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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).

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This 1935 reference also uses this convention: ncbi.nlm.nih.gov/pmc/articles/PMC1076540/pdf/pnas01754-0040.pdf –  François G. Dorais May 18 '13 at 19:06
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Ahem. $\:$ If countable choice then second countability is a stronger condition than separability. $\;\;$ –  Ricky Demer May 19 '13 at 6:16
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This might not be the right decade to make this comment, but separability seems to be just the poor man's version of overtness, something Paul Taylor has been pointing out. If this is indeed the case, then separability would indeed be a suboptimal notion. –  Andrej Bauer May 19 '13 at 7:07
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And this would be Paul Taylor: paultaylor.eu The photo is a bit blurry, but that's how I percieve Paul most of the time anyway. –  Andrej Bauer May 19 '13 at 12:42
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Thank you, Andrej, but you took the photo! I am a long way from being able to discuss separability and second countability, but for a published introduction to overtness see A Lambda Calculus for Real Analysis www.paultaylor.eu/ASD/lamcra or if you are keen to see an incomplete attempt to introduce the idea to "ordinary mathematicians" you could look at Overt Subspaces of $R^n$ www.paultaylor.eu/drafts/overtrn.pdf . I am not sure that the nLab account is correct. –  Paul Taylor May 19 '13 at 14:59
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3 Answers

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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.

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That is indeed remarkable. Although it is a little bit exotic, Suslin’s condition is a well-known, classical property of the real numbers and its finite products and the theorem you mentioned features separability prominently. I think this is closest to those kinds of theorems I was looking for. –  The User May 24 '13 at 23:12
    
Thanks, Ramiro. –  The User May 24 '13 at 23:14
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Separability can be used to study the Stone-Cech compactification of a countable discrete space. Recall that if $X$ is a discrete space, then the Stone-Cech compactification $\beta X$ of $X$ is precisely the set of ultrafilters on $X$. We can therefore use separability to prove facts about ultrafilters without mentioning ultrafilters.

First, we use separability to observe that $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$. Since $\beta\mathbb{N}\subseteq P(P(\mathbb{N}))$ as the set of ultrafilters on $\mathbb{N}$, we have $|\beta\mathbb{N}|\leq|P(P(\mathbb{N}))|=2^{2^{\aleph_{0}}}$. To prove the other direction, let $I$ be a set of cardinality continuum. Then since the product of continuumly many separable spaces is separable, the product space $\{0,1\}^{I}$ is separable. Therefore let $A\subseteq\{0,1\}^{I}$ be a countable dense subset. Then there is a surjective function $f:\mathbb{N}\rightarrow A$. Therefore the function $f$ extends to a continuous function $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$. Since the image $\overline{f}[\beta\mathbb{N}]$ is a compact set, the set $\overline{f}[\beta\mathbb{N}]$ is a closed subset of $\{0,1\}^{I}$, so $\overline{f}[\beta\mathbb{N}]=\{0,1\}^{I}$ since $A\subseteq\overline{f}[\beta\mathbb{N}]$. Since $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$ is surjective, we have $2^{2^{\aleph_{0}}}=|\{0,1\}^{I}|\leq|\beta\mathbb{N}|$, so $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$.

Separability may also be used to prove facts about the Rudin-Keisler ordering. The Rudin-Keisler ordering is the preordering $\leq_{RK}$ on the class of ultrafilters where if $\mathcal{U}\in\beta X,\mathcal{V}\in\beta Y$ are ultrafilters, then $\mathcal{U}\leq_{RK}\mathcal{V}$ if there is a continuous $f:\beta Y\rightarrow\beta X$ with $f(\mathcal{V})=\mathcal{U}$ and $f[Y]\subseteq X$. The motivation for the notion of the Rudin-Keisler ordering is that the Rudin-Keisler ordering measures the size of an ultrapower. In particular, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\mathcal{A}^{\mathcal{V}}$ for each first order structure $\mathcal{A}$.

The Rudin-Keisler ordering is a pre-ordering on $\beta\mathbb{N}$. One can use separability to show that every subset of $\beta\mathbb{N}$ of size at most continuum has an upper bound in $\beta\mathbb{N}$. Assume that $I$ is an index set of cardinality at most continuum and $x_{i}\in\beta\mathbb{N}$ for $i\in I$. Then $\mathbb{N}^{I}$ is separable since the product of at most continuumly many separable spaces is separable, so there is a countable dense subset $A\subseteq\mathbb{N}^{I}$. Therefore let $f:\mathbb{N}\rightarrow A$ be a surjective function. Then $f$ extends to a unique continuous function $\overline{f}:\beta\mathbb{N}\rightarrow(\beta\mathbb{N})^{I}$. The function $\overline{f}$ is clearly surjective, so there is some $x\in\beta\mathbb{N}$ with $\overline{f}(x)=(x_{i})_{i\in I}$. Therefore if $\pi_{i}:(\beta\mathbb{N})^{I}\rightarrow\beta\mathbb{N}$ is the projection mapping, then $\pi_{i}\overline{f}(x)=x_{i}$ for $i\in I$, so $x_{i}\leq_{RK}x$ for $i\in I$.

It should be noted that there are very similar proofs of the above two results using independent sets and independent partitions (the proofs using independent sets and independent partitions are essentially the same proof. See Andreas Blass's comment below). Furthermore, the two above results can be generalized to larger cardinals with the same proofs. In particular, if $X$ is a discrete space, then $|\beta X|=2^{2^{|X|}}$. Furthermore, every subset of $\beta X$ of cardinality at most $2^{|X|}$ has an upper bound in $\beta X$. To prove these facts, one uses a generalization of the notion of separability called the density and the generalized proof is very similar to the original proof. If I remember correctly, the book The Theory of Ultrafilters by Comfort and Negrepontis also gives generalizations of these facts to large cardinals such as compact cardinals.

I hope the above results clear up any confusion about the importance of separability in non-metrizable spaces.

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Concerning "the Rudin-Keisler ordering measures the size of an ultrafilter": Did you mean "the size of an ultrapower"? –  Andreas Blass May 19 '13 at 18:40
    
I'd go a bit further than "there are very similar proofs ... using independent sets and independent partitions." I think of the existence of continuum many independent partitions of $\mathbb N$ and the separability of a product of continuum many separable spaces as being essentially the same theorem. Each is deducible easily from the other. Being more combinatorial than topological, I tend to view the former (and its generalizations to higher cardinals) as the main point, and to view separability as a nice way to make it look topological for those whose tastes differ from mine. –  Andreas Blass May 19 '13 at 18:45
    
Yes. I did mean the size of an ultrapower. –  Joseph Van Name May 19 '13 at 19:31
    
It is a very nice and interesting answer, but I accepted Ramiro’s answer, because it describes more that kind of property I was looking for. Thanks. –  The User Jun 5 '13 at 20:24
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I second the idea that second countability is more fundamental than separability --- topologies are defined in terms of open sets, not points, and second countability is the natural "countability" condition on the family of open sets. It just says that the topology is countably generated.

Important theorems where separability is crucial: the basic example is that a continuous image of any separable space is separable. Even a quotient of a second countable space need not be second countable.

Is the popularity of the word/concept of "separability" just due to the special case of metric spaces? Yes, I think so. But that's a pretty important special case! I just finished writing a book on measure theory and functional analysis, and I found that by restricting attention to separable Banach spaces and their duals I was able to get by just fine without mentioning generalized convergence (nets/filters). For instance, the weak* topology is metrizable on the unit ball of the dual of a separable Banach space, which is good enough for most purposes by the Krein-Smulian theorem.

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What makes the simple fact that images of separable spaces are separable an important theorem? –  Martin May 18 '13 at 19:31
    
@Martin That was indeed not that kind of theorems I was looking for (a property does not get important by being well-behaved when taking images). But I think it is worth mentioning (I have also mentioned it in my question). –  The User May 18 '13 at 19:50
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@Martin: do you wish to tell me that this simple fact is not important? –  Nik Weaver May 18 '13 at 20:50
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No, I was asking an honest question. I agree that it is a useful fact, but it seems more related to continuity than separability per se: It is a characteristic property of continuous functions that the image of the closure is contained in the closure of the image. Thus, dense sets are mapped to dense sets. What makes the instantiation of this observation to countable dense sets particularly noteworthy and important? –  Martin May 19 '13 at 3:53
    
It sounds like you are asking why separability is important. Speaking from my area of expertise, I could tell you any number of basic theorems about separable C*-algebras which fail in the nonseparable case. –  Nik Weaver May 19 '13 at 6:08
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