The Topology of Higher Dimensions: interior, closure and boundary

We are now going to 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: