Interior of a Set¶

Definition¶

For a metric space $(X,d)\text{,}$ and a subset $E \subseteq X\text{,}$ a point $x_0$ is an interior point of $E$ if there exists some $\varepsilon > 0$ such that $B(x_0,\varepsilon) \subseteq E\text{.}$ We denote the set of interior points of $E$ as $\operatorname{int} E\text{.}$

Interpretation¶

An interior point of $E$ is a point that lies in $E$ with room to spare: some ball around it, however small, stays inside $E$ . From an interior point one can move a little in every direction without leaving the set.

A shaded region E with a point x0 inside it and a small ball fully contained in E, so x0 is interior, and a point on the edge whose every ball leaves E.

The green ball lies entirely inside $E$ , so its centre $x_0$ is an interior point. Every ball around the red edge point leaves $E$ , however small its radius, so that point is not interior but a boundary point. This does not depend on whether the edge point itself belongs to $E$ .

$E$ is the open unit disc together with one extra point $p$ . Drag $x_0$ and shrink $\varepsilon$ : an interior point has some ball inside $E$ .

The radius $\varepsilon$ may depend on the point. In $E=(0,1)$ the point $0.01$ is interior, but only balls of radius at most $0.01$ fit around it; what matters is that some positive radius works. Collecting all interior points gives the largest open subset of $E$ , and $E$ is open if and only if $E=\operatorname{int}E$ $int E$ . Interior points of the complement are the exterior points of $E$ .

A common slip is to forget the ambient space. As a subset of $\mathbb{R}$ , $E=[0,1]$ has interior $(0,1)$ . As a subset of the metric space $X=[0,1]$ with $d(x,y)=|x-y|$ $y ∣$ , every point is interior, because there $B(0,\tfrac{1}{2})=[0,\tfrac{1}{2})\subseteq E$ $E$ .

This idea returns later: derivatives of functions of several variables, such as the gradient, are taken at interior points of the domain, where the function is defined in every direction around the point.

Key Results¶

An interior point $x_0$ of $E$ is always in $E$ since balls centred at $x_0$ include $x_0\text{.}$
The interior of $E$ is the largest open subset of $E$ .

Examples¶

For $E = (0,1) \subseteq \mathbb{R}$ $R$ , every point in $(0,1)$ is interior. Pick a small enough interval around any point and it stays inside $(0,1)$ .
For $E = \{1,2,3\} \subseteq \mathbb{R}$ $R$ , no point is interior. Any ball around $1$ contains numbers other than $1,2,3$ .

Prerequisites¶

Open Metric Ball

Exterior of a Set

Sources¶

Math III: Analysis · Lecture 5

Aliases: interior point Symbols: epsilon, real numbers, subset

Last updated: 20 October 2026 03:02