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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 ${\displaystyle \operatorname {bd} (S),\operatorname {fr} (S),}$ and ${\displaystyle \partial S}$.

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]

Definitions

There are several equivalent definitions for the boundary of a subset ${\displaystyle S\subseteq X}$ of a topological space ${\displaystyle X,}$ which will be denoted by ${\displaystyle \partial _{X}S,}$ ${\displaystyle \operatorname {Bd} _{X}S,}$ or simply ${\displaystyle \partial S}$ if ${\displaystyle X}$ is understood:

1. It is the closure of ${\displaystyle S}$ minus the interior of ${\displaystyle S}$ in ${\displaystyle X}$:
   $${\displaystyle \partial S~:=~{\overline {S}}\setminus \operatorname {int} _{X}S}$$

   
   where ${\displaystyle {\overline {S}}=\operatorname {cl} _{X}S}$ denotes the closure of ${\displaystyle S}$ in ${\displaystyle X}$ and ${\displaystyle \operatorname {int} _{X}S}$ denotes the topological interior of ${\displaystyle S}$ in ${\displaystyle X.}$
2. It is the intersection of the closure of ${\displaystyle S}$ with the closure of its complement:
   $${\displaystyle \partial S~:=~{\overline {S}}\cap {\overline {(X\setminus S)}}}$$

   
3. It is the set of points ${\displaystyle p\in X}$ such that every neighborhood of ${\displaystyle p}$ contains at least one point of ${\displaystyle S}$ and at least one point not of ${\displaystyle S}$:
   $${\displaystyle \partial S~:=~\{p\in X:{\text{ for every neighborhood }}O{\text{ of }}p,\ O\cap S\neq \varnothing \,{\text{ and }}\,O\cap (X\setminus S)\neq \varnothing \}.}$$

   

A boundary point of a set is any element of that set's boundary. The boundary ${\displaystyle \partial _{X}S}$ defined above is sometimes called the set's topological boundary 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.

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

Properties

The closure of a set ${\displaystyle S}$ equals the union of the set with its boundary:

where ${\displaystyle {\overline {S}}=\operatorname {cl} _{X}S}$ denotes the closure of ${\displaystyle S}$ in ${\displaystyle X.}$ A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary. The boundary of a set is closed;^[3] this follows from the formula ${\displaystyle \partial _{X}S~:=~{\overline {S}}\cap {\overline {(X\setminus S)}},}$ which expresses ${\displaystyle \partial _{X}S}$ as the intersection of two closed subsets of ${\displaystyle X.}$

("Trichotomy") Given any subset ${\displaystyle S\subseteq X,}$ each point of ${\displaystyle X}$ lies in exactly one of the three sets ${\displaystyle \operatorname {int} _{X}S,\partial _{X}S,}$ and ${\displaystyle \operatorname {int} _{X}(X\setminus S).}$ Said differently,

and these three sets are pairwise disjoint. Consequently, if these set are not empty^{[note 1]} then they form a partition of ${\displaystyle X.}$

A point ${\displaystyle p\in X}$ is a boundary point of a set if and only if every neighborhood of ${\displaystyle p}$ contains at least one point in the set and at least one point not in the set. The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set.



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