Closure (topology)

Closure

A closure C(S) of a set S is the 'smallest' closed set that contains S. This smallest set C(S) can be constructed by intersecting all closed sets which contain S.

A topology is defined by defining what a closed set is and what an open set is. See the definition of topology for details.

In a metric space such as the n-dimensional Euclidean space, the closure of a set S is the set of all limits of all converging sequences of points in S.

Border

The difference C(S)/S is often called the border of S. E.g. the border of a square S in the Euclidean plane are the four sides of S.