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.ko:경계 (위상수학)