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
, and so this is taken to be the definition of closed sets in topological spaces
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!