Given a topological space $X$ and a subset of it $S$, $S$ is compact iff for every open cover of $S$, there is a finite subcover of $S$.
Just curiosity: I've done some search in Internet why compact sets are called compact, but it doesn't contain any good result. For someone with no knowledge of the topology, Facing compactness creates the mentality that a compact set is a compressed set!
Does anyone know or have any information on the question?
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I like to think of compact spaces as those which admit a "compact" description. So then the usual characterization of compactness can be interpreted along the lines of: whenever you have an infinite amount of information to describe a space, you really only need some finite reduction of that information.
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If you are curious about the history of compact sets (the definition of which dates back to Fréchet as mentioned in another response) and you can read French, then I suggest checking out the following historical article:
Pier, J. P. (1980). Historique de la notion de compacité. Historia mathematica, 7(4), 425-443. Retrieved from http://www.sciencedirect.com/science/article/pii/0315086080900063.
I came across this article when posting a response on MO here.
Frechet oiginally coined the term in 1904 to refer to a space where every sequence had a limit point. Thus, the space was 'compact' because there was no room for the sequence to escape (my interpretation).