Exterior of a Set¶

Definition¶

For a metric space $(X,d)\text{,}$ and a subset $E \subseteq X\text{,}$ a point $x_0$ is an exterior point of $E$ if $B(x_0,\varepsilon) \cap E = \varnothing$ for some $\varepsilon > 0\text{.}$ We denote the set of exterior points of $E$ as $\operatorname{ext} E\text{.}$

The Open Metric Ball is a neighbourhood that avoids $E$; the empty Intersection of Sets expresses that no point lies in both the neighbourhood and $E$.

Interpretation¶

An exterior point of $E$ is a point that is not only outside $E$ but keeps some distance from it: a small open ball around it misses $E$ completely. The exterior collects the points that are clearly outside $E$, as opposed to outside points that $E$ still touches.

In the figure the shaded region is $\operatorname{int}E$, the red curve is $\partial E$, and the rest of the rectangle $X$ is $\operatorname{ext}E$. The marked point $x_0$ has a ball that does not meet $E$, exactly as the definition requires. The radius may be as small as needed, so a point close to the red curve but not on it is still exterior, only with a smaller ball.

The condition $B(x_0,\varepsilon)\cap E=\varnothing$ says the same as $B(x_0,\varepsilon)\subseteq X\setminus E$$E$, so the exterior points of $E$ are the interior points of its complement: $\operatorname{ext}E=\operatorname{int}(X\setminus E)$$E)$. Every point of $X$ is of exactly one of three kinds, so $X$ is the disjoint union $\operatorname{int}E\cup\partial E\cup\operatorname{ext}E$$\mathrm{ext}E$. The points that are not exterior are precisely the adherent points of $E$, which is why the closure is $\overline{E}=X\setminus\operatorname{ext}E$$\mathrm{ext}E$.

A common slip is to take every point outside $E$ as exterior. For $E=(0,1)$ in $\mathbb{R}$, the point $0$ is not in $E$, but every interval $(-\varepsilon,\varepsilon)$ contains points of $E$, so $0$ is a boundary point. For $E=\mathbb{Q}$ in $\mathbb{R}$ there are no exterior points at all, since every interval contains rational numbers.

Key Results¶

• An exterior point $x_0$ of $E$ can never be in $E\text{.}$

Examples¶

• For $E = (0,1) \subseteq \mathbb{R}$$\mathbb{R}$, the point $x = 2$ is an exterior point. A small ball around $2$, say $(1.9,2.1)$, avoids $(0,1)$.
• For $E = \{(x,y)\in\mathbb{R}^2\colon x^2 + y^2 \leq 1\}$$1\}$, the point $(3,0)$ is an exterior point. A small disc around it stays entirely outside the unit disc.

Prerequisites¶

• Open Metric Ball
• Intersection of Sets

Related Concepts¶

• Interior of a Set

Sources¶

• Math III: Analysis · Lecture 5

Aliases: exterior point Symbols: element of, epsilon, intersection, partial, rational numbers, real numbers, subset, union

Last updated: 20 October 2026 03:02