Interior, Boundary, and Exterior Points in Euclidean Space

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: 