Separable Metric Spaces

# Separable Metric Spaces

Recall from the Dense Sets in a Metric Space page that if $(M, d)$ is a metric space then a subset $S \subseteq M$ is said to be dense in $M$ if for every $x \in M$ and for all $r > 0$ we have that:

(1)
\begin{align} \quad B(x, r) \cap S \neq \emptyset \end{align}

In other words, $S$ is dense in $M$ if every open ball contains a point of $S$.

We will now look at a special type of metric space known as a separable metric space which we define below.

 Definition: A metric space $(M, d)$ is said to be Separable if there exists a countable dense subset $S$ of $M$.

For example, consider the metric space $(\mathbb{R}, d)$ where $d$ is the usual Euclidean metric defined for all $x, y \in \mathbb{R}$ by $d(x, y) = \mid x - y \mid$. Then the subset $\mathbb{Q}$ is dense in $\mathbb{R}$ since every open interval contains rational numbers.

In fact, in general, the metric space $(\mathbb{R}^n, d)$ where $d$ is the usual Euclidean metric defined for all $\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ by:

(2)
\begin{align} \quad \| \mathbf{x} - \mathbf{y} \| = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + ... + (x_n - y_n)^2} \end{align}

Then it can be shown similarly that the following set is dense in $\mathbb{R}^n$:

(3)
\begin{align} \quad \mathbb{Q}^n = \{(x_1, x_2, ..., x_n) \in \mathbb{R}^n : x_1, x_2, ..., x_n \in \mathbb{Q} \} \end{align}