The Cantor Set, Redux

It is well known that set theory was the solution to many technical problems posed by Newton and Leibniz’s calculus. In the process of developing what is now called real analysis, many strange counterexamples to visual intuition were found. Often they had a single source: the Cantor set. Pictured below are three such counterexamples.

The Cantor set is formed by taking the unit interval $I = [0,1]$ and successively removing open middle thirds of the remaining intervals. One lets $C^0 = I$, then $C^1 = [0,1/3] \cup [2/3,1]$, $C^2 = [0,1/9]\cup [2/9, 1/3]\cup [2/3, 7/9]\cup [8/9, 1],\ldots$ Note $C^0\supset C^1 \supset C^2\supset\cdots$, and we define

\[C = \bigcap_i C^i\]

to be the Cantor set. $C$ has Lebesgue measure $0$ as $\lambda(C^n) = (2/3)^n$ and $\lambda(C)\le \lambda(C^n)$. $C$ is a perfect set, meaning that every single point in $C$ is a limit point of $C$, which implies it is uncountable. One can also establish this uncountability directly. Writing our numbers in base $3$, $C$ consists of precisely the numbers between $0$ and $1$ not containing the digit $1$. Each middle third removal removes the set of numbers which has digit $1$ in the $n$th digit. So, we still have a set that is in bijection with $2^\BN$. By composing this bijection with the map $2^\BN\rightarrow I$ by turning the $0$-$1$ sequences into a binary expansion, we can form $\iota: E\rightarrow 2^\BN\rightarrow I$. Given a ternary sequence of $0$s and $2$s, $\iota$ turns the $2$s into $1$s and treats the new sequence as a binary expansion. $.020202020\ldots_3$ would map to $010101010\ldots_2\in I$.

With this map $\iota$ we can form the devil’s staircase above. For $x\in I$, define \(f(x) = \sup_{0\le y\le x, y\in C} \iota(y)\). That is, $f(x)$ is what $y$ corresponds to in the unit interval for $y$ the largest element of $C$ below $x$. You will get flat staircases in the middle thirds where no elements of $C$ lives. This gives us an everywhere continuous function, with derivative $0$ almost everywhere, yet it still increases. Going one step further, we form $g(x) = (f(x)+x)/2$ which will be a homeomorphism which identifies a subset of $C$, a set of measure $0$, with one of measure $1/2$. With a homeomorphism, we can create measure out of no measure. This rounds out the classic Cantor-set counterexamples one would see in a standard analysis education. Lets see the sequel.

Cantor Sets Two

Fourier series were very important to mathematics in the 19th century (and for that matter in the 20th and 21st century). An early question “uniqueness.” That is, if we have numbers $c_n\in \BC$, indexed by $n\in \BZ$, such that $\sum_n c_n e^{2\pi inx}$ converges point-wise outside of a set $U$, can we determine the coefficients $c_n$? Otherwise put, if we have a function which converges point-wise to $0$ outside $U$, is every coefficient $0$?

A decent amount of work was put into establishing $U = \emptyset$ is a set of uniqueness. A clever set of arguments from Riemann and Cantor. Jumping to $U$ finite was not too hard but proving all countable closed sets are sets of uniqueness required the advent of transfinite induction via the Cantor-Bendixson process.¹ One could wonder, would any null set (a set with $0$ measure) be a set of uniqueness? Well, that brings us to our first novel Cantor set from Menshov.

Similar to the cantor set, one forms a sequences of sets $M^i$ with $M^0 = I$ by iteratively removing open intervals from each closed interval in their $n$th stage. However, instead of always removing the middle third. One removes the middle $1/(n+1)$st open interval at stage $n$. The middle half, then third, fourth, etc. \(M = \bigcap_i M^i\) still has measure $0$ as $\lambda(M^i) = 1/(n+1)$. Interestingly enough though, we can leverage this to create a nonzero sequence ${c_n}$ such that $\sum_n c_n e^{2\pi inx}$ converges to $0$ away from $M$.

Just like previous Cantor sets, this set has a natural bijection with $2^\BN$ and one can take the coin-flip measure of $2^\BN$ and its pushforward along $\iota_M:2^\BN\rightarrow M\rightarrow I$ to yield a Borel measure $\mu_M$ on the interval. That is, if I have a (Borel) subset $B\subset I$, $\mu_M(B)$ is the coin measure on $\iota_M^{-1}(B)$. This gives us a probability measure on $I$, called $\mu_M$, and we can thus we can form its Fourier coefficients via

\[\hat{\mu}_M(n) = \int_0^1 e^{-2\pi inx}\mathrm{d}\mu_M(x)\]

$\sum_n \hat{\mu}_M(n)e^{2\pi inx}$ converges for $x\not\in M$ but the coefficients $\hat{\mu}_M(n)$ are not all $0$. This might not be entirely unintuitive. As $\mu_M$ is a pushforward that factors through $M\subset I$, all of the mass of it lies on $M$. So, if Fourier theory were to work perfectly nicely, you might expect it to converge back to something like $\mu_M$ and thus be $0$ away from $M$. Though, if you do this same procedure with the Cantor set, $\hat{\mu}_C(n)$ won’t even converge to $0$ as $n\rightarrow \infty$. In fact, the Cantor set is a set of uniqueness. We can find a further set of generalizations of the Cantor set which sometimes are a set of uniqueness and sometimes aren’t.

1/ξ φ² 3 1 + √2 custom

φ²; ξ=.382

1/ξ = φ² ≈ 2.618

We generalize cantor sets to $E_\xi$ for $0< \xi<1/2$. $E_\xi^0 = I$ and each successive $E^i_\xi$ is obtained by removing the central open interval of relative length $1-2\xi$ from each remaining closed interval, leaving two closed intervals of relative length $\xi$. $E^i_\xi$ will be a disjoint union of $2^i$ closed intervals and we define $E_\xi = \bigcap_i E^i_\xi$. The following remarkable theorem determines when $E_\xi$ is a set of uniqueness:

The Salem-Zygmund Theorem: $E_\xi$ is a set of uniqueness if and only if $1/\xi$ is a Pisot-Vijayaragavhan number.

I have a few previous posts on Pisot-Vijayaraghavan (PV) numbers. PV numbers are algebraic integers all of whose conjugates have modulus less than one. This has the side-effect that their powers get closer and closer to being integral—a property that is known to be rare because of Koksma’s theorem. In my previous posts I excluded positive integers from the class of PV numbers. Here they count. Namely, the Cantor Set is a set of uniqueness and furnishes our first example of an uncountable set of uniqueness. Somehow, this odd number theoretic property determines whether one is a set of uniqueness or not for these generalized Cantor sets.

Thoughts on Counterexamples

I’m often not sure what to make of counterexamples. In a standard mathematical education, you are showered with definitions of mathematical objects like they fell from the sky. You are then shown wonderful connections, properties, and theorems involving them. Upon playing around with them, you find many results smoothly emit from them. However, this is not how mathematics was developed. Newton did not play with limits as we known them, and Galois had no notion of a group. They did not have these refined objects to to create their theories. The definitions came last.

Similarly, many definitions did not survive. Often, they did not survive because they did not do a good job of fitting the theory. Lots of those misfits they had looked a lot like counterexamples. So, this leads me to the question I often have with counterexamples: is the counterexample telling me something is off with the theory or with the objects we’ve chosen?

Let’s talk about the topologist’s sine curve pictured above. The topologist’s sine curve is defined as:

\[T = \{0\}\times [-1,1]\cup \{(x, \sin(1/x))\mid x>0\}\]

This is a classic counterexample presented in topology. It is a connected space that isn’t path-connected. That is, there is no path from the left-side interval component to the $\sin(1/x)$ looping curve on the right. Yet, they are still connected as the sine portion of the curve has limit points in the left-side interval. So, now we have learned that we can have connected spaces where things aren’t path connected. Although, did we?

In topology, very often the full unabridged class of topological spaces is inconvenient. We often specialize to other classes of spaces for our purposes. The subject of the recent work by Dustin Clausen and Peter Scholze is all about fixing up the inconveniences of topological spaces. Following a similar, but simplified line, we can remove the topologist’s sine curve counterexample as well.

Instead of working with topological spaces, we can consider the full sub-category of $\BR$-generated spaces. To do this, we map topological spaces to their sheaf on the category of domains in $\BR^n$. That is, we replace a topological space $X$, with $\Hom_{\cat{Top}}(-, X)$. This gives us a functor \(\cat{Top}\rightarrow \cat{Shv}(\cat{Dom})\) By a colimit construction, we can furnish a left-adjoint which gives us a functor back $\cat{Shv}(\cat{Dom})\rightarrow \cat{Top}$. After going back and forth through this construction, we end up replacing a topological space $X$ by the re-topologized $X_\BR$ wherein $U\subset X$ is open if and only if $f^{-1}(U)$ is open for every continuous map $f: D\rightarrow X$ from a domain to $X$. The image of this construction is $\BR\cat{Top}$. We also get a map $X_{\BR}\rightarrow X$ which is the counit of the adjunction.

Outside of all of this category theory language, what does this mean? Well, in essence it says that in relation to domains of $\BR^n$, which is enough to determine maps between topological manifolds, one can not tell the difference between $X_\BR$ and $X$. How does this remove the pathology of the topologist’s sine curve? Well, let’s look at $T_\BR$.

I claim that $T_\BR$ is homeomorphic to $[0,1]\cup (2,3)$—a disjoint union of two intervals. The $[0,1]$ corresponds to the ${0}\times [-1,1]$ and the $(2,3)$ corresponds to the sine part of the curve. As these are the two path components, and all domains are path connected, the image of a domain must be contained completely in one or the other. Thus, their preimages will always be empty or the entire domain which means that the two halves of the curve become open. The effect of this $\BR$-generation is to gently separate the topologist sine curve to make it properly disconnected.

All of this is to make the point that counterexamples can often be removed. One can tell a similar story with the Hawaiian ear ring not being semilocally simply-connected and thus does not admit a universal covering space. However, if you specialize to CW-complexes, or manifolds, two common classes of spaces, you find that non-semilocally simply-connected spaces don’t come up. So is this counterexample one that should change our conception of spaces or one that should change our definition of spaces?

Like all things, the answer to these questions is contextual. It depends upon what you are interested in and what you want from your theory. For me, I do not have settled answers for these topological examples. I’m much less settled with regards to the analysis counterexamples discussed above. Are these counterexamples curiosities or do they manifest themselves in meaningful theory? If they do manifest themselves, where does that happen? If you have any thoughts, email me.

Footnotes: