In a metric space, a set S is said to be closed if, for every convergent sequence of points (x_n) 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!