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!