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) or <math>\partial S<math>.

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

Define the boundary of S to be the intersection of the closure of S with the closure of its complement.
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.
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).

Examples

If <math>X=[0,5)<math>, then <math>\partial X = \{0,5\}.<math>
<math>\partial \overline{B}(\mathbf{a}, r) = \overline{B}(\mathbf{a}, r) – B(\mathbf{a}, r)<math>
<math>\partial D^n \simeq S^{n-1}<math>
<math>\partial \emptyset = \emptyset<math>
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.

Categories: General topology