The Stone-Čech Compactification

Mathematics is full of surprising connections between two seemingly unrelated topics. It’s one of the things I like most about maths. Over the next few blog posts I hope to explain one such connection which I’ve been thinking about a lot recently.

The Stone-Čech compactification connects the study of C*-algebras with topology. This first blog post will set the scene by explaining the topological notion of a compactification. In the next blog post, I’ll define and discuss C*-algebras and we’ll see how they can be used to study compactifications. In the final post we will look at a particular example.

Compactifications

Let $X$ be a locally compact Hausdorff space. A compactification of $X$ is a compact Hausdorff space $K$ and a continuous function $\phi : X \rightarrow K$ such that $\phi$ is a homeomorphism between $X$ and $\phi(X)$ and $\phi(X)$ is an open, dense subset of $K$ . Thus a compactification is a way of nicely embedding the space $X$ into a compact space $K$ .

For example if $X$ is the real line $\mathbb{R}$ , then the circle $S^1$ and the stereographic projection is a compactification of $X$ . In this case the image of $X$ is all of the circle apart from one point. Since the circle is compact, this is indeed a compactification. This is an example of a one-point compactification. An idea which we will return to later.

Comparing Compactifications

A space $X$ will in general have many compactifications and we would like to compare these different compactifications. Suppose that $(K_1,\phi_1)$ and $(K_2, \phi_2)$ are two compactifications. Then a morphism from $(K_1,\phi_1)$ to $(K_2,\phi_2)$ is a continuous map $f : K_1 \rightarrow K_2$ such that $\phi_2 = f \circ \phi_1$ . That is, the below diagram commutes:

Hence we have a morphism from $(K_1,\phi_1)$ to $(K_2,\phi_2)$ precisely when the embedding $\phi_2 : X \rightarrow K_2$ extends to a map $f : K_1 \rightarrow K_2$ . Since $\phi_1(X)$ is dense in $K_1$ , if such a function $f$ exists, it is unique.

Note that the map $f$ must be surjective since $\phi_2(X)$ is dense in $K_2$ and $g(K_1)$ is a closed set containing $\phi_2(X)$ . We can think of the compactification $(K_1,\phi_1)$ as being “bigger” or “more general” than the compactification $(K_2, \phi_2)$ as $f$ is a surjection onto $K_2$ . More formally we will say that $(K_1,\phi_1)$ is finer than $(K_2, \phi _2)$ and equivalently that $(K_2, \phi _2)$ is coarser than $(K_1, \phi _1)$ whenever there is a morphism from $(K_1, \phi_1)$ to $(K_2,\phi_2)$ . Note that the composition of two morphism of compactifications is again a morphism of compactifications. Thus we can talk about the category of compactifications of a space. The compactifications $(K_1, \phi _1)$ and $(K_2, \phi _2)$ are isomorphic if there exists a morphism $g$ between $(K_1, \phi _1)$ and $(K_2, \phi _2)$ such that $g$ is a homeomorphism between $K_1$ and $K_2$ .

For example again let $X = \mathbb{R}$ . Then the closed interval $[-1,1]$ is again a compactification of $\mathbb{R}$ with the map $(x,y) \mapsto \frac{x}{1+|x|}$ which maps $\mathbb{R}$ onto the open interval $(-1,1)$ . We can then create a morphism from $[-1,1]$ to the circle by sending the endpoints of $[-1,1]$ to the top of the circle and sending the interior of $[-1,1]$ to the rest of the circle. We can perform this map in such a way that the following diagram commutes:

Thus we have a morphism from the compactification $[-1,1]$ to the compactification $S^1$ . Thus the compactification $[-1,1]$ is finer than the compactification $S^1$ .

Now that we have a way of comparing compactifications of $X$ we can ask about the existence of extremal compactifications of $X$ . Does there exists a compactification of $X$ that is the coarser than any other compactification? Or one which is finer than any other? From a category-theory perspective, we are interested in the existence of terminal and initial objects in the category of compactifications of $X$ . We will first show the existence of a coarsest or “least general” compactification.

The one point compactification

A coarsest compactification would be a terminal object in the category of compactifications. That is a compactification $(\alpha X, i)$ with the property that for all compactification $(K, \phi)$ there is a unique extension $g : K \rightarrow \alpha X$ of $i : X \rightarrow \alpha X$ . If such a coarsest compactification exists, then it is unique up to isomorphism. Thus we can safely refer to the coarsest compactification.

The one point compactification of $X$ is constructed by adding a single point denoted by $\infty$ to $X$ . It is defined to be the set $\alpha X = X \sqcup \{ \infty\}$ with the topology given by the collection of open sets in $X$ and sets of the form $\alpha X \setminus K$ for $K \subseteq X$ a compact subset. The map $i : X \rightarrow \alpha X$ is given by simply including $X$ into $\alpha X = X \sqcup \{\infty\}$ .

The one point compactification is the coarsest compactification of $X$ . Let $(K,\phi)$ be another compactification of $X$ . Then the map $g : K \rightarrow \alpha X$ given by

$g(y) = \begin{cases} i(\phi^{-1}(y)) & \text{if } y \in \phi(X),\\ \infty & \text{if } y \notin \phi(X), \end{cases}$

is the unique morphism from $(K,f)$ to $(\alpha X, i)$ .

A Stone-Čech compactification of $X$ is a compactification $(\beta X,j)$ which is the finest compactification of $X$ . That is $(\beta X,j)$ is an initial object in the category of a compactifications of $X$ and so for every compactification $(K,f)$ there exists a unique morphism from $(\beta X,j)$ to $(K,f)$ . Thus any embedding $f : X \rightarrow K$ , has a unique extension $g : \beta X \rightarrow K$ . As with coarest compactification of $X$ , the Stone-Čech compactification of $X$ is unique up to isomorphism and thus we will talk of the Stone-Čech compactification.

Unlike the one point compactification of $X$ , there is no simple description of $\beta X$ even when $X$ is a very simple space such as $\mathbb{N}$ or $\mathbb{R}$ . To show the existence of a Stone-Čech compactification of any space $X$ we will need to make a detour and develop some tools from the study of C*-algebras which will be the topic of the next blog post.