I must confess that I used to feel a vague hostility towards the long line. But I decided to give it another chance after seeing this tweet from Mike Lawler:
my love for you is like the long line - similar to real love in most respects, just longer. #inspirationaltopology— Mike Lawler (@mikeandallie) September 2, 2015
In other words, the long line is the cheesy Valentine’s Day card of topological spaces, and if there’s anything I can get behind, it’s finding the love in mathematics, preferably in the cheesiest way possible.
As the name suggests, the long line is a really long line, somehow “longer” than the regular number line. We can think of the regular number line as a bunch of unit-long intervals laid end to end. Specifically, there’s one interval for every integer. The long line is the same thing, except there’s one interval for every real number instead.
At least it would be nice if that were true. The truth is weirder, and it takes us on a journey through some of the intricacies of set theory and infinity that many claim drove Georg Cantor crazy. You’ve been warned.
To define the long line, we need to talk about different sizes of infinity. When mathematicians talk about size, or cardinality, they use the idea of bijection: two sets are the same size if you can pair each element in the first set with exactly one element in the second set, and vice versa. In other words, instead of counting our fingers, we line up our two thumbs, two index fingers, and so on, to conclude that both of our hands have the same number of fingers.
When we get to infinite sets, strange things happen. There are the same number of integers as even integers, even (heh) though the even integers are a subset of the integers. We can line all the integers up on the left and the even integers on the right and pair them up so the number n on the left is paired with 2n on the right. We’ve found a bijection, so these two sets have the same size. With finite sets, on the other hand, you can’t find a bijection between a set and one of its subsets.
The set of all real numbers is certifiably larger than the set of integers, so we know there are at least two different sizes of infinity. In fact, there is a way to get a bigger infinity from a smaller infinity, so we can generate an infinity of infinities just starting from the infinity of the integers, called countable infinity.
What does this have to do with the long line? The long line isn’t actually defined as the concatenation of one unit-length interval for each real number. Instead, we find the smallest uncountable infinity and string that many intervals together.
At this point, we run smack into the continuum hypothesis. The continuum hypothesis states that the infinity of the reals is the smallest uncountable infinity. So my original description of the long line is accurate enough if the cardinality of the real numbers is the smallest uncountable infinity. If not, there is some infinity in between the integers and the real numbers, and the long line is made using that infinity instead. (For more on the continuum hypothesis and the long line, check out this pdf by Richard Koch.)
So is the continuum hypothesis true? Good news: you can have it either way! In 1963, Paul Cohen proved that the continuum hypothesis does not violate the Zermelo-Fraenkel axioms that form the foundations of mathematics. The negation of the continuum hypothesis does not violate them either. In other words, the continuum hypothesis is independent of the foundations of mathematics. You can’t prove either it or its negation using the other axioms of mathematics. Some people believe that means we haven’t found the right foundations yet, but I tend to sympathize with those who think it means we have the freedom to choose between different valid systems.
Whether or not we decide to accept the continuum hypothesis, we have a lot of long line to deal with. What is it good for? Like many of my other favorite spaces, it’s a counterexample, a space that was concocted to show exactly where you can break your favorite math tools. In this case, the long line shows us the dangers of having too much of a good thing. Basically, the long line is too big to do calculus on.
For reasons that get awfully technical, especially after you’ve just bent your brain trying to think about the smallest uncountable infinity, it’s easier to do calculus on spaces that satisfy three conditions: they “look” locally like Euclidean space of some dimension; they are Hausdorff, meaning you can tell points in them apart; and they are second countable, meaning you can build the space from a smallish (i.e. countable) number of sets. The long line violates this last requirement. Even though you might think it's basically the same as the real line, it has fundamental differences just because it's so long.
"How do I love thee? Let me count the ways..." doesn't sound so impressive when you think about the long line. "How do I love thee? I cannot count the ways because, like the sets that build the long line, they are truly uncountable" is a more romantic, though less poetic, way to express your affection.
Read about more of my favorite spaces:
The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Infinite Earring
The Line with Two Origins
The House with Two Rooms
The Fano Plane
The Möbius Strip