closed set (metric)

A closed set is one that already contains every point it gets arbitrarily close to: if a sequence inside the set converges, its limit cannot escape. A closed set keeps hold of its own edge — nothing leaks out in the limit.

There are two equivalent definitions. First: a set F is closed if it contains all its limit points, that is, every point approached arbitrarily closely by points of F already lies in F. Second: F is closed if its complement X minus F is open. These two descriptions always coincide, and either can serve as the definition.

Closedness is exactly the property that makes limits stay put, which is why it appears whenever we want a process to converge to something inside the set. By taking complements, the rules for open sets flip: the empty set and X are closed; any finite union of closed sets is closed; and any intersection, finite or infinite, of closed sets is closed.