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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?2012-06-01
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    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.2012-06-01

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