Boundary (topology)

For a different notion of boundary related to manifolds, see that article.

In topology, the boundary of a subset S of a topological space X is the set's closure minus its interior. An element of the boundary of S is called a boundary point of S. Notations used for boundary of a set S include bd(S), fr(S), and $\partial S$.

There are two other common (and equivalent) approaches to defining the boundary of S and the boundary points of S.

1. Define the boundary of S to be the intersection of the closure of S with the closure of its complement.
2. Define p in X to be a boundary point of S if every neighborhood of p contains at least one point of S and at least one point not in S. Then define the boundary of S to be the set of all boundary points of S.

Properties

• The boundary of a set is closed.
• The boundary of a boundary of a set, being closed, contains the boundary of the set, but need not be empty, unlike the case of manifolds with boundary, for which the boundary of a boundary is always empty.
• p is a boundary point of a set iff every neighborhood of p contains at least one point in the set and at least one point not in the set.
• The boundary of a set equals the intersection of that set's closure with the closure of its complement.
• A set is closed iff the boundary of the set is in the set, and open iff it is disjoint from its boundary.
• The boundary of a set equals the boundary of its complement.
• The closure of a set equals the union of the set with its boundary.
• The boundary of a set is empty iff the set is both closed and open (i.e. a clopen set).
• In $\mathbb {R} ^{n}$, every closed set is the boundary of a set.

Examples

• If $X=[0,5)$, then $\partial X=\{0,5\}.$
• $\partial {\overline {B}}(\mathbf {a} ,r)={\overline {B}}(\mathbf {a} ,r)-B(\mathbf {a} ,r)$
• $\partial D^{n}\simeq S^{n-1}$
• $\partial \emptyset =\emptyset$
• In R³, if Ω=x²+y² ≤ 1, ∂Ω = Ω, but in R², ∂Ω = {(x, y) | x²+y² = 1}. So, the boundary of a set can depend on what set it lies in.