Open and Closed Balls in Metric Spaces

# Open and Closed Balls in Metric Spaces

Recall from the Open and Closed Balls in Euclidean Space page that if $\mathbf{a} = (a_1, a_2, ..., a_n) \in \mathbb{R}^n$ then the open ball centered at $\mathbf{a}$ with radius $r > 0$ is as the set of points $\mathbf{x} = (x_1, x_2, ..., x_n) \in \mathbb{R}^n$ contained in:

(1)
\begin{align} \quad B(\mathbf{a}, r) = \left \{ \mathbf{x} \in \mathbb{R}^n : \| \mathbf{x} - \mathbf{a} \| < r \right \} = \left \{ \mathbf{x} \in \mathbb{R}^n : \sqrt{(x_1 - a_1)^2 + (x_2 - a_2)^2 + ... + (x_n - a_n)^2} \leq r \right \} \end{align}

Similarly, the closed ball centered at $\mathbf{a}$ with radius $r \geq 0$ is the set of points $\mathbf{x} \in \mathbb{R}^n$ contained in:

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

Now notice that $\| \mathbf{x} - \mathbf{a} \|$ is simply the Euclidean distance function $d : \mathbb{R}^n \to [0, \infty)$ for the metric space $(\mathbb{R}^n, d)$. We can extend the concept of open and closed balls to any metric space with its own defined metric as defined below.

 Definition: If $(M, d)$ is a metric space, $a \in M$, and $r > 0$ then the Open Ball centered at $a$ with radius $r$ is defined to be the set $B(a, r) = \{ x \in M : d(x, a) < r \}$.

For example, consider the metric space $(\mathbb{R}^2, d)$ where the metric $d : \mathbb{R}^2 \times \mathbb{R}^2 \to [0, \infty)$ for all $\mathbf{x} = (x_1, x_2), \mathbf{y} = (y_1, y_2) \in \mathbb{R}^2$ by:

(3)
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \mid x_1 - y_1 \mid + \mid x_2 - y_2 \mid \end{align}

Then for each $\mathbf{a} = (a_1, a_2) \in \mathbb{R}^2$, we can define the open ball centered at $\mathbf{a}$ and with radius $r > 0$ to be the set of all points $\mathbf{x} \in \mathbb{R}^2$ contained in:

(4)
\begin{align} \quad B(\mathbf{a}, r) = \{ \mathbf{x} \in \mathbb{R}^2 : d(\mathbf{x}, \mathbf{a}) < r \} = \{\mathbf{x} \in \mathbb{R}^2 : \mid x_1 - a_1 \mid + \mid x_2 - a_2 \mid < r \} \end{align}

We can likewise define the closed ball centered at $\mathbf{a} \in M$ for some metric space $(M, d)$ as follows.

 Definition: If $(M, d)$ is a metric space, $a \in M$, and $r > 0$ then the Closed Ball centered at $a$ with radius $r$ is defined to be the set $\bar{B}(a, r) = \{ x \in M : d(x, a) \leq r \}$.

Note the subtle but important difference between the definitions of open and closed balls centered at $a$ with radius $r$.

From the example above, we can define the closed ball centered at $\mathbf{a}$ and with radius $r > 0$ to be the set of all points $\mathbf{x} \in \mathbb{R}^2$ contained in:

(5)
\begin{align} \quad \bar{B}(\mathbf{a}, r) = \{ \mathbf{x} \in \mathbb{R}^2 : d(\mathbf{x}, \mathbf{a}) \leq r \} = \{\mathbf{x} \in \mathbb{R}^2 : \mid x_1 - a_1 \mid + \mid x_2 - a_2 \mid \leq r \} \end{align}