Why are compact sets called "compact" in topology?
Solution 1:
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).
Solution 2:
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.