Closed Set

Closed sets admit several equivalent characterizations. Let be a subset of a metric space. A set is closed if

1. The complement of is an open set,

2. is its own set closure,

3. Every limit in the ambient metric space of a sequence, net, or filter in belongs to ,

4. Every point outside has a neighborhood disjoint from .

The point-set topological definition of a closed set is a set which contains all of its limit points. Therefore, a closed set is one for which, whatever point is picked outside of , can always be isolated in some open set which doesn't touch .

The most commonly encountered closed sets are the closed interval, a closed path, the closed disk, the region enclosed by a Jordan curve together with the curve itself, and the closed ball. The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points (and is nowhere dense and has Lebesgue measure 0).

It is possible for a set to be neither open nor closed, e.g., the half-closed interval .