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!