In case you were curious about Cantor Sets

This is my paper that I wrote for my masters, and I’m going to try to bring it back down to Earth. I’m going to try to give links to relevant topics that I think maybe need a bit more explaining.

::Let me first suggest that you read this other cute little write-up I did about sizes of infinity::.  Important concepts from that are ideas of uncoutable infinities and the notion of a bijective function.

First, we need to discuss a few “basic” concepts.  First, the question to answer, I think, is ::”What is Topology?”::  Topology is the study of shape without respect to distance or angles.  The classic example (read: joke) is that a Topologist cannot tell the difference between a donut and a coffee mug since both of them have one hole, and if you had a super-moldable coffee mug, you could change it into a donut without ripping or tearing the material.  Mathematically, we say that the donut and the coffee mug are ::homeomorphic::.  This means there is a bijective function that maps each point of the coffee mug to a point of the donut, and in addition, the map must be continuous (analogous to saying no ripping or tearing allowed) and furthermore, the inverse function must be continuous, where the inverse function sends points in the donut back to the points of the coffee mug that got sent to it.

Okay, so now we have some idea what it means for two shapes to be “the same.”  In contrast, a donut hole (a solid sphere) is not the same as a donut, since trying to mold a donut from a donut hole involves ripping and tearing.

For those who are not faint of heart when it comes to some abstraction, we can apply these concepts to arbitrary sets endowed with particular structure called a topology, which essentially determines which functions are continuous by defining what the word “nearby” means while still skirting any concept of “distance.”  Anyway, what I’m trying to say is that we can still talk about topology on sets that are so abstract that their “shape” is inconceivable to us three-dimensional Euclidean creatures.

Next, a tool that we use is ::homotopy theory::.  In a nutshell, this deals with equivalence of closed loops.  By a closed loop, informally, I mean take a long piece of malleable wire and bend it and arrange it in space so that the two ends meet.  Formally, a closed loop is a parameterization r(t), where t ranges from 0 to 1 and r(0) = r(1).  That is, we map a line segment into space so that the starting point and ending points are the same.  One particularly nice brand of closed loops are those that are homotopic to a point.  These are paths that can be crumpled to a point.  Imagine a tabletop.  If you draw any closed loop, it can be shrunk to a point.  Such loops are called null-homotopic.  In contrast, imagine a tabletop with a hole in the middle, say like the ones they use to eat monkey brains. If you draw a closed loop that encircles the hole in the middle of the table, there is no hope of shrinking the loop because the hole gets in the way.  In the same way, two different paths are homotopic to each other if they can be continuously deformed into one another.  Finding distinctions between homotopy classes of paths are an important tool in distinguishing surfaces.  For example, what kinds of paths can we have on the surface of a sphere? Well, any path that we draw can easily be contracted to a point because there is nothing getting in the way.  What about on the surface of a donut (henceforth referred to as a torus)? Well, there are paths that go around the hole, there are paths that go through the hole, and there are paths that do neither.  A little bit of envisioning these paths should be enough to convince you that none of them are homotopic to each other.  Hence, the torus has 3 different types of loops where the sphere only has 1.  This shows that the torus and sphere are not homeomorphic.  (This sounds like sort of a dumb example because, um, duh, the torus has a hole and the sphere doesn’t, but trust me it gets more complicated and subtle than that when doing more topology.)

Lastly, just a small bit of set theory. This first bit, I’m pretty sure everyone intuitively understands, but I’m going to mention it anyway.  Given two sets, we define the intersection of these sets to be the collection of points that are in both sets.  This should immediately conjure up visions of venn-diagrams where the intersection is the part where the circles overlap.  In fact, you can take intersections of any number of sets — even an infinite number of sets!  It just amounts to figuring out what points are in every single set.

Let me state a rigorous definition real quick:
Def: A set is perfect if for every point, x,  in the set and any arbitrarily small number e, necessarily there is another point, y, so that the distance between x and y is less than e.  This means that if you are standing at any point in the set, there is automatically another point as close to you as you’d like.  For instance, the real numbers (i.e. any decimal number) are perfect.  Say you are at the point 0 and you choose e = .00000001.  Then .000000001 is within distance e to you.  Its easy to see that this is true for any small number.  In contrast, the natural numbers (N={0, 1, 2, 3, 4, …}) are not perfect.  Say you are standing again at the point 0.  Set e = 1/2.  Surely there is not a natural number that is distance 1/2 away from 0.  Hence, the natural numbers are not perfect.

I think I should only have to give one more tedious definition.  Here goes:
A set is totally disconnected if the largest connected components are points.  In the examples above, the natural numbers are totally disconnected since they are just points on a line, whereas the real numbers are a continuum.

It may be tempting to think that any set that is perfect is not totally disconnected and visa-versa by the examples above, but that isn’t the case.  A set being totally disconnected says more about the points that are not in the set.  There are enough points that are not in the set so that there isn’t a “continuum”, or line-segment in the set.  For example, the rational numbers (fractions) are a totally disconnected set since between any two rational number there is an irrational number.  The rationals are also perfect since there is a rational number between any two real numbers.  We’ll see that this distinction is one of the things that makes the Cantor Set so cool.

Okay, now we start.
The Cantor Set was discovered by a guy named Smith in 1873, then by Cantor almost a decade later.  The one that you usually run into first is called the standard middle-thirds Cantor set.  It’s constructed like this:
1. Draw a line segment
2. Divide the segment into thirds
3. Delete the middle third
4. Rinse, Lather and Repeat steps 2-4 for each of the remaining line segments.

Here’s a picture:


In the language of above, The Cantor Set is the intersection of each stage.

Are there any points in the Cantor Set? Well, consider endpoints of the intervals.  At the next stage, only the insides of the intervals are removed and the endpoints are left, and they never get touched after that. So, at the very least, the endpoints of each interval is in the Cantor set.

Let’s look at some of it’s really crazy properties.
First, recall (from the infinity paper I suggested that you glance at), there is uncountably many points in our original line segment.  Turns out, that there are still uncountably many points in the Cantor Set.  To see this, we give each point an address.  Pick a point in the Cantor Set.  At stage one, is this point in the left division or the right division?  At stage two, is this point in the left or right subdivision of the previous subdivision? And so on.  Every time your point is in the left subdivision, write down the number 1.  Every time that your point is in the right subdivision, write down a 2.  From this rule, its “not difficult” to show that the Middle-thirds Cantor Set is homeomorphic to the set of all possible strings of 1’s and 2’s.  This is a prime example when we can look at what the Cantor set looks like, but can’t imagine what the latter set “looks like,” yet we can draw an equivalence between them.  Once we shown this, we can use a diagonalization argument (pg of infinity paper) to show that the Cantor set is uncountable as well.  That is, the Cantor set has the same number of points as the line segment we started with!!  How crazy is that!?  If your answer was really crazy, buckle your seatbelt and put on a helmet.

As we have it, the total length of the line segments we removed is equal to the original line segment.  At stage 1, we took away one segment of length 1/3.  At stage 2, we took away two segments of length 1/9. At stage 3, we took away four line segments of length 1/27.  In general, at stage n, we take away 2^n line segments of length 1/3^{n+1}, so the total length removed is \sum_{n=1}^\infty (1/3)(2/3)^n.  If you’re particularly savvy with your geometric sums, you’ll notice that this is equal to 1!  So what have we done? We have removed the “entire” line segment, but there are still the same number of points left.  Don’t worry, if that makes your head hurt, its okay.  Infinity is crazy!

Remember when we talked about perfect and totally disconnected sets?  Well the Cantor set is both of those things.  If you’re interested, here are the proofs below.

First, notice that at stage n, all of the line segments are of length 1/3^n.  Also, remember that whenever a point is an endpoint of a line segment at any stage, it is automatically in the Cantor Set.  So pick some arbitrary point x in the Cantor set and some arbitrary e.  Now pick n large enough so that 1/3^n < e.  Then at stage n, x is contained within a line segment of length less than e, and therefore, is within distance e to both endpoints in the line segment, and is therefore within distance e to at least one other point in the Cantor Set.  Since both x and e were arbitrary, we have that the Middle-Thirds Cantor Set is perfect.

The argument that the Cantor Set is totally disconnected has a similar flavor.  Suppose that x and y are in the same connected component of the Cantor set.  You can think of this as being able to draw a line from x to y without leaving the Cantor set.  Now, there is some distance between x and y, call it D.  Pick n such that 1/3^n < D.  Then at stage n, all of the line segments are of length 1/3^n and so it is impossible for x and y to be in the same line segment, which renders it impossible for x and y to be in the same connected component of the Cantor Set.  Since x and y were arbitrary, NO TWO POINTS are in the same connected component, and hence it is totally disconnected.

Phew, that got a little technical, but hopefully nothing too bad.  The point is that the Cantor Set has these two seemingly contradictory properties.

As it turns out, all totally disconnected perfect sets are homeomorphic to each other.  And also, anything that the Cantor set is homeomorphic to, is also perfect and totally disconnected.  So really, in some sense there is really only one Cantor set.  But in some other sense, there are as many different Cantor sets as there are points in the Cantor Set.  Enter Louis Antoine, a blind mathematician.  In 1924, he introduced the first example of a Cantor set that is non-standardly embedded, or a bit more colloquially known as a WILD CANTOR SET!!  I’ll explain what that means in a bit.

Antoine’s Necklace


Consider a solid torus in space.  Now, on the inside of that torus, place a chain of at least four linked tori.  On the inside of each one of those tori, place a chain of at least four linked tori.  Rinse, lather, repeat infinitely many times and then take the intersection of each stage.  By similar arguments as above, what we get is a totally disconnected, perfect set — essentially donut dust suspended in space.  By our comments above, this is a Cantor Set.  This set is homeomorphic to our original middle-thirds Cantor set.  That is, there exists a bijective function from the middle thirds Cantor set to Antoine’s necklace which is continuous and has continuous inverse.  Antoine’s Necklace, however, is wild because informally it does not “sit in space” in the same way as the middle thirds Cantor set.  The linked tori give it a strange, pathological /embedding/ into three-dimensional space.  The way one would show this is to note that if you thread a path through the center of the biggest donut, you cannot contract the path to a point (it may be helpful to go back and remind yourself about what path homotopies are).  By contrast, any closed path in three dimensional space that does not touch the middle thirds Cantor set can easily be contracted to a point by simply pulling the loop through the center gap of the middle thirds.

The stuff beyond the research that I did is generalizations of Antoine’s necklace.  One can construct Cantor Sets with several interesting embedding properties just by varying the number and arrangement of tori, or the number of holes in the tori at each successive stage.

As closing remarks, Cantor sets are of interest by themselves as really cool sets.  One place they show up in a useful applied sense is as the invariant sets of dynamical systems.  These are important systems of study on their own, as well as an important tool for applied mathematicians to solve partial differential equations, which are the math-stuffs of a lot of the physical phenomenon we experience in the world around us.

Anyway, If you are a friend, I hope you enjoyed this.  If you happen to stumble upon this for some reason and have questions, feel free to shoot me an email.

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