What's the difference between open and closed sets? What's the difference between open and closed sets?
Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!

I will not reiterate the very nice definitions found in the other answers, however I think that these "practical" definitions might help you as well on an intuitive level.
Open sets are typically used as domains for functions, as they are more useful for analysing "continuous" properties like differentiability. Also they don't have nasty borders (hence you don't have to deal with functions which are well behaved only on one side of the edge).
Closed sets are useful because, if they are limited, they are compact.
