Boundary (Analysis)

analysis topology

Definition

Boundary

The boundary $\partial D$ of a set $D$ is the set of all points that belong to the closure of $D$ but not to its interior.

$$\partial D=\bar{D}\setminus \mathring{D}$$

Formally, a point $x$ lies in the boundary $\partial D$ if and only if every neighbourhood of $x$ contains at least one point in $D$ and at least one point not in $D$:

$$x\in \partial D⟺\forall \epsilon >0:(U\ast \epsilon (x)\cap D\ne \varnothing )\wedge (U\ast \epsilon (x)\cap D^c\ne \varnothing )$$

where $D^c$ denotes the complement of $D$. Consequently,$\partial D=\bar{D}\cap \overline{D^c}$.

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 $D^c$.

The set of all these boundary points forms the boundary of the circular disk.