Interior and Boundary Points of a Set in a Metric Space

Recall from the Interior, Boundary, and Exterior Points in Euclidean Space that if $S \subseteq \mathbb{R}^n$ then a point $\mathbf{a} \in S$ is called an interior point of $S$ if there exists a positive real number $r > 0$ such that the ball centered at $a$ with radius $r$ is a subset of $S$.

Furthermore, a point $\mathbf{a}$ is called a boundary point of $S$ if for every positive real number $r > 0$ we have that there exists points $\mathbf{x}, \mathbf{y} \in B(\mathbf{a}, r)$ such that $\mathbf{x} \in S$ and $\mathbf{y} \in S^c$.

We will now generalize these definitions to metric spaces $(M, d)$.

Definition: Let $(M, d)$ be a metric space and let $S \subseteq M$. A point $a \in S$ is said to be an Interior Point of $S$ if there exists a positive real number $r > 0$ such that the ball centered at $a$ with radius $r$ with respect to the metric $d$ is a subset of $S$, i.e., $B(a, r) \subseteq S$. The set of all interior points of $S$ is called the Interior of $S$ and is denoted $\mathrm{int} (S)$

In shorter terms, a point $a \in S$ is an interior point of $S$ if there exists a ball centered at $a$ that is fully contained in $S$. Note that from the definition above we have that a point can be an interior point of a set only if that point is contained in $S$. Therefore $\mathrm{int} (A) \subseteq A$.

Definition: Let $(M, d)$ be a metric space and let $S \subseteq M$. A point $a \in M$ is said to be a Boundary Point of $S$ if for every positive real number $r > 0$ we have that there exists points $x, y \in B(a, r)$ such that $x \in S$ and $y \in S^c$. The set of all boundary points is called the Boundary of $S$ and is denoted $\partial S$ or $\mathrm{bdry} (S)$.

A point $a \in M$ is said to be a boundary point of $S$ if every ball centered at $a$ contains points in $S$ and points in the complement $S^c$. Notice that from the definition above that a boundary point of a set need not be contained in that set.