Intuition for separable spaces?

What should I think of when I read that a space is separable? Is it a 'nice' property? Why would I prefer a separable space over a non-separable space or vice versa?

When I think of compact spaces I think of spaces that are in some sense finite. Is there a similar analogy for separable spaces?

Solution 1:

A dense set $D$ of $X$ is such that the closure of $D$ equals $X$. Or equivalently, every non-empty open set contains a point of $D$. So the points of $D$ are in a sense "close" to all points of $X$, we can "approximate" points of $X$ by points in $D$.

The name separable is somewhat unfortunate (what can be separated, exactly?). It probably has an historic origin in some context. We can more neutrally say that $d(X)$, the so-called density of $X$, is countable. Here density $d(X)$ of $X$ is defined as the minimal cardinality of a dense subset of $X$ (well-defined by set-theory arguments), and if it happens to be finite we round it up to $\aleph_0$, the first infinite cardinal. By definition then separable is $d(X) = \aleph_0$, and this is already special as it's the minimal value.

So having countable density means we can "approach" all points by at most countably many points (of a dense set), and this also happens to bound the number of points we can have: a Hausdorff (we need some separation axiom, or we use infinite indiscrete topologies, or cofinite ones) separable space has size at most $2^{|\mathbb{R}|}$, which can be generalised to $|X| \le 2^{2^{d(x)}}$ for Hausdorff $X$, and there are separable spaces that reach that size, like $\mathbb{R}^\mathbb{R}$ in the product topology.

Also, in normed spaces we often have (not always) that we have a natural countable Schauder base (all $\ell_p$ spaces, etc.) and all such spaces are separable. So they occur very frequently in applications as well.