What well known results with countability assumptions can be naturally extended to uncountable settings?

Question

In many of the common categories of spaces (or algebras) in mathematics, one often restricts attention to those spaces or algebras which are "countable" or "countably generated" in some sense. For instance:

• When studying topological spaces, it is common to restrict attention to those spaces that are metrizable, separable, first or second countable, or Polish.
• When studying function spaces or operator algebras, it is common to restrict attention to those spaces that are separable (in either a weak or strong topology), or that are naturally associated to a separable auxiliary space.
• When studying measurable spaces, it is common to restrict attention to standard Borel spaces; similarly, when studying measure spaces, it is common to restrict attention to Lebesgue spaces (i.e., standard Borel spaces equipped with a Borel measure), or standard probability spaces.

One reason for this is that many of the standard theorems of analysis are stated only in such "countable" settings, with well known counterexamples showing that the "naive" version of these theorems can fail in "uncountable" settings. For instance, the assertion that a topological space is compact if and only if it is sequentially compact is true in the metrizable case, but not in general.

However, it seems to me that in many cases, a basic theorem in analysis which is commonly presented only in the countable case can actually be extended to uncountable settings, after making some natural changes to the setup in order to avoid the standard counterexamples. Let me illustrate this phenomenon with some examples:

1. As mentioned previously, in general it is not the case that a topological space $X$ is compact if and only if every sequence has a convergent subsequence. However, it is true that $X$ is compact if and only if every net has a convergent subnet. (Among other things, this can be used to give a short proof of Tychonoff's theorem.)
2. It is well known that a metric space is compact if and only if it is complete and totally bounded [side question: what is the name of this theorem? It is not the Heine-Borel theorem]. But this claim is in fact also true in the larger "uncountable" category of uniform spaces, which include in particular topological groups and compact Hausdorff spaces as important special cases.
3. The Baire category theorem is often stated for complete metric spaces, but also applies (by the same argument) for locally compact Hausdorff spaces (which doesn't quite contain complete metric spaces as a subclass, but which I still think of as an "uncountable" analogue of that latter class).
4. The Kolmogorov extension theorem is usually stated in the case when the factor spaces are Euclidean spaces, or more generally Polish spaces, but in fact is true in arbitrary factor spaces as long as all measures are required to be inner regular Borel.
5. If a topological group $G$ is metrisable, then the multiplication operation $\cdot: G \times G \to G$ is Borel measurable if we equip $G \times G$ with the product of the Borel $\sigma$-algebras of $G$ (which, in the metrisable case, agrees with the Borel $\sigma$-algebra on $G \times G$). However, in the non-metrisable case this statement fails even if $G$ is compact (this is known as the "Nedoma pathology"). Nevertheless, one can restore measurability (in the compact Hausdorff case at least) if one replaces the Borel sigma-algebra with the slightly smaller Baire sigma-algebra (the sigma-algebra generated by $C(X)$). (Related to this, the Baire sigma algebra of even uncountably many compact Hausdorff spaces agrees with the product of the individual Baire sigma algebras, whereas this claim in the Borel case is only true in general for metrisable spaces and for at most countable products.)
6. Many theorems in ergodic theory restrict attention to actions by countable groups, to take advantage of the fact that the countable union of null sets is still null. (For instance: if the measure-preserving action of a countable group on a probability space is non-ergodic, then there exists an invariant set of measure strictly between zero and one, since one can start with a set which is invariant up to null sets and then delete the orbit of these null sets to get a genuinely invariant set.) However, these sorts of issues tend to go away if one works in a "point-free" fashion by replacing the underlying measure space with its measure algebra. (For some examples of this, see the papers An uncountable Moore–Schmidt theorem and An uncountable Mackey–Zimmer theorem by Jamneshan and myself.)

My motivation in extending countable results to the uncountable setting is because there are some natural uncountable spaces that arise from basic constructions, such as ultraproducts, Stone–Čech compactifications, or Gelfand duals of $L^\infty$-type algebras, to which standard theorems in analysis often do not appear to directly apply.

Anyway, my question is this:

What are some other examples of standard theorems in analysis (or other areas of mathematics) which are commonly only stated in "countable" settings, but which can in fact be extended to "uncountable" settings, possibly after a suitable natural reformulation of the theorem?

Answer 1

Here is an example from graph theory.

Theorem. A graph has an edge-decomposition into cycles if and only if it does not contain an odd cut.

This is very easy in the finite case, fairly easy in the countable case, and suprisingly difficult (but true) in the uncountable case. A deep theorem of Laviolette reduces the uncountable case to the countable case. Recently, Thomassen has given a short proof of Laviolette's theorem. See Nash-Williams’ cycle-decomposition theorem.

Answer 2

The basic mathematical results about the foundations of quantum mechanics.

When von Neumann published in 1932 the first book about the mathematical study of quantum mechanics, he choose to work inside a complex separable Hilbert space.

Now we know that almost everything in the basic mathematical theory of Hilbert spaces does not need separability and many things do not require complex scalars (among the few well known and elementary stated counter-examples: the invariant subspace problem for bounded linear operators; existence of orthonormal bases for pre-Hilbert spaces).

However, at that time (and in particular when the "rings of operators" were conceived in 1929) practically everyone worked only in the separable complex case. Even today, in physics, one (usually) works only in that setting.

For the founders of the theory the restriction was possibly quite natural: "please, young man, say infinite matrix" once said (in German) E. Schmidt (and "matrix mechanics" was the term in physics). Infinite sequences with a converging sum are much more intuitive that uncountable summable families, and even more are matrices with such indexes. The modern definitions (independent from a basis) were needed to abstract away from separability without hurting intuition too much.

However, in 1936 - 1937, when von Neumann was very active in operator algebras, continuous geometries, quantum logic he discovered that it is better to drop both conditions of separability and complex scalars for the foundations (i.e. to axiomatize the subject with a few axioms that are more directly intuitive, in the same way as the basic facts of Euclidean geometry are more intuitively presented to people with general culture, with primitive concepts of points, lines, planes, incidence, orthogonality rather than algebraically with "a vector space with a simply transitive action on a set of points, and a scalar product on the vector space": yes, computations and proofs come out better in the algebraic settings, but the geometric language is what one uses for understanding of the statements).

The output was his manuscript "Continuous geometries with a transition probability" (posthumously published by I. Halperin in 1981 as Memoirs AMS Vol. 252) where a equivalence (yes, a true categorical equivalence, and more) was established between three kinds of structures (except types I$_1$, I$_2$, I$_3$ i.e. the cases representable in Hilbert dimension at most 3):

1. rings with involution abstractly isomorphic to "finite factors" in Hilbert spaces (real or complex, of arbitrary dimension; such rings have a unique compatible real algebra structure and a unique compatible C$^*$-norm; they might have or not $i$, a anti-hermitian central square root of $-1$ [i.e. a complex algebra structure], but even when they have one [and then exactly two $i$ in the factorial case, always more if not], it is not required to fix it)

2. "continuous geometries with a transition probability": these are a set of "quantum mechanical propositions" structured by a binary order relation (implication) and a unary operation (negation), and a function that to a pair of propositions associates the probability that, when the physical system is in a state such that the first proposition is surely true, a measurement to test the second proposition would give "true" as result. The axioms to define such structures are clearly discussed by von Neumann for their mathematical and physical meaning; the transition probability is in principle redundant (there is only one compatible with a the ortholattice structure of the propositions and the wanted axioms) but its presence permits a compact statement of the axioms and their meanings.

3. Certain regular rings with involution. They are exactly the (possibly) unbounded operators affiliated with the factor (structured by the usual algebraic operations), and also the coordinatizing ring (with involution) of the lattice (with involution) of propositions.

To see how a crystal clear concept of categorical equivalence von Neumann had in 1936 - 1937, even if not in the language of categories, read the two pages that introduce the coordinatization theorem in the book "continuous geometries", with the distinction between cases $n\geq 4$ (existence), $n\geq 3$ (unicity up to isomorphisms), $n\geq 2$ (when the isomorphism exists it is unique).

Generalization to non-factorial cases and to include semi-finite and properly infinite components are possible (see Type III factor representation ); other equivalent structures are known (the Jordan ring structure, which is then a real [JBW] algebra in a unique way, on the self-adjoint part of the $*$-ring of bounded operators, or also the one for the $*$-ring of the affiliated unbounded ones; the "unsharp quantum logic" of "effects" i.e. the structure of self-adjoint operators with spectrum in the real interval $[0,1]$ instead of the set of two values $\{0,1\}$: projections are the "sharp" propositions, "effects" are the "fuzzy" ones, in the same way as, for the classical case, boolean algebras are generalized by MV-algebras). Further generalizations are possible, but von Neumann's axiomatization is (or at least should be in my opinion) the archetype. Without the countability (and the complex scalars) assumption because (as A. Weil once wrote for completely different matters about "basic number theory") they would be "unnecessary machinery on a ship which seemed well equipped for this particular voyage; instead of making it more seaworthy, it might have sunk it" [One more axiom in the foundation of quantum mechanics means even more disputes ...]

PS: to see how around the same years another giant was fighting against countability assumptions, one can read in A. Weil, Collected papers, Vol I, pag. 540 writing in 1978 commenting the 1937 work about uniform spaces (original french, I do not attempt a pseudo-translation):

Avec le recul que donnent les quarante dernières années, on sourira sans doute du zèle que j'apportais alors à l'expulsion du dénombrable; chassé par la porte, il a fini par rentrer par la fenêtre, avec les espaces paracompacts, les espaces polonais, etc.

[read all that commentary ...]

EDIT. Small notes:

(a) the common setting for completely metrizable spaces and locally compact T$_2$ spaces is that of Čech-complete spaces (G$_\delta$ subspaces of T$_2$ compact spaces). However, they do not cover all spaces such that the usual proof of Baire theorem goes on (locally countably compact does not imply completely regular but implies Baire) and they do not cover all Baire completely regular (or even metrizable) spaces.

(b) the usual separability condition for von Neumann algebras (faithful representability with a separable Hilbert space i.e. norm-separable predual) is equivalent to: countably generated (i.e. being the double commutant of a countable [self adjoint] subset) and orthoseparable (any set of [nonzero] mutually orthogonal projections is countable). As already noted in the comments, commutative cases exist with the first condition but not the second. Finite factors always have the second condition, but not always the first: take the ultraproduct in Connes' embedding problem [and yes, by von Neumann's equivalence one can restate it using continuous orthogeometries or complete $*$-regular rings].

(c) As implicitly noted by others, often the extension to uncountable cases reduces to a proof of reduction to the countable subcase (elementary example: infinite summable families. Another example, related to this answer: Soler's theorem). This is not so in the cases of these answers (see also Weil's comments about uniform spaces as common generalization of metric spaces and topological groups).

Answer 3

The unit ball of an Hilbert space is sequentially compact for the weak topology.

People often think that separability is needed here (this is equivalent to the existence of a countable Hilbert basis) and that without it, the result does not hold and one must use the characterization of compactness by open covers together with Tykhonov theorem + Axiom of choice to get a somewhat weaker statement, namely the existence of a cluster point.

But in fact, one can work in the closed separable subspace generated by the elements of the sequence and proceed by diagonal extraction. Everything is zero in the orthogonal of that subspace and nothing has to be done there.

There are a few applications of that result in ergodic theory.

Answer 4

Topology/Geometry

• Partitions of unity vs paracompactness. Either in the setting of topological manifolds or the setting of smooth manifolds, many learned the results under the assumption that the space in question is second countable, while the more natural setting is of course the result of Dieudonné stating that for Hausdorff topological spaces, existence of partitions of unity subordinated to any open covering is equivalent to the space is paracompact. Proof of this is quite natural, and can be adapted to smooth manifolds (not assume to be second-countable) without any essential difficulty.

• Continuous extension. Let $X$, $Y$ be metric spaces, $A$ a dense subspace of $X$, a continuous map $f : A \to Y$ can be extended to a continuous map $F: X \to Y$ if and only if for each sequence $(a_n)$ in $A$ converging to some $x \in X$, the sequence $(f(a_n)$ converges in $Y$ (the limit being $F(x)$ of course). More natural and general result is stated as follows: let $X$ be an arbitrary topological space, $Y$ a regular Hausdorff space, $A$ a dense subspace, a continuous map $f : A \to Y$ extends to a continuous map $F : X \to Y$ if and only if $\lim_{a \in A, a \to x}f(a)$ exists in $Y$ for every $x \in X$. Here the limit is taken along the trace on $A$ of the filter of neighborhoods of $x$ in $X$. Once formulated in this way, the proof is almost transparent. Equivalently, one can replace filters by nets, but interestingly, the argument is less transparent than using the filter of neighborhoods.

Riemann surface

• Automatic second countability from connectedness. This is traditionally defined as a connected $1$-dimensional complex manifold that is second countable (everything is assumed to be Hausdorff). But by Rado's theorem, the assumption on the second countability is redundant, as long as we require the manifold is connected. Interestingly, this is no longer true for higher dimensional connected complex manifolds (they might well be not second-countable).

Measure theory

• Monotone convergence. Consider Radon measures on a locally compact Hausdorff space $X$ which we identify as a positive functional $I$ on $C_0(X)_+$ via Riesz's representation theorem. The monotone convergence theorem holds for arbitrary increasing nets of lower semi-continuous functions on $X$ that is non-negative outside some compact subsets. The proof easily reduces to the case of compactly supported real-valued continuous functions on $X$ via Daniell's approach to integration, in which case it follows quite naturally from Dini's theorem, which in turn is simply a consequence of the finite intersection property of closed sets in compact spaces.

• Fubini's theorem beyond the $\sigma$-finite setting. I know two different approaches, which do not give consistent results, but both are useful in different contexts. One approach is to consider only outer measures, and construct the natural outer product measure (see e.g. Federer or Evans & Gariepy). Another approach only applies to Radon measures on locally compact Hausdorff spaces. The idea is to show that, iterated integration of compact supported continuous functions on the product space with respect to the Radon measure on each of the factor space, does not depend on the order of the iterated integral, hence yields a well-defined positive linear functional on $C_0(X \times Y)$ by standard approximation arguments.

• Convolution of $L^1(G, \mathrm{d} \mu).$ This is usually done for second countable locally compact Hausdorff group $G$ with a left Haar measure $\mu$, due to measure theoretic complications. Without the second-countability assumption, two main problems arises: one is the Fubini theorem beyond $\sigma$-finite case which is tackled above; another is that the product $\sigma$-algebra of Borel algebras on $G$ is strictly smaller than the Borel algebra of the product on $G \times G$. Also, two natural approaches to the general setting can be adopted: one is to consider only Baire measurable functions, so the product of the $\sigma$-algebras of Baire measurable sets match nicely (e.g. in Loomis's book); another is to use the full power of the Fubini type theorem and related constructions of Radon measures mentioned above (e.g. in Hewitt & Ross).

• Radon-Nikodym theorem(s). For people working on operator algebras and related fields, perhaps it is reasonable to say that the more natural setting is normal functionals/weights on von Neumann algebras. There are so many powerful results along this line (Sakai, Connes, Pedersen etc.), which not only extend Radon-Nikodym for Radon measures on second countable (or $\sigma$-compact) locally compact Hausdorff spaces to arbitrary locally compact Hausdorff spaces, but much more importantly, broadly and profoundly, to the non-commutative case.

Operator algebras (Some results are already mentioned, here are some more)

• Pedersen's up-down theorem vs up-down-up theorem. Let $A$ be a concrete $C^*$-algebra acting non-degenerately on a Hilbert space $H$, and $A''$ the von Neumann algebra generated by $A$. For $X \subseteq B(H)_{sa}$, let $X_\delta$ (resp. $X_m$) denote the set of limits of bounded decreasing sequences (resp. nets) in $X$, and $X^\sigma$ (resp. $X^m$) the set of limits of bounded increasing sequences (resp. nets) in $X$. Pedersen's up-down theorem says that if $H$ is separable, or more generally, if $A''$ is $\sigma$-finite, then $(A_{sa}^\sigma)_\delta = A''_{sa}$. Pedersen's up-down-up deals says that in general, one has $(((A_{sa})^m)_m)^m = A''_{sa}$. Techniques for the proofs both versions are similar and the latter can be considered as a natural complement of the former (with slightly weaker result of course).

• Normal faithful functionals vs normal faithful semi-finite weights. Theory around these notions are motivated by and intertwined with the development of the modular theory of von Neumann algebras (Tomita-Takesaki theory). The former is sufficient for $\sigma$-finite von Neumann algebras, but the latter is needed for the general case, which can be seen as a natural extension of the former.

Haar measures

• Haar measure for locally compact groups. Haar established the existence of Haar measure for second-countable locally compact groups. Later van Kampen and Weil (independantly?) removed the second-countability restriction.
• Haar measure for compact quantum groups. Originally, Woronowicz proved the existence for separable compact quantum groups. The general case is established later by van Daele.

Answer 5

It is a frequently mentioned basic fact of left-ordered groups that, assuming they are countable, they admit a faithful action by homeomorphisms on the real line. But it is easy to show that any left-ordered group admits a faithful order-preserving action on a linearly ordered set, and conversely. As Navas says in the introduction of Navas - On the dynamics of (left) orderable groups: “Quite surprisingly, this very simple remark has not been exploited as it should have been ….”

(I am brushing under the carpet the fact that from a countable group acting faithfully on a linearly ordered set one can produce a faithful action by homeomorphisms on $\mathbb{R}$.)

Answer 6

Here are some examples from algebra where finiteness assumptions can be removed. In the first two, the statement of the more general result is unchanged, but the third result has to be expressed in a new way in order to have a chance of being true without the finiteness condition.

1. From the classification of finitely generated modules over a PID, every submodule of a finitely generated free module over a PID is free. That consequence is also true without finite generatedness: every submodule of a free module over a PID is free. See https://math.stackexchange.com/questions/162945.

2. A finitely generated projective module over a local ring is free, but this result is also true without finite generatedness. See https://en.wikipedia.org/wiki/Kaplansky%27s_theorem_on_projective_modules.

3. The normal basis theorem says that for a finite Galois extension $L/K$, there is some $\alpha \in L$ such that the set of its $K$-conjugates $\{\sigma(\alpha) : \sigma \in {\rm Gal}(L/K)\}$ is a $K$-basis of $L$ (called a "normal basis"). If $L/K$ is an infinite Galois extension, then the normal basis theorem as described above does not make sense since $\{\sigma(\alpha) : \sigma \in {\rm Gal}(L/K)\}$ is a finite set for each $\alpha$ and thus could not be a $K$-basis when $L/K$ is infinite. Lenstra found a way to reformulate the definition of a normal basis so it does make sense for infinite Galois extensions and the normal basis theorem is then true in that setting. See https://pub.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1985c/art.pdf. The basic idea is that the normal basis theorem for finite Galois extensions is about a comparison between ${\rm Map}(G,K)$ and $L$, where $G = {\rm Gal}(L/K)$ and ${\rm Map}(G,K)$ is the set of all maps $G \to K$. When $L/K$ is an infinite Galois extension, we should replace ${\rm Map}(G,K)$ with the set $C(G,K)$ of all continuous maps $G \to K$, where $G$ has the Krull topology and $K$ has the discrete topology. Lenstra's version of the normal basis theoem reduces to the usual version of the normal basis theorem when $L/K$ is finite since $C(G,K) = {\rm Map}(G,K)$ when $L/K$ is finite, as $G$ in that case has the discrete topology.

Answer 7

This is a riff on this query, concentrating on aspects of functional analysis and related topics. It is formulated in a conversational manner but is based on the concepts of ind and pro categories which were introduced in the 50´s by Grothendieck and collaborators as a method to add limits and colimits to a given category and can be made precise in this context. I have added the intermediates cases of limits for countable spectra in the pirit of this question.

In fact, I think that it is sometimes advantageous to consider a threefold transition--from the finite to the countable, then to the uncountable.

Thus we can start with the category of finite dimensional Banach space (with linear contractions as morphisms)- This leads successively to those of separable, resp. non-separable spaces (ind-categories).

Dually, one gets the separable Waelbroeck spaces, resp. Waelbroeck spaces (H.Buchwalter and L. Waelbroeck).

Starting with the Banach spaces (now with continuous linear mappings as morphisms), one proceeds to Frechet spaces, respectively locally convex spaces as projective limits. Dually we obtain, with the pro-construction, convex bornological spaces of countable type (closely related to Grothendieck´s $DF$-spaces) and convex bornological spaces.

If one applies the pro construction to the category of Banach spaces, with contractions as morphisms, one gets the class of Saks spaces, initially those with metrisable unit balls.

Analogue constructions can be carried out in the categories of Banach algebras (resp. $C^\ast$-algebras), commutative or non-commutative, with or without units.

In topology, one can start with the compact subsets of euclidean pace as the finite objects, then proceed to the compact metrisable spaces, followed by the compact spaces.

Using the ind constructionand starting with the category of compact spaces, one obtains the compactologies, initially those of countable type (again Buchwalter and Waelbroeck). These are a natural parallel to the category of topological spaces but have more convenient properties. One can move back and forward between these two categories so that they coincide for a large fraction of the topological spaces which occur in practice--metrisable space, locally convex spaces, more genereally $k$-spaces wich means that their theory coincides in many contexts.

Touching on an example mentioned in the query, one can start with metric spaces with Lipschitz functions as morphisms--the pro-construction turns up the category of metrisable uniform spaces, followed by all such spaces.

This is a small selection of the kind of spaces which arise in this manner. It could be extended almost to infinity. I have concentrated on examples which have already been studied intensively in their own right but there are many where this is not the case, but perhaps should be.

Of course, the thrust of the question is on the extension of results and proofs from the countable to the uncountable case. There are so many such examples in the above situations that I can only realistically with one and I have chosen that of tensor products. Suppose that we have some notion of a tensor product for finite Banach spaces (I am thinking of the inductive and projective norms but the following holds much more generally). Then one can extend it successively to the categories of Banach spaces, of locally conovex spaces and of convex bornological spaces in the natural manner.

Another recurrent situation is to take a standard duality from a special situation (typical examples--Riesz representation or Gelfand-Neumark duality for the case of compact spaces) and extend it to one for general topological spaces (more precisely compactologies but this can be used to deduce results for the topological case using the above mechanism)