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:
- 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.)
- 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.
- 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).
- 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.
- 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.)
- 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?
Best Answer
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.