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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (syntax error): {\displaystyle r > 0} that is entirely contained in $S$.

<img src="http://mathonline.wdfiles.com/local--files/interior-boundary-and-exterior-points-in-euclidean-space/Screen%20Shot%202015-09-13%20at%205.43.29%20PM.png" alt="Screen%20Shot%202015-09-13%20at%205.43.29%20PM.png" class="image" />

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 Failed to parse (syntax error): {\displaystyle r &gt; 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:

<img src="http://mathonline.wdfiles.com/local--files/interior-boundary-and-exterior-points-in-euclidean-space/Screen%20Shot%202015-09-13%20at%206.02.25%20PM.png" alt="Screen%20Shot%202015-09-13%20at%206.02.25%20PM.png" class="image" />

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:

<img src="http://mathonline.wdfiles.com/local--files/interior-boundary-and-exterior-points-in-euclidean-space/Screen%20Shot%202015-09-13%20at%206.02.07%20PM.png" alt="Screen%20Shot%202015-09-13%20at%206.02.07%20PM.png" class="image" />