# Open and Closed Balls in Euclidean Space

Definition: Let $\mathbf{a} \in \mathbb{R}^n$. The Open Ball Centered at $\mathbf{a}$ with Radius $r > 0$ denoted $B(\mathbf{a}, r)$ is defined to be the set of points $\mathbf{x} \in \mathbb{R}^n$ such that $\| \mathbf{x} - \mathbf{a} \| < r$, that is, $B(\mathbf{a}, r) = \{ \mathbf{x} \in \mathbb{R}^n : \| \mathbf{x} - \mathbf{a} \| < r \}$. |

If $n = 1$ then for each $a, r \in \mathbb{R}$ where $r > 0$ we have that the ball centered at $a$ with radius $r$ is:

(1)Therefore, $B(a, r)$ is the open interval centered at $a$ with radius $r$.

If $n = 2$ then for each $\mathbf{a} = (a_1, a_2) \in \mathbb{R}^2$ and $r \in \mathbb{R}$ where $r > 0$ we have that the ball centered at $a$ with radius $r$ is:

(2)Therefore, $B(\mathbf{a}, r)$ is the open disk (the interior of a circle not containing points on its perimeter) in the plane centered at $\mathbf{a} = (a_1, a_2)$ with radius $r$.

If $n = 3$ then for each $\mathbf{a} = (a_1, a_2, a_3) \in \mathbb{R}^3$ and $r \in \mathbb{R}$ where $r > 0$ we have that the ball centered at $a$ with radius $r$ is:

(3)Therefore, $B(\mathbf{a}, r)$ is the open ball (The interior of a sphere not containing points on its surface) in the plane centered at $\mathbf{a} = (a_1, a_2, a_3)$ with radius $r$.

As you can see, for the cases when $n = 1, 2, 3$ the name "open ball" makes intuitive sense. Of course, since we can't visualize $\mathbb{R}^n$ when $n \geq 4$ we define open balls in higher dimensions analogously. We can also define closed balls in $\mathbb{R}^n$ too.

Definition: Let $\mathbf{a} \in \mathbb{R}^n$. The Closed Ball Centered at $\mathbf{a}$ with Radius $r > 0$ denoted $\bar{B}(\mathbf{a}, r)$ is defined to be the set of points $\mathbf{x} \in \mathbb{R}^n$ such that $\| \mathbf{x} - \mathbf{a} \| \leq r$, that is, $\bar{B}(\mathbf{a}, r) = \{ \mathbf{x} \in \mathbb{R}^n : \| \mathbf{x} - \mathbf{a} \| \leq r \}$. |

*Notice the subtle difference between the definition of an open ball and the definition of a closed ball.*

The only difference between an open ball in $\mathbb{R}^n$ and a closed ball in $\mathbb{R}^n$ is that closed balls contain all "exterior" points while open balls do not. I.e., a closed ball in $\mathbb{R}^3$ centered at $\mathbf{a}$ with radius $r$ contains all points on the surface of and in the sphere centered at $\mathbf{a}$ with radius $r$.