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 be a locally compact Hausdorff space. A ** compactification** of is a compact Hausdorff space and a continuous function such that is a homeomorphism between and and is an open, dense subset of . Thus a compactification is a way of nicely embedding the space into a compact space .

For example if is the real line , then the circle and the stereographic projection is a compactification of . In this case the image of 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 will in general have many compactifications and we would like to compare these different compactifications. Suppose that and are two compactifications. Then a ** morphism **from to is a continuous map such that . That is, the below diagram commutes:

Hence we have a morphism from to precisely when the embedding extends to a map . Since is dense in , if such a function exists, it is unique.

Note that the map must be surjective since is dense in and is a closed set containing . We can think of the compactification as being “bigger” or “more general” than the compactification as is a surjection onto . More formally we will say that is ** finer **than and equivalently that is

**than whenever there is a morphism from to . 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 and are**

*coarser***if there exists a morphism between and such that is a homeomorphism between and .**

*isomorphic*For example again let . Then the closed interval is again a compactification of with the map which maps onto the open interval . We can then create a morphism from to the circle by sending the endpoints of to the top of the circle and sending the interior of 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 to the compactification . Thus the compactification is finer than the compactification .

Now that we have a way of comparing compactifications of we can ask about the existence of extremal compactifications of . Does there exists a compactification of 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 . 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 with the property that for all compactification there is a unique extension of . 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 is constructed by adding a single point denoted by to . It is defined to be the set with the topology given by the collection of open sets in and sets of the form for a compact subset. The map is given by simply including into .

The one point compactification is the coarsest compactification of . Let be another compactification of . Then the map given by

is the unique morphism from to .

## The Stone-Čech compactification

A * Stone-Čech compactification* of is a compactification which is the finest compactification of . That is is an initial object in the category of a compactifications of and so for every compactification there exists a unique morphism from to . Thus any embedding , has a unique extension . As with coarest compactification of , the Stone-Čech compactification of is unique up to isomorphism and thus we will talk of

*the*Stone-Čech compactification.

Unlike the one point compactification of , there is no simple description of even when is a very simple space such as or . To show the existence of a Stone-Čech compactification of any space 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.

## 2 thoughts on “The Stone-Čech Compactification – Part 1”