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The seemingly simple definition:

A topological space $X$ is compact, if every open cover of $X$ has a finite sub-cover;

turns out to be quite central one in topology. It's seems like once we are granted the compactness of the space at hand, a whole bunch of good things can be said and done about it, but I am not exactly sure why? Many thanks.

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    You might find the following American Mathematical Monthly article interesting (requires JStor access): http://www.jstor.org/discover/10.2307/2309166?uid=3738232&uid=2129&uid=2&uid=70&uid=4&sid=21101250017113 The basic thesis of the article is that compactness is in a sense a generalization of finiteness: there are many propositions in analysis that are (1) trivial for finite sets, (2) true and relatively easy to prove for compact sets, (3) false or difficult to prove for non-compact sets.2012-11-05
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    You should check out wikipedia for the history of the concept.2012-11-05
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    One could think of compactness as a topological equivalent to finiteness (and that is why everything is simpler). If the topology is given by a distance and $X$ is compact, then for any $\epsilon$, you can cover $X$ with a finite number of open balls of radius $\epsilon$. You can see that as "finite up to $\epsilon$, for all $\epsilon > 0$" (but topology is interested in properties that hold "up to $\epsilon$ for all $\epsilon > 0$"). In general topology, replace open balls with open sets, and "up to $\epsilon$, for all $\epsilon > 0$" with "up to an open set, for all open sets".2012-11-05

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Some examples of the usefulness of compactness, following what James Fennell said:

One of the more important consequences is that (on a metric space) compactness implies sequential compactness: every sequence in the space has a subsequence that converges in that space. Additionally, some theorems can be proven by first proving them for compact sets (which is easier) and extending to more general sets. Compactness implies a number of other useful results, for example, continuous functions on a compact set are uniformly continuous.