Boundary (topology)

topology, the boundary of a subset S of a topological space X is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement.

We define a boundary point of S as a point P of X such that every neighborhood N of that point contains at least one point of S and at least one point not in S. Then an equivalent definition is that the set of all boundary points forms the boundary of S.

The boundary of a set S is denoted by bd S, or <math>\partial S<math>.

A set is closed if the boundary of the set is in the set, and open if it disjoint from its boundary.

Examples

• bd [0, 5] = { 0, 5 }
• <math>\partial \overline{B}(\mathbf{a}, r) = \overline{B}(\mathbf{a}, r) - B(\mathbf{a}, r)<math>.
• The boundary of the empty set is empty.
• 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.

For the different usage applied to manifolds, see boundary.