The Stone-Čech Compactification – Part 1

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.


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).

The Stone-Čech compactification

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.

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