Definition
Boundary
The boundary
of a set is the set of all points that belong to the closure of but not to its interior. Formally, a point
lies in the boundary if and only if every neighbourhood of contains at least one point in and at least one point not in : where
denotes the complement of . Consequently, .
Intuition
Imagine a circular disk in the plane that contains all points inside a circle as well as the points on the circle’s circumference.
- Interior Points: For each of these points, one can draw an arbitrarily small circle around it that still lies completely within the original disk.
- Boundary Points: No matter how small a circle is drawn around such a point, it will always contain both points that belong to the disk and points that lie in the complement
.
The set of all these boundary points forms the boundary of the circular disk.