# Definition:Separable Space

## Definition

A topological space $T = \struct {S, \tau}$ is **separable** if and only if there exists a countable subset of $S$ which is everywhere dense in $T$.

### Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $Y \subseteq X$ be a subset of $X$.

Let $Y$ be (everywhere) dense in $X$.

Let $Y$ be a countable set:

- $Y = \set {y_1, y_2, \ldots}$

Suppose:

- $\forall x \in X : \forall \epsilon \in \R : \epsilon > 0 : \exists y_{n \mathop \in \N} \in Y : \norm {y_n - x} < \epsilon$

Then $X$ is **separable**.

## Also see

- Results about
**separable spaces**can be found here.

## Linguistic Note

The thinking behind applying the word **separable** to the concept of a **separable space** arises from the idea of denseness in the context of the real number line.

For example, a subset $S \subseteq \R$ is dense if and only if any two distinct real numbers $a, b \in \R$ such that $a < b$ can be **separated** by an element of $S$ in the sense that there exists $s \in S$ such that $a < s < b$.

The term seems to have been coined by Maurice René Fréchet in $1906$.

Modern approaches sometimes call into question the usefulness of the term **separable** in this context, as it is not immediately intuitively obvious.

However, the name has stuck, and we now have to live with it.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Countability Properties - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Countability Axioms and Separability