Adherent Points of Subsets in Euclidean Space

Definition: Let $S \subseteq \mathbb{R}^n$. An Adherent Point to $S$ is a point $\mathbb{x} \in \mathbb{R}^n$ such that there exists a point $\mathbf{s} \in S$ such that $\mathbf{s} \in B(\mathbf{x}, r)$ for all positive real numbers $r > 0$. The set of all adherent points to $S$ is called the Closure of $S$ and is denoted $\bar{S}$.

Alternatively, an adherent point $\mathbf{x} \in \mathbb{R}^n$ to $S$ is a point such that every ball centered at $\mathbf{x}$ contains at least one element from $S$.

The simplest examples of adherent points are points $\mathbf{x} \in S$ because every ball centered at $\mathbf{x}$ contains $\mathbf{x} \in S$, so $\mathbf{x}$ is an adherent point of $S$.

For another example, consider the subset $S \subseteq \mathbb{R}^2$ as the open disk of radius $1$ centered at the origin $\mathbf{0} = (0, 0)$:

We claim that any point on the unit circle is an adherent point. Geometrically, this is intuitively obvious as all disks centered at a point $\mathbf{x} \in \mathbb{R}^2$ on the unit circle will intersect the disk $S$ (or contain all of $S$) and hence there exists a $\mathbf{s} \in S$ such that $\mathbf{s} \in B(\mathbf{x}, r)$ for every positive real number $r > 0$.