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?

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    $\begingroup$ Complete normed spaces cannot have countable dimension. Separability is however a sort of "morally of countable dimension" criterium, ie separable Hilbert spaces have countable ONBs. $\endgroup$
    – s.harp
    Apr 9 '16 at 13:49
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    $\begingroup$ Separablility roughly means there's not "too much" stuffed in the space. It is a nice property. $\endgroup$
    – Qiyu Wen
    Apr 9 '16 at 13:58
  • $\begingroup$ Thanks, these are exactly the kind of answers I was looking for! ...one thing though, aren't the reals separable because the rationals form a countable dense subspace of $\mathbb{R}$...'dense' makes it sounds like there is 'a lot of stuff' in the space? $\endgroup$
    – csss
    Apr 9 '16 at 14:06
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    $\begingroup$ A metric space is separable if and only if it is a Lindelöf space (every open cover has a countable subcover). $\endgroup$
    – Math1000
    Apr 9 '16 at 14:21

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.

  • $\begingroup$ @"The name separable is somewhat unfortunate (what can be separated, exactly?)" . You might want to check out this mathoverflow post. $\endgroup$
    – Ramanujan
    Apr 10 '19 at 9:42

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