Closed set

In topology, a branch of mathematics, a closed set is a set that contains all of its boundary points. An example is the closed interval ${\displaystyle [a,b]}$, which is closed in the real line because it includes both points ${\displaystyle a}$ and ${\displaystyle b}$ of its boundary. A point is on the boundary if every neighbourhood of it meets both the set and its complement. A set is thus closed if it is equal to its closure, the set obtained by adjoining all boundary points to it.

Closed sets are defined as subsets of topological spaces. The topology of a space is usually described in terms of its open sets, which determine what counts as a "neighborhood" of its points. A set is closed if it is the complement of an open set. In metric spaces, a set is closed if and only if the limit of every convergent sequence of elements in the set has limit in this set; thus a closed set is a set that includes all of its limit points. Because the limits of convergent sequences do not escape a closed set, they are important in many areas of mathematics where limiting arguments are used.

A closed set is distinct from the notion of a closed curve or closed manifold. Those are also closed in the topological sense, but the term there usually means that the boundary in the relevant sense is empty. Likewise a closed differential form is not a set at all, but a form that has zero coboundary. Closed is thus used in a different, but related, sense in homology and cohomology.