In topology and mathematicsin general, the boundary of a subset S of a topological spaceX is the set of points which can be approached both from S and from the outside of S. More precisely, it 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 bd, fr, and ∂S. Some authors use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory ofmanifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, the term frontier has been used to describe the residue of S, namely S \ S. Felix Hausdorff named the intersection of S with its boundary the border of S. A connected component of the boundary of S is called a boundary component of S.
Common definitions
There are several equivalent definitions for the boundary of a subset S of a topological space X:
the closure of ' minus the interior of :
the intersection of the closure of ' with the closure of its complement:
the set of points such that every neighborhood of contains at least one point of ' and at least one point not of ':
These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. In the space of rational numbers with the usual topology, the boundary of, where a is irrational, is empty. The boundary of a set is a topological notion and may change if one changes the topology. For example, given the usual topology on R2, the boundary of a closed disk Ω = is the disk's surrounding circle: ∂Ω =. If the disk is viewed as a set in R3 with its own usual topology, i.e. Ω =, then the boundary of the disk is the disk itself: ∂Ω = Ω. If the disk is viewed as its own topological space, then the boundary of the disk is empty.
p is a boundary point of a set if and only if every neighborhood of p contains at least one point in the set and at least one point not in the set.
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 closure of a set equals the union of the set with its boundary. S = S ∪ ∂S.
The boundary of a set is empty if and only if the set is both closed and open.
The interior of the boundary of the closure of a set is the empty set.
Boundary of a boundary
For any set S, ∂S⊇ ∂∂S, with equality holding if and only if the boundary of S has no interior points, which will be the case for example if S is either closed or open. Since the boundary of a set is closed, ∂∂S = ∂∂∂S for any set S. The boundary operator thus satisfies a weakened kind of idempotence. In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the singular homology rests critically on this fact. The explanation for the apparent incongruity is that the topological boundary is a slightly different concept from the boundary of a manifold or of a simplicial complex. For example, the boundary of an open disk viewed as a manifold is empty, while its boundary in the sense of topological space is the circle surrounding the disk.
