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

(3)