Ultrafilters II: Origin Story

The day after publishing my first post in the Ultrafilter saga, I attended an excellent seminar talk by Maryanthe Malliaris, which, fittingly, was largely about ultrafilters. During the talk, she told a story about the development of filters by Bourbaki (they had been independently developed by at least one other mathematician around the same time) that I will share, with some additions, here, with the caveat that, as this will be a remembered account from a story told in an informal seminar talk about a group that, over the years, has been heavily mythologized, the details may not all be 100 percent accurate. But origin stories are rarely 100 percent accurate, so let’s push onward.

We should first say a few words about Nicolas Bourbaki, the pseudonym of a semi-secret collective founded in the mid-1930s by nine young French mathematicians. The formation of the group was prompted by the fact that all of the members were dissatisfied with the lack of rigor in the mathematical analysis textbooks in use at the time, and the initial goal was the composition of a new textbook. They quickly realized that, to do this in what they saw as the correct way, they would need to start at the very beginning and reformulate all of mathematics on a rigorous, axiomatic basis. They began to produce a series of books, called Elements of Mathematics, providing a self-contained exposition of what they considered the fundamental areas of mathematics. Volume 1 was on set theory, Volume 2 on algebra, Volume 3 on general topology. A new volume on algebraic topology was published in 2016.

It is hard to overstate the influence of Bourbaki on the mathematics of the twentieth century. A number of the most prominent mathematicians of the century were members (initial members include, for example, André Weil, Henri Cartan, and Claude Chevalley; later members include Laurent Schwartz, Jean-Pierre Serre, and Alexander Grothendieck, among others). They introduced a staggering number of definitions and notations that are now standard throughout mathematics (for example, the use of \emptyset to denote the empty set). Most importantly, Bourbaki pushed the mathematical community towards rigor, abstraction, and generality, which are central to their mission.

Now on to filters. One fine summer (I have no idea if it was actually summer or if the summer was fine, but let’s assume it was) in the mid-1930s, the members of Bourbaki convened at the country house of Chevalley’s parents in central France for one of their famous retreats. One day, their topic of discussion was the possibility of freeing the notion of ‘limit’ from the countable.

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Members of Bourbaki at Chevalley’s parents’ house, with special guest Simone Weil, sister of André. 

Let’s step aside for a moment to try to gain at least an imprecise idea of what this means. To illustrate, we will consider sequences in the set of rational numbers, \mathbb{Q}, and the set of real numbers, \mathbb{R}. By a sequence, we mean a collection of objects indexed by the natural numbers. So, for example, a sequence of real numbers is an object of the form \langle a_0, a_1, a_2, \ldots \rangle = \langle a_n \mid n \in \mathbb{N} \rangle, where, for all n \in \mathbb{N}, a_n \in \mathbb{R}. If \langle a_n \mid n \in \mathbb{N} \rangle is a sequence of real numbers and b \in \mathbb{R}, then we say \langle a_n \mid n \in \mathbb{N} \rangle converges to b, or that b is the limit of \langle a_n \mid n \in \mathbb{N}\rangle, if, as n gets arbitrarily large, a_n gets arbitrarily close to b. (I am of course being imprecise here; the rigorous definition can easily be found elsewhere.) So, for example, the sequence \langle 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \rangle has a limit of 0.

We can also talk about a sequence being convergent without talking about limits. Let us say that a sequence \langle a_n \mid n \in \mathbb{N} \rangle is convergent if, as the sequence goes on, its elements get closer and closer to one another, or, slightly more precisely, as n gets arbitrarily large, the distance between a_{m_0} and a_{m_1} for arbitrary m_0, m_1 \geq n gets arbitrarily small.

It is easy to show that every sequence that has a limit is convergent. Now the question naturally arises: does every convergent sequence have a limit? For sequences of real numbers, the answer is yes; this property of the real numbers is known as completeness. However, let’s step back for a moment to the set of rational numbers. Consider the sequence \langle 3, 3.1, 3.14, 3.141, \ldots \rangle, in which the n^{\mathrm{th}} element is the n-digit expansion of \pi. This is a sequence of rational numbers, and it is easily seen to be convergent. However, since \pi is irrational, it does not have a limit in the set of rational numbers (of course, it has a limit in the set of real numbers, namely \pi). Thus, if you somehow lived in a world in which rational numbers existed but irrational real numbers did not and someone handed you this sequence, you would be able to see that it is convergent, but if you looked at the point to which it apparently converges, there would be nothing there! Or, more precisely but less colorfully, \mathbb{Q} is not complete. (In fact, one of the earliest rigorous definitions of the set of real numbers is as the completion of the set of rational numbers.)

These notions of limits and convergence can be generalized to a class of structures known as metric spaces, which are special examples of topological spaces, but the collections that converge or have limits are always countable (namely, they’re always sequences indexed by the set of natural numbers). This was seen as unnecessarily restrictive in the setting of a general topological space, though, so, in the spirit of generality, Bourbaki sought a definition of limits and convergence that did not depend on the cardinality of the convergent collection.

It was not clear what the correct generalization should be, so, on this particular day, having made little progress, the members of Bourbaki decided to go for a walk. Henri Cartan, though, stayed behind, and, by the time the others returned, he had developed the notion of a filter. He started explaining his ideas, the mathematicians became increasingly excited, and, at one point, one of them, recognizing that this was something truly important, shouted, “Boom!” The objects became known as booms, and their study as boomology. Only when Cartan was later writing about them in a formal paper did he refer to them as filters, the name by which they are known today.

In our last post, we introduced filters as a way of specifying subsets of a given set that should be considered “large.” But, in the context of topological spaces, filters can also be seen as generalizations of sequences and, as such, we can define what it means for a filter to converge to a limit. (For those interested, here’s the technical definition. If (X, \tau) is a topological space, \mathcal{F} is a filter on X, and x \in X, then \mathcal{F} converges to x if, for every U \in \tau with x \in U, there is Y \in \mathcal{F} such that Y \subseteq U.) And it turns out that the compactness of a topological space has a very appealing characterization in terms of convergence of filters: a topological space (X, \tau) is compact if and only if every ultrafilter on X converges to an element of X.

We finally connect these ideas to the Stone-Čech compactification, presented on Monday. Recall that, if (X, \tau) is a discrete topological space, i.e. \tau = \mathcal{P}(X) and X is infinite, then the Stone-Čech compactification of X, \beta X, is the space of ultrafilters on X, where a point x \in X is associated with the principal ultrafilter at x, \mathcal{U}_x, where \mathcal{U}_x is the collection of all subsets of X containing x as an element.

Now, it can be shown that, if (X, \tau) is a discrete topological space and \mathcal{U} is an ultrafilter on X, then \mathcal{U} converges if and only if the intersection of all of the sets in \mathcal{U}, denoted \bigcap \mathcal{U}, is non-empty. Moreover, this happens if and only if \mathcal{U} is a principal ultrafilter, in which case, if \mathcal{U} = \mathcal{U}_x for some x \in X, then \bigcap \mathcal{U} = \{x\} and \mathcal{U} converges to x. If X is infinite, there are nonprincipal ultrafilters on X which therefore do not converge. This can be seen as an explanation for why (X, \tau) is not compact and gives a slightly different picture of the Stone-Čech compactification of (X, \tau). To make (X, \tau) compact, we want to make sure that every ultrafilter \mathcal{U} on X converges. If \mathcal{U} is a principal ultrafilter, then it already converges. If \mathcal{U} is non-principal, though, it does not, and, just as we add irrational numbers like \pi to \mathbb{Q} to make sure that all convergent sequences have a limit, so we add new points to X to make sure that all non-principal ultrafilters converge. Namely, for all non-principal ultrafilters \mathcal{U}, we add a new point x_{\mathcal{U}}, which can be considered to be the point at infinity that lies in  \bigcap \mathcal{U} .

P.S. A slight disclaimer here. The analogy between \mathbb{Q} \mapsto \mathbb{R} and X \mapsto \beta X is perhaps not as nice as I have made it out to be. \mathbb{R} is not a compact space, precisely because, while every convergent sequence from \mathbb{R} has a limit, there are sequences from \mathbb{R} with no convergent subsequence (such as \langle 0, 1, 2, 3 \ldots \rangle). \mathbb{R} is a metric space, and it turns out that a metric space is compact if and only if every sequence from the space has a subsequence that converges to a limit in that space; this can be seen as a special case of the fact, mentioned above, that a topological space is compact if and only if every ultrafilter converges to a limit in the space.

 

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