Definition:Separable Space

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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 dense in $X$.

Let $Y$ be a countable set:

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

Suppose:

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


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