# Two Compactification Theorems

A Brief History of Spread-out Spaces.

Unless you’ve studied some topology, you probably haven’t heard of a “compact space.” This is a type of space (I’m leaving that term vague for now) that has a lot of really nice properties.

In the early 1900’s, people wanted to understand how to generalize many of the most useful theorems from Calculus to more general situations, such as the Extreme Value Theorem (continuous functions on closed intervals attain their maximum and minimum values).

Compact spaces tend to take up a major portion of a first course on topology, so we definitely will not get into all the subtleties. The point is that tons of theorems have been proved about compact spaces, and they’re generally easier to work with.

But most spaces aren’t compact, and so there’s a whole class of theorems trying to figure out when and how you can “compactify” them, and what sorts of properties are preserved after doing this. I’ll give more details on that later.

This article will provide some general idea of what a compact space is, and then go through the two most famous compactification theorems.

# Compact Spaces

The definition of a compact space is quite abstract if you look it up online or in a textbook. The definition gives you pretty much no sense of what it is. We’ll use a definition that is almost equivalent (it’s equivalent in all but the most bizarre situations).

We’ll say a space X is ** compact** if any infinite collection of points accumulates somewhere (in the space). In other words, if you try to “spread out” infinitely many points, you can’t do it. They will bunch up somewhere.

This way of thinking of it actually makes the term “compact” sound like the right term.

Here are some examples.

The real number line is not compact because you can spread out infinitely many points by putting them on the integers.

A finite line segment containing the endpoints is compact. It should be intuitively obvious that if you try to put infinitely many points in a finite space like that, then they will bunch up somewhere. Proving it is a little subtle (and would require a more rigorous definition of “accumulation point”).

A finite line segment not containing the endpoints (an open interval) is not compact. Take for example (0,1).

The reason this is not compact is that the infinite collection of points given by 1/n (starting at n=2) accumulates to 0, but 0 isn’t in the space!

The set {1/2, 1/3, 1/4, 1/5, …} stays spread out in the context of this space in the following sense. Given any point of (0,1), I can find a small area of the space surrounding that point containing no element of the set.

Notice we can’t do that with the closed interval that contained the endpoints because 0 is in the space and no matter how small the area I pick near 0, it will contain ** infinitely** many points of {1/2, 1/3, …}.

Now, I know someone will object here because (0,1) doesn’t ** feel **spread out in any traditional sense. That’s true. This is just trying to give you a sense of the definition of compact.

It turns out that when you make spaces out of the standard real number coordinate system, ℝⁿ, there’s a very simple way to classify compact spaces. They are precisely the ** closed** and

**subsets.**

*bounded*** Bounded** just means you can contain the whole thing is some big ball. It makes sense that bounded is needed for compact because if the subset went off to infinity, we could spread out our points in that direction without them accumulating.

** Closed** is a subtler topological notion. It basically means that any time some points accumulate, that accumulation point is actually in the set.

When you use these definitions, it’s basically a tautology that compact is equivalent to closed and bounded. But I’ll take this moment to remind you again that these aren’t quite the definitions you’ll find in textbooks.

In the real world, many “spaces” that we care about don’t come about by carving out a set in ℝⁿ. I wrote about many of these here:

The Brilliance of the Yoneda LemmaA beautiful revolution of the concept of “space”medium.com

In these more abstract cases, it can be hard to tell if a space is compact. Figuring out equivalent and easy-to-check conditions for compactness is a big chunk of many topology textbooks, so let’s move on.

Most spaces aren’t compact, and so we want to figure out how we can put the space inside of a compact space in an understandable way. This process is called ** compactification**.

# One-Point Compactification

There’s a fairly simple compactification process called the ** one-point compactification**. It has some serious downsides, but it’s a good place for us to get started.

It is exactly what it sounds like. It’s a way to add a single point to a space to turn it into a compact space. The standard example is to think about how to do this for the real number line.

Recall, we said this wasn’t compact because it goes off to infinity. Well, what if we just add a point at infinity? There’s a great way to visualize this by turning it into a circle. You wrap the number line around the circle.

There’s a formula to make this precise and see that we’ve actually only added a single point. It’s called ** stereographic projection**.

If you understand the picture, you should be convinced, though. When you draw a straight line from the top of the circle and find where it intersects the circle, you will get every single point of ℝ exactly once and then you just add the point at infinity. This shows the one-point compactification of ℝ is a circle.

The more fun part is to think about why the circle is actually compact. Try to think of a way to put infinitely many points on the circle without them bunching up. As you try to do this, you’ll start to get a feel for compact spaces better.

Given some mild conditions on the space X, it turns out you can always add one point to X, to get a compact space. Pavel Alexandroff proved this in 1924, and so this is sometimes called the ** Alexandroff extension** of X, though most people will just say

**.**

*one-point compactification*Even though this is the minimal construction possible, it is actually not very good for technical reasons having to do with how strange topological spaces can get. But that’s beyond the scope of this article.

# Stone-Čech Compactification

The Stone-Čech compactification was first explicitly written down in 1937 by Eduard Čech and then refined the same year to the form that is usually presented by Marshall Stone, though Tychonoff refers to such a construction as early as 1930.

The idea is quite simple and clever, though hashing out the details is extremely subtle and difficult (it earned Čech a paper in the *Annals of Mathematics*).

Here’s a key preliminary fact: Taking a product of two compact spaces is still compact. This isn’t very hard to prove.

You know about product spaces if you understand ℝ², which is just a copy of the real numbers ℝ with itself. Product spaces basically let you make a new space with the old spaces as the “axes.”

As we said, ℝ² is not compact because it is not bounded.

But the simple example we started with of the closed interval [0,1] *is* compact. Taking the product of it with itself, [0,1]², gives a solid, closed square if you can imagine that. Then [0,1]³ is a solid 1x1x1 cube.

Here’s the leap. It turns out that any arbitrary product of compact spaces is still compact (called the *Tychonoff Theorem*). This theorem is extremely hard to prove, so don’t be fooled by how “obvious” it might sound.

Given some space X, we cleverly find a copy of X inside an incredibly huge product of [0,1] with itself (it’s a product over an uncountably infinite set). Then we just take the closure of X in this. Since a closed subset of a compact set is compact, we know the end result is compact.

You might be thinking this is a horrible way to do it compared to the one-point compactification, but it has a really nice property. It turns out to be a “universal” construction. Every other compactification is essentially found inside this one.

This is amazing, because it says that even if you find some other nice method to compactify your space, it has a very close relation to the Stone-Čech compactification.

Now that we’ve seen the minimal and maximal compactifications of a space, you should have a better idea of what they are and how they come about.