What is the geometric interpretation of a separable space?

Question

What is the geometric interpretation of a separable space?

I know the definition of a separable space and I can give some examples.

Answer 1

A separable space possesses a countable, dense subset, so it is intuitively a restriction on the size of the space. Any finite or countable set is clearly a countable, dense subset of itself, and any finite dimensional vector space over the real or complex numbers possesses a basis which, when combined with the rationals (a countable, dense subset), makes it separable. And finite/countable sets are 'small' relatively speaking.

'Big' spaces are things like $\ell_\infty({\mathbb R})$, the set of all bounded sequences of real numbers (equipped with the supremum norm) and $C_b({\mathbb R})$, the space of all real functions of bounded variation, and neither of these is separable. If we take $C_b({\mathbb R})$ and equip it with the BV-norm then the family of functions $g_r(x):=\sin(r\cdot x\pi)$ where $r\in {\mathbb R}$ are such that $\|g_{r_1}(x) - g_{r_2}(x)\|_{BV} = 2$ for all $r_1 \not= r_2$. This tell us that there is an entire subspace the size of the continuum which is discrete sitting inside $C_b({\mathbb R})$, and so it can never be separable.

Separability thus tells you that the space is 'small enough' that all the elements of it are 'close enough together', and while I don't know if that was ever the intent of using the word dense in the definition it is what it should suggest to you: the elements of the space are tightly packed.

As a side-note: size is quite a tricky concept when you get to infinite-dimensions. Separability gives you a notion of it, as does Baire Category and do measure-zero sets but they generally disagree in lots of place over whether something is 'big' or 'small'.