On Cantor Sets (Summary)

To earn my M.S. at Oregon State, I produced an expository paper under the advice of ::Dr. Dennis Garity::, a topologist. My paper is about a special kind of set of points with some very interesting and surprising properties. On this page, I will try to summarize the big ideas. Click the link below to read the full paper.

Before we start, if you're not quite sure what is up with the concept of Infinity, you may want to leaf through this other cute little ::write up:: that I made about sizes of Infinity. There are important concepts about countable and uncountable infinities, as well as information about functions. Also, as convention, I will set aside some of the more complex stuff in italics.

What is Topology

Here are a few 'basic' ideas. First, we should talk about ::Topology::. Topology is not mapmaking, that's topography, but the root words are the same. Topography is pictures of maps, or surfaces, whereas topology is study of surfaces, and shapes in general. In topology, we don't care about angles or distances, all we care about is shape. The classic example (read: joke) is that a topologist cannot tell the difference between a coffee mug and a donut. Both of them have one hole, and if they were made out of silly putty, we could transform one into the other without breaking or tearing the putty. In fancy language, 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 is continuous. Furthermore, the inverse function must be continuous. This just says that I can make this transformation from coffee mug to donut nicely, and I can go backwards also from donut to coffee mug. So in the homeomorphic sense, these objects are 'the same'. In contrast, I can't make a pancake into a coffee mug because I have to rip a hole in it, so these objects are not 'the same'.

For those not faint of heart when it comes to a bit of abstraction, we can apply these same concepts to arbitrary sets endowed with a particular structure called a topology, which essentially determines which functions are continuous by defining what the workd 'nearby' means, while still skirting any concept of 'distance'. Anyway, what I'm trying to say is that we can still talk about topology of sets that aren't easily thought of to have 'shape'. In this way, topology is truly interested in shapes of things.

Homotopy

Next, we explore the equivalence of loops on surfaces. This is called ::Homotopy Theory::. Consider a piece of paper. Draw a dot on the paper, then draw any loop on the paper that starts and finishes at that dot, but does not intersect itself. We like to explore the feasibility of 'retracting', or shrinking this loop back to a point, sort of like a very small rubber band. If it can be contracted, then the loop is called 'null-homotopic'. But now imagine that your paper has a hole in the middle, and a rule that the loop can't be pulled accross the hole. Any loop that doesn't go around the hole can be contracted, but any loop that goes around the hole can't be. So these two types of paths are different. Two loops that can be transformed into one another are considered to be the same, homotopically speaking. This is a way to decide whether shapes are different.

For example, consider the surface of a sphere. Any path that we draw on it can easily be contracted back to a point because there is nothing in the way. What about on the surface of a donut (aka torus)? Well, we have 3 different kinds of loops. One brand of loop goes through the center hole. One brand goes around the center hole. The third does neither. It should just take a bit of pondering at a ::Top Pot:: donut that none of the three can be transformed into one of the others. This shows that the Torus and the Sphere are different, because they have a different set of kinds of loops that live on their surfaces. So in the language of above, a torus and a sphere are not homeomorphic. However, a donut hole and a donut are equally delicious.

A Bit of Set Theory

First, we define a set as a collection of objects. A set of people, a set of baseball cards, a set of moral values, etc. Given two sets we can look at what is in the intersection, that is, what is in both sets. This should probably conjure up images of venn diagrams. The intersection is the bit where the circles all overlap.

Here's a rigorous definition: Definition: A set is perfect if for every point x in the set, and any arbitrarily small number e, necessarily there is another point in the set, 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 set of real numbers (any decimal) is perfect. Pick a number, any number. Now find a number that is within e = .000000002 away from that number. You can always do it no matter how small e is. On the contrary, the counting numbers are not perfect. Pick your favorite counting number. Find another counting number that is only a distance of e = 1/2 away from it. You can't do it!

Here's one more definition: Definition: A set is totally disconnected if the largest connected components are points.

In the examples above, the counting numbers are totally disconnected, it's just a bunch of points, whereas the real numbers are connected, since they are a continuum.

It may be tempting to think that a set that is perfect (i.e. points as close together as you can imagine) cannot possibly be disconnected and visa-versa. But, it turns out this is not true. A set being totally disconnected says more about the points that are not in teh set. There are enough points that are not in the set so there is not a 'continuum', or line segment in the set.

Consider the rational numbers. They are totally disconnected because between every two rational numbers, I can find an irrational number. However, I can find rational numbers as close together as I'd like. So the rational numbers are both totally disconnected and perfect.

We will see that this property is one of the things that makes the Cantor Set so cool.

The first question to ask is maybe a dumb one, are there even any points in the Cantor Set? Surprisingly, yes. Notice that the two endpoints of the first stage never get removed. Then the new endpoints of the second stage never get removed in any subsequent stage. Its easy to see that once you're an endpoint, you're an endpoint all the way, and you never get taken out. So Yes. There are points left, and infinitely many at that.

Lets look at some of its crazy properties. In our original line segment, the amount of points is infinte, and it is actually a type of Infinity known as 'uncountable'. For more info see this ::write-up:: on infinity. It turns out that not only are there infinitely many points in the Cantor Set, but there are uncountably many. That is a whole lot of points. That's as many points as there are real numbers!

To see this, we can give each point an address, or a set of directions to follow to arrive at this point. At stage one, we must either go into the left or the right third. If we have to go left, write '0' and if we have to go right, write '1'. At each stage we must make a similar decision, so we can write down a digit. We churn out an infinite string of 0's and 1's, each of them unique. From this rule, it is 'not difficult' to show that the middle thirds Cantor Set is homeomorphic to the set of words in Binary Code, which can represent all real numbers. This is an example where, by way of topology, look at the 'shape' of binary code! Once we draw this relationship, we can show that the Cantor Set is uncountable, and thus has the same size as the oringal line segment. How crazy is that?

You might say, "But we took some of it out!!" Let's find out how much we actually took out. Notice at stage 1 we removed 1 piece of length 1/3. At stage 2 we removed 2 pieces of length 1/9. At stage 3, we removed 4 pieces of length 1/27. You begin to see a pattern that at stage n, we remove 2^{n-1} pieces of length 1/3^n. If you're savvy with your geometric sums, you'll remember that

∑(2^{n-1}/3^n) = ∑(1/3)(2/3)^n = (1/3)*(1/(1-2/3)) = 1.

We've removed the entire line segment. But we still have the same number of points left. How's your head?

Further, the Cantor Set is both Totally Disconnected and Perfect. That is, there is nothing bigger than a point, but all of the points are as close as you'd like to other points. Here are some proofs:

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 in the set, x, and some arbitrary small length, e. Now pick K large enough so that each of the line segments are smaller than e. That is, pick K so that (1/3)^K < e. Then, at stage K, x is definitely within e distance of an endpoint. Since x and e were arbitrary, that is, we didn't impose any conditions on them other than x is in the set, this is true for ANY point in the set. So the Cantor Set is Perfect.

Now we would like to show that the Cantor Set is totally disconnected (i.e., nothing bigger than points.) Suppose that there is something in the Cantor Set bigger than a point, i.e. a line segment. Suppose that x and y are two points such that they are in a line segment together. If they are different points, then there is a distance between them, call it D. Choose K so that all of the segments in stage K are smaller than D. That is, pick K big enough so that (1/3)^K < D. But that makes it impossible for x and y to be in the same line segment. Since x and y were arbitrary, we have that NO TWO POINTS can live in the same line segment. Hence, the Cantor Set is totally disconnected.

So that was a bit technical, but the point is that the Cantor Set has these two seemingly contradictory properties.

As it turns out, all sets that are both totally disconnected and perfect are homeomorphic to eachother (read: topologically the same). Also, anything to which the Cantor Set is homeomorphic also has these two properties. So in some sense, there is really only ONE Cantor Set. Enter Louis Antoine, a blind Mathematician active in the early 20th Century. In 1924 he introduced a version of the Cantor Set that has meaningful differences from the Middle-Thirds Cantor Set. The way it is embedded in space is more pathological, and it is colloquially known as a 'wild' Cantor Set.