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How can one know if a set is compact? From the definition, a set is compact if for any open cover, there exist a finite subcover. However, it is not possible to list out all the open covers to a set. So is there any way one can know if it is a compact set? In the case $R^n$ it is easy as one only needs to check the set is closed bounded but what about other metric space or other spaces?

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    @Mathematics: See [here](http://www.mast.queensu.ca/~speicher/Chapter10.pdf) for examples.2012-12-07

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An equivalent condition in the case of metric spaces is sequential compactness: Every sequence in the space has a convergent subsequence.

There exist a number of other characterizations, but they are usually not that easy to verify (such as being complete and totally bounded).

Often, one can use basic properties of compactness to show a given space is compact. For example, a closed subset of a compact set is compact and the forward image of a compact set under a continous function is compact.

There are many results for showing that some spaces are compact. These include the Arzelà–Ascoli theorem, the Banach–Alaoglu theorem, and a number of other results. These results are important precisely because it is in general often nontrivial to show that some space is compact.