TITLE: How can one know if a set is compact?
QUESTION [6 upvotes]: 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?

REPLY [8 votes]: 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.