For me second-countability always felt like to be the more important and fundamental concept from general topology than separability. I wonder whether there are any points which can be made for the importance of separability.
Let me subsume the situation: Both notions are intended to guarantee smallness known from classical spaces, from geometry and analysis. Second-countability is a stronger condition, but for metrisable spaces both conditions are equivalent—the word “separable” seems to be more popular in these cases (for example in functional analysis and descriptive set theory). However, under most weaker conditions than metrisability, second-countability is not guaranteed by separability (and in fact metrisability is implied by second-countability and regularity, that is Urysohn’s metrisation theorem): For example the space $[0,1]^{\mathbb{R}}$ with the product topology is separable, but not second-countable, although it is compact and even a product of compact Lie-groups (are there nicer spaces?). There are even locally euclidean, separable spaces which are not second-countable, as required in the usual definition of a topological manifold (see this question). In the locally compact case second-countability implies $\sigma$-compactness, which is useful for integration theory, and the space $X$ is second-countable if and only if the space $C_0(X)$ of continuous numerical functions vanishing at infinity is second-countable. For metrisability there are no analoga. For locally compact groups the second-countability is equivalent to the second-countability of $L^2$ with respect to the Haar measure (it should also hold more generally for certain non-degenerate Borel measures on general locally compact spaces).
Some classical analytic methods using sequences can be used for second-countable spaces: For second-countable spaces compactness is equivalent to countable compactness and sequential compactness. In first-countable Hausdorff spaces you can choose convergent subsequences from every convergent net. Especially in first-countable topological vector spaces (or abelian groups) the convergence of the net of all finite partial sums of a set of vectors is equivalent to the unconditional convergence of a series (the series converges independently of the order). In the separable function space $\mathbb{R}^\mathbb{R}$ this does not work.
Some more general points: Second-countability imposes a strict smallness condition (the cardinality of the topology and in the Hausdorff case the cardinality of the space must be at most the cardinality of the continuum), while separable Hausdorff spaces might consist of $\beth_2$ points. Separability has the advantage of being preserved under continuous images–however, it has the big disadvantage of not being preserved under taking subspaces.
Do you know of any important theorems/theories where separability is crucial—not second-countability? Which generalisations of important concepts from classical analysis only depend on separability? Is the popularity of the word/concept of “separability” just due to the special case of metric spaces? I have even seen some authors using the word “separable” instead of “second-countable” (which sounds reasonable, since “second-countable” sounds cumbersome).
Best Answer
An arbitrary product of separable spaces satisfies Suslin´s condition (i.e. any disjoint family of open sets is countable). I find this result remarkable since separability is not preserved under (large) products while Suslin´s condition might or might not be preserved under (even finite) products, depending on the underlying axioms of set theory.