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Concept
Boundary (topology)
Summary
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include and . Some authors (for example Willard, in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.
A connected component of the boundary of S is called a boundary component of S.
There are several equivalent definitions for the of a subset of a topological space which will be denoted by or simply if is understood:
It is the closure of minus the interior of in :
where denotes the closure of in and denotes the topological interior of in
It is the intersection of the closure of with the closure of its complement:
It is the set of points such that every neighborhood of contains at least one point of and at least one point not of :
A of a set refers to any element of that set's boundary. The boundary defined above is sometimes called the set's to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few examples.
The closure of a set equals the union of the set with its boundary:
where denotes the closure of in
A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary.
Official source
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