Boundary of a Set¶

Definition¶

For a metric space $(X,d)\text{,}$ and a subset $E \subseteq X\text{,}$ a point $x_0$ is a boundary point of $E$ if it's neither an interior nor an exterior point of $E\text{.}$ We denote the set of boundary points of $E$ as $\partial E\text{.}$

Interpretation¶

The boundary is the dividing line between inside and outside: it consists of the points that $E$ neither surrounds nor avoids. Relative to $E$ , every point of $X$ is of exactly one of three kinds: interior, exterior or boundary.

A shaded region E with a point on its edge and a small ball around it that contains points both inside and outside E, marking it as a boundary point.

Each ball around the point $x_0$ in the figure, however small, contains points of $E$ and points of $X\setminus E$ $E$ , so no ball fits inside $E$ and no ball avoids $E$ . The boundary consists of points that are neither interior nor exterior: $\partial E=\overline{E}\setminus\operatorname{int}(E)$ $int (E)$ , the adherent points that are not interior.

On the unit circle, and at the isolated point $\boldsymbol p$ , every ball meets both $E$ and its complement, however small $\varepsilon$ is.

The condition is symmetric in $E$ and its complement, so $\partial E=\partial(X\setminus E)$ $E)$ . A boundary point may belong to $E$ or not, and this is what separates the kinds of sets: $E$ is closed exactly when it contains its whole boundary, and open exactly when it contains none of it.

Boundary points matter in optimization. A global extremum on a domain can lie at an interior point, where a derivative test applies, or at a boundary point, which has to be checked separately.

A common slip is to picture the boundary as a thin curve around the set. It depends on the set and on the metric: in the plane, an isolated point of $E$ such as $\boldsymbol{p}$ in the interactive example is a boundary point, while with the discrete metric every set has empty boundary, because the ball of radius $\tfrac{1}{2}$ around any point contains only that point.

Key Results¶

A boundary point $x_0$ of $E$ may or may not be in $E\text{.}$
For a boundary point $x_0$ of $E\text{,}$ any balls centred at $x_0$ will contain points both inside and outside $E\text{.}$

Examples¶

The boundary of $(0,1)$ is $\{0,1\}$ . Every neighbourhood around $0$ contains points in $(0,1)$ and points not in $(0,1)$ .
The boundary of $\mathbb{Q}$ in $\mathbb{R}$ is all of $\mathbb{R}$ .

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 5

Aliases: boundary point, frontier, topological boundary Symbols: epsilon, partial, rational numbers, real numbers, subset

Last updated: 20 October 2026 03:02