Boundary (topology) - Properties

Properties

• The boundary of a set is closed.
• The boundary of a set is the boundary of the complement of the set: ∂S = ∂(SC).

Hence:

• 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 (that is, a clopen set).
• In Rn, every closed set is the boundary of some set.

Conceptual Venn diagram showing the relationships among different points of a subset S of Rn. A = set of limit points of S, B = set of boundary points of S, area shaded green = set of interior points of S, area shaded yellow = set of isolated points of S, areas shaded black = empty sets. Every point of S is either an interior point or a boundary point. Also, every point of S is either an accumulation point or an isolated point. Likewise, every boundary point of S is either an accumulation point or an isolated point. Isolated points are always boundary points.