Topology is sometimes described as geometry with beer goggles or geometry without your glasses on. Geometry is the study shapes: the way they lie in space, the way they interact with themselves and each other. Doing geometry usually requires measuring distance in some way. In topology, on the other hand, you can stretch or squish things with impunity, so the exact distance between any two points in a space doesn’t really matter. But sometimes geometry pops up where you don’t expect it. The geometry of an object can affect its topology.
The infinite earring, sometimes called the Hawaiian earring, is one of my favorite examples in topology because it illustrates some of this delicate interplay between geometry and topology.
To build the infinite earring, you start with the two-dimensional Euclidean plane and add a circle with center at the point (1,0) and radius 1. Now add the circle of radius 1/2 with center (1/2,0). Now add the circle of radius 1/3 and center (1/3,0). This pattern continues: the infinite earring consists of all the circles of radius 1/n and center (1/n,0) for all positive integers n.
In the end, it’s a bunch of nested circles that all touch at exactly one point, (0,0). But it’s more than that. It’s instructive to think of the infinite earring by thinking of what it is not. For example, it’s not the same as an infinite collection of circles that all meet in a point (we’ll call that a wedge or bouquet of countably many circles). This fact may come as a surprise because, after all, we could just make the wedge of circles into the infinite earring by shrinking the circles by different amounts and shoving them into the plane, or we could go the other way by blowing up the small circles of the infinite earring. Even though those transformations are pretty tame, the two spaces are not completely topologically equivalent.
The circle is a topological space in and of itself, and when we talk about an infinite collection of circles, we aren’t requiring them to sit in any particular ambient space. The fact that the infinite earring sits in the plane affects its topology. We can’t fit the wedge of countably many same-sized circles in the plane because the circles will start overlapping in ways we don’t want.
Proving that two spaces are topologically the same is tough, but it only takes one difference to prove that they are different. The lowest-hanging fruit for proving that the infinite earring is different from the bouquet of countably many circles is zooming in on points and seeing what you get. If the two spaces were equivalent, then when you zoomed in on corresponding points, you’d see things that were topologically equivalent. The specialest point in either space is the wedge point where all the circles come together. If we just look at a small region around that point, we’ll find our difference.
In the bouquet of circles, a small neighborhood of the wedge point is just made up of small neighborhoods of one point on each circle. It’s like a handful of spaghetti. (A lot of spaghetti.)
A small region around the point (0,0) in the infinite earring is a small circle in the Euclidean plane. It has some finite radius, and that radius is larger than some of the radii of circles in the earring. In fact, it’s larger than an infinite number of them. So the circle contains infinitely many circles from the earring. In pasta, it’s a finite number of spaghetti noodles with infinitely many (very tiny) Spaghetti-O’s thrown in.
The difference between the infinite earring and the bouquet of circles manifests itself in other complicated and delightful ways, but I think I’ll save those for later. Right now, I want to think about another space that isn’t the same as the infinite earring, this time a space that's also sitting in the plane.
If we build the circles out instead of in, so they have integer radii 1, 2, 3, and so on (perhaps we should call it the more infinite earring?), we get another space that is fundamentally different from the infinite earring. There are several topological properties that distinguish the two spaces. The easiest to see is that the infinite earring is bounded—it all fits in a box in the plane—and the more infinite earring is not.
Another difference between the two infinite earrings is that one space is closed and one isn’t. There are several equivalent definitions of being a closed set in the plane. One of them is that any point that is really close to points in the set is itself in the set. The more infinite earring doesn’t contain any points on the y-axis other than (0,0), but as the circles in the earring get bigger and bigger, they look more and more like a straight line, so they get closer and closer to the y-axis. When we take the whole infinite space, all the points on the y-axis are extremely close to points in the more infinite earring, but they never quite touch them.
The less infinite earring, on the other hand, doesn’t have this problem. The circle of radius one is the largest one in the bunch, so that circle puts a limit on how close the y-axis can get to the circles.
It took me a while to accept the fact that the infinite earring was really so different from the bouquet of circles and the more infinite earring, that the geometry of a space could really have such an effect on the topology. I think I am not alone among topology students in feeling like I was taking someone else’s word for it instead of grokking it fully. But in times like that, I comfort myself with a quote from John von Neumann: “in mathematics, you don’t understand things. You just get used to them.”
Read about more of my favorite spaces:
The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Line with Two Origins
The House with Two Rooms
The Fano Plane
The Möbius Strip
The Long Line