In a metric space, a set S is said to be closed if, for every convergent sequence of points (xn) lying wholly in S, its limit point is also in S.
You can prove that in all metric spaces, a set is closed iff its complement is open, and so this is taken to be the definition of closed sets in topological spaces in general.
N.b. A set can be both open and closed (eg. R, the real numbers), or neither open nor closed (eg. (0,1]). Do not let sine1 fool you! | 
