Boundary (Analysis)

analysis

Definition

Rand

Boundary

The boundary $\partial D$ of a set $D$ is defined as the set of all point that lie in the closure but not in the interior of $D$.

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

Therefore, a point $x$ lies the the boundary $\partial D$ of set $D$ if and only if it is an adherent point but not an interior point:

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

where $D^c$ denotes the complement of $D$ and $U_{\epsilon }(x)$ denotes the epsilon neighbourhood of $x$. Thus:

$$\partial D=\bar{D}\cap \overline{D^c}$$

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.

• The points lying strictly inside are the interior points. For each of these points, we can draw an arbitrarily small circle around it that still lies completely with the original disk.
• The points on the circumference are the boundary points*. No matter how small a circle we draw around such a point, it will always contain both points that belong to the disk and points that lie outside of it.

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