theorem says that a partially ordered set of finite width $w$ can be covered by $w$ chains, which can actually be chosen to be pairwise disjoint. There is some interest in theoretical computer science in algorithms for carrying out partitions into as few chains as possible when the partially ordered set is presented one element at a time (together with its relationships, if any to previously presented elements), and the elements are irrevocably assigned to chains as they are presented. In general the Dilworth limit cannot be achieved, and the on-line partitioning problem for partially ordered sets is to establish the best achievable value.

I would expect other applications of covers to be similarly specific to the types of structures in which they’re used.

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When we thinks of characterizations of compact set in $R^{n} $ ,we may have many equivalent charizations: such as

  • bounded and closed
  • very family of open over of the set has finite open sub cover and so on. According to our experience, the compactness is the best gift from God, we want to keep it into the more general cases, such as infinite diemsional space, or some other topological spaces.

When it comes think to compact topological space, or compact subset. we may need have the essential charization of compact sets, so we choose to use open cover to characterize compact set.

I think the conception of 'totally bounded' and 'finite $ \epsilon$ net', also can been seen as cover used in functional analysis.

Now I will give you nontrival applications of conver

  • Adler's definition of topological entropy

http://www.ams.org/journals/tran/1965-114-02/S0002-9947-1965-0175106-9/S0002-9947-1965-0175106-9.pdf

  • Bowen's definition of metric entropy

http://www.jstor.org/stable/10.2307/1996403

  • Also some other sequence entropy in topolgical dynamics.
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