Open and Closed Balls in Euclidean Space

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)
\begin{align} \quad B(a, r) = \{ x \in \mathbb{R} : \| x - a \| < r \} = \{ x \in \mathbb{R} : \mid x - a \mid < r \} = \{ x \in \mathbb{R} : a - r < x < a + r \} = (a - r, a + r) \end{align}

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)
\begin{align} \quad B(\mathbf{a}, r) = \{ \mathbf{x} \in \mathbb{R}^2 : \| \mathbf{x} - \mathbf{a} \| < r \} = \left \{ \mathbf{x} \in \mathbb{R}^2 : \sqrt{(x_1 - a_1)^2 + (x_2 - a_2)^2} < r \right \} \end{align}

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)
\begin{align} \quad B(\mathbf{a}, r) = \{ \mathbf{x} \in \mathbb{R}^3 : \| \mathbf{x} - \mathbf{a} \| < r \} = \left \{ \mathbf{x} \in \mathbb{R}^3 : \sqrt{(x_1 - a_1)^2 + (x_2 - a_2)^2 + (x_3 - a_3)^3} < r \right \} \end{align}

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 $r$.