Interior, Boundary, and Exterior Points in Euclidean Space

Before we look much further into Euclidean space, we will need discuss some important classifications of points regarding a subset $S$ of $\mathbb{R}^n$ which we define below.

Definition: Let $S \subseteq \mathbb{R}^n$. A point $\mathbf{a} \in \mathbb{R}^n$ is said to be an Interior Point of $S$ if there exists an $r > 0$ such that $B(\mathbf{a}, r) \subseteq S$, i.e., there exists an open ball centered at $\mathbf{a}$ for some positive radius $r$ that is a subset of $S$. The set of all interior points of $S$ is denoted by $\mathrm{int} (S)$.

In the case where $n = 2$ and we have some subset $S \subseteq \mathbb{R}^2$ (like the one illustrated below), then we say that a point $\mathbf{a} \in \mathbb{R}^2$ is an interior point if there exists an open disk of some positive radius $r > 0$ that is entirely contained in $S$.

In the illustration above, we see that the point on the boundary of this subset is not an interior point. If we take a disk centered at this point of ANY positive radius then there will exist points in this disk that are always not contained within the pink region.

Definition: Let $S \subseteq \mathbb{R}^n$. A point $\mathbf{a} \in \mathbb{R}^n$ is said to be a Boundary Point of $S$ if for every for every $B(\mathbf{a}, r)$ with $r > 0$ there exists $\mathbf{x}, \mathbf{y} \in B(\mathbf{a}, r)$ such that $\mathbf{x} \in S$ and $\mathbf{y} \in S^c$, i.e., in every ball centered at $\mathbf{a}$ there exists a point contained in $S$ and a point contained in the complement $S^c$. The set of all boundary points of $S$ is denoted $\mathrm{bdry} (S)$.

For $n = 1$, $\mathrm{bdry} (S)$ comprises the endpoints of $S$. For $n = 2$, $\mathrm{bdry} (S)$ comprises the border of $S$ as illustrated below:

For $n = 3$, $\mathrm{bdry} (S)$ comprises the surface of $S$.

Definition: Let $S \subseteq \mathbb{R}^n$. A point $\mathbf{a} \in \mathbb{R}^n$ is said to be an Exterior Point of $S$ if $\mathbf{a} \in S^c \setminus \mathrm{bdry} (S)$. The set of all exterior points of $S$ is denoted $\mathrm{ext} (S)$.

For $n = 2$, a visualization of some exterior points of a set of points (in green) is illustrated below: