Boundary (topology)

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 {\displaystyle \operatorname {bd} (S),\operatorname {fr} (S),}{\displaystyle \operatorname {bd} (S),\operatorname {fr} (S),} and {\displaystyle \partial S}\partial S. 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.[1] Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.[2]

A connected component of the boundary of S is called a boundary component of S.

A set (in light blue) and its boundary (in dark blue).

Contents
1 Common definitions
2 Properties
3 Examples
3.1 Characterizations and general examples
3.2 Concrete examples
3.3 Boundary of an open ball vs. its surrounding sphere
4 Boundary of a boundary
5 See also