You Can Count on It
By Bobby Neal Winters
I’ve been revisiting an old friend this semester, or should I say and old opponent; when you get older sometimes the two are the same. This isn’t a man or a woman. It’s a book: Topology by J. R. Munkres. I could quote Heraclites here and it would be half true. Even though it is a second edition, the part of the book I covered in class hadn’t changed. I have.
I only have a tithe of the energy I had in 1983 when I took the course from this book, and my mind is not as quick as it was in days of old. I do, however,have almost 30 more years of experience now so my energy is focused better. It’s like the joke told about the old bull and the young bull.
The young bull said, “Let’s run down the hill, jump over the fence, and breed a couple of those cows.”
The old bull countered this, “Let’s walk down the hill, go through the gate, and breed all of those cows.”
It’s about planning and priority.
I suppose my lower energy level makes my mind go more slowly. I pause over ideas I would’ve pushed on by before. I try to get the point of what I understood only on a technical level before.
Recently my mind has been focusing on the concept of the uncountably infinite.
Infinity is one of those concepts we really don’t get even when we “get it.” There is a gestalt of a sort when we play games with children asking them to name the biggest number they can.
“A thousand million billion trillion zillion,” they say.
“I can name a bigger one,” you reply.
“Uh, uh,” they reply.
“Oh, yeah,” you return. “What about a thousand million billion trillion zillion plus one?”
You better be careful doing this, because if they don’t have the gestalt the game can go on a long, long time.
Uncountability goes beyond this game of always being able to add one to get a bigger number. It is a stranger critter.
The first kind of infinity we encounter is the infinity of the so-called natural numbers, the numbers we use for counting: 1, 2, 3, and so forth. Sets that can be listed out, each element having a unique natural number for a label, are said to be countably infinite. We can count the numbers and will eventually get to every one of them even if there will never be a time when we counted all of them.
Look back at the last sentence. If you’ve read it carefully and are not a mathematician, you may feel a little green, but it’s written as I meant to write it.
A set is uncountable if you can’t list them in such a way that you will eventually get to each particular one of them. This can all be put in precise, technical language, I assure you.
Don’t make me do it.
To show that an uncountable set exists, all one has to do is construct a set all of whose elements can’t be listed. My favorite such set is the set of all sequences of zeros and ones.
A sequence is a list itself. A sequence of zeros and ones would be like the following: 1,0,1,0,0,1,0, 0, 0, and so forth. Note that here I’ve attempted to make something with a pattern. First a 1 and then one zero; then another 1 followed by two zeros; then 1 followed by three zeros. Continue in this way. These sequences needn’t have a pattern. I could have a sequence: 1,1, 0, 1,0, 1,1,1,1,0 and so for with no pattern. These sequences would be different.
Now I claim that it is impossible to list out all sequences like this even if the list goes on forever. The way I show that making this list is impossible is through the method of proof by contraction. I assume that I can but the show there is always at least one left over.
Assume there is such a list. Then make another sequence of zeros and ones whose nth item disagrees with the nth item on the nth list. Such an element is clearly not on the list and since the list was assumed to be exhaustive there is a contradiction.
You say you could just put it on top of the list. I say, “Bah! By your own lying words you claimed if was already there! Die you varlet.”
Sometimes doing this stuff makes me sound like I am talking to a pirate. Let’s just push on.
What I am going to say now will sound strange. What else is new? Anyway, I like this example because it is so concrete. Seriously. It is just lists of zeros and ones. School children can make lists of zeros and ones. I appreciate this now, better than when I was a kid those three decades ago, because of a theorem from Munkres I am going to teach my students tomorrow if they don’t derail class by bringing donuts or something, which is a constant risk with these youngsters.
The theorem states that if a topological space is compact and Hausdorff and contains no isolated points then it is uncountable.
You are no doubt saying to yourself, “Make sense to me.”
Okay, let me gloss that a bit. A topological space is a set with structure: its open sets. Compactness is a technical condition that gives us a certain type of control. Hausdorff, other than being the German work describing a dorff belonging to a particular haus, is a condition that gives us another sort of control. Again, I could make this even less readable by rolling out the technical definitions, but my point is these are simply abstract conditions, as is the condition of having no isolated points.
It is at the opposite end of the abstractness spectrum of our example of the set of sequences of ones and zeros, but, and here is the kicker, the spirit of the proof is the same. You assume the points of the space are in a list and then construct a point that is not in the list.
More interesting yet is the fact that--if I had a taser and another semester with these kids--we could use the same technique to prove the Baire Category Theorem which, to mind young, energetic mind of three decades ago was not related to either of these results.