Boundary

The set of points where every neighborhood meets both the set and its complement.

Boundary

Let $(X,d)$ be a metric space and let $A\subseteq X$ . The boundary of $A$ , denoted $\partial A$ , is

\partial A := \overline{A}\setminus \operatorname{int}(A).

Equivalently,

\partial A = \overline{A}\cap \overline{X\setminus A}

(see closure and interior ). Equivalently again, $x\in\partial A$ iff every open ball $B(x,r)$ meets both $A$ and $X\setminus A$ .

Boundaries isolate the “edge” of a set and play a key role in topology and analysis (e.g., in describing discontinuity sets and in integration theory).

Examples:

In $\mathbb{R}$ , $\partial(0,1)=\{0,1\}$ .
In $\mathbb{R}^2$ , the boundary of the open unit disk $B(0,1)$ is the unit circle $S(0,1)$ .
If $A=\mathbb{Q}\subseteq\mathbb{R}$ , then $\partial A=\mathbb{R}$ (every interval meets both $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ ).