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

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 ${\displaystyle \partial _{X}S,}$ or simply ${\displaystyle \partial S}$ if ${\displaystyle X}$ is understood:

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.

Consider the real line ${\displaystyle \mathbb {R} }$ with the usual topology (that is, the topology whose basis sets are open intervals) and ${\displaystyle \mathbb {Q} ,}$ the subset of rational numbers (whose topological interior in ${\displaystyle \mathbb {R} }$ is empty). Then in ${\displaystyle \mathbb {R} }$ we have

These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. They also show that it is possible for the boundary ${\displaystyle \partial S}$ of a subset ${\displaystyle S}$ to contain a non-empty open subset of ${\displaystyle X:=\mathbb {R} }$; that is, for the interior of ${\displaystyle \partial S}$ in ${\displaystyle X}$ to be non-empty. However, a closed subset's boundary always has an empty interior.

The notation ${\displaystyle \partial _{X}S}$ is used because the boundary of a set ${\displaystyle S}$ crucially depends on the surrounding topological space ${\displaystyle X}$ that's considered. Take for instance the set ${\displaystyle S=\{r\in \mathbb {Q} \mid 0<r<{\sqrt {2}}\}}$. Considered as a subset of ${\displaystyle \mathbb {R} }$, its boundary is the closed interval ${\displaystyle [0,{\sqrt {2}}]}$; considered as a subset of ${\displaystyle \mathbb {Q} }$ (where ${\displaystyle \mathbb {Q} }$ is given its usual topology, the subspace topology inherited from ${\displaystyle \mathbb {R} }$), the boundary of ${\displaystyle S}$ is ${\displaystyle \{0\}}$; and considered as a subset of ${\displaystyle X=S}$ itself, its boundary is empty.