Boundary (topology)

:For a different notion of boundary related to manifolds, see that article.

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In topology, the boundary of a subset S of a topological space X is the set's closure minus its interior. An element of the boundary of S is called a boundary point of S. Notations used for boundary of a set S include bd(S), fr(S), and partial S.

Related Topics:
Topology - Topological space - Closure - Interior

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There are two other common (and equivalent) approaches to defining the boundary of S and the boundary points of S.

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• Define the boundary of S to be the intersection of the closure of S with the closure of its complement.
• Define p in X to be a boundary point of S if every neighborhood of p contains at least one point of S and at least one point not in S. Then define the boundary of S to be the set of all boundary points of S.
 
Boundary (topology): Properties ►