Importance of separability vs. second-countability

Question

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).

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

Let me elaborate a little on your observation:

Separability has the advantage of being preserved under continuous images–however, it has the big disadvantage of not being preserved under taking subspaces.

Even hereditary separability does not imply second-countability (the Sorgenfrey Line would be a counterexample). It is however consistent with the axioms of set theory that every hereditarily separable space enjoys the following weakening of second-countability: every open cover has a countable subcover. This is known as the Lindelof property.

THEOREM (Todorcevic) Assume the Proper Forcing Axiom. Then every hereditarily separable space is Lindelof (and hence hereditarily Lindelof).

Whether hereditarily separable implies hereditarily Lindelof is however independent of the axioms of set theory. For instance, assuming as little as the Continuum Hypothesis there is a counterexample which even has very good extra topological properties (it's compact, first-countable and homogeneous). See:

de la Vega, Ramiro; Kunen, Kenneth, A compact homogeneous S-space., Topology Appl. 136, No. 1-3, 123-127 (2004). ZBL1048.54013.

Answer 5

So suppose that a topological space $X$ is separable (or that it at least contains a dense subset which is bijective to an ordinal). Then it is selective, which in this context means that if $(U_α)_{α \in A}$ is a family of open subsets of $X$, then $$ \prod_{α \in A} U_α \neq \emptyset. $$ This is the axiom of choice for open subsets of $X$, which is required for the implication second-countability $\Rightarrow$ separability.