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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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    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.