**The Stone-Čech compactification: a reminder.**

Remember from Exercise 6.7.23 that every T_{3 1/2} topological space *X* can be embedded in a compact T_{2} space, called its Stone-Čech compactification ß*X*. This has the universal property of being the *free* compact T_{2} space over *X* as a T_{3 1/2} space.

More elementarily, every subspace of a compact T_{2} space must be T_{3 1/2}, and every T_{3 1/2} space embeds into a compact T_{2} space. ß*X* is in a sense the largest such completion, and *X* is dense in ß*X*.

As a free object, ß*X* is necessarily unique up to isomorphism, but can be implemented in several ways, at least in principle. Exercise 6.7.23 gives one possible implementation of ß*X*, as a subspace of the compact space obtained as the product of as many copies of [0, 1] as there are continuous maps from *X* to [0, 1]. This is natural considering the definition of complete regularity. There are other implementations, based on filters, which we shall explore.

**Spaces of ultrafilters.**

The most well-known implementation of this program is by building the space **U***X* of all ultrafilters of subsets of *X*. This will work out as expected, and provide another implementation of ß*X*, only when *X* is a discrete space. But that is a good start, and anyway the case *X*=**N** (the set of natural numbers) is already an intriguing beast. I will not even try to explain what ß**N** looks like; Jan van Mill calls it the `three-headed monster’, and this should be enough to scare you away from trying to understand it finely.

So let us fix a discrete space *X*; in other words, a set, with the discrete topology. For a subset *U* of *X*, let us write *U*^{♯} for the subset of **U***X* of all those ultrafilters *F* of subsets that contain *U*. In other words, *F* is in *U*^{♯} if and only if *U* is in *F*. The intersection of finitely many sets *U _{i}*

^{♯}is (⋂

*)*

_{i}U_{i}^{♯}, so the subsets

*U*

^{♯}form a base for a topology.

I claim that with this topology turns **U***X* into a compact T_{2} space. This can be checked by hand:

- [T
_{2}] If*F*and*F’*are two distinct ultrafilters, then there is a subset*A*in*F*and outside*F’*, say. Then*F*is in*A*^{♯},*F’*is in (*X*\*A*)^{♯}, and*A*^{♯}and (*X*\*A*)^{♯}are disjoint open subsets; so**U***X*is T_{2}. - [Compact] we shall use a handy gadget: the
*Kowalsky sum*operation. This is a map ♭ :**U****U***X***→ U***X*, defined by*F*^{♭}={*A*⊆*X*|*A*^{♯}is in}. Yes, I know, this is hard to digest:*F*is an ultrafilter of subsets of the set*F***U***X*of ultrafilters of subsets (!).

This map ♭ has plenty of properties, but let us concentrate on one in particular: for every ultrafilterof subsets of*F***U***X*,*F*^{♭}is a limit of. Indeed, it suffices to show that any basic open neighborhood*F**U*^{♯}of*F*^{♭}is in… but this is completely mechanical: since*F**F*^{♭}is in*U*^{♯}, U is in*F*^{♭}[definition of #], so*U*^{♯}is in[definition of*F**F*^{♭}].

In particular, every ultrafilterof subsets of*F***U***X*has a limit. We have seen in Filters, part I that that meant that**U***X*was compact.

We can embed *X* into **U***X*, too. Define η : *X* → **U***X* by: η(*x*) is the principal ultrafilter at *x*, namely the filter of all the subsets of *X* that contain *x*. (I had written it (*x*) in previous posts, but this would pose some readibility problems here. A standard notation is *x* with a dot above it.) Check that this is a continuous map; in fact, the inverse image of *U*^{♯} is just *U*, which immediately shows that η is almost open; since η is easily seen to be injective, η is an embedding.

Before we proceed, we should mention some of the other properties of ♭. I will use some of them below (exercise!):

- ♭ :
**U****U***X***→ U***X*is continuous: the inverse image of*U*^{♯}is*U*^{♯♯}; **U**is a functor; on morphisms (functions)*f*:*X***→***Y*, it acts by:**U***f*:**U***X***→****U***Y*maps every ultrafilter*F*in to its image filter*f*[*F*];- ♭ is natural in
*X*, that is, ♭ o**UU***f*=**U***f*o ♭ for every map*f*; - in fact η is also natural in
*X*:**U***f*o η = η o*f*; - The composite ♭ o η :
**U***X***→ U***X*is equal to the identity map; - The composite ♭ o
**U**η :**U***X***→ U***X*is also equal to the identity map; - The composites ♭ o ♭ and ♭ o
**U**♭ :**U**U**U***X***→ U***X*are the same map.

All this means that (**U**, η, ♭) is a monad on the category of sets. A very nice structure, but this would carry us away somehow… apart from the fact that I’ll use the equations above anyway.

To show that **U***X* is also (isomorphic to) the Stone-Čech compactification ß*X*, it suffices to show that it satisfies the universal property of being the free compact T_{2} space over *X*, since all solutions to a universal property are isomorphic. Consider any continuous map *g* : *X* → *Y*, where *Y* is compact T_{2}. We must show that it extends to a unique continuous map *g’* from **U***X* to *Y*, namely that *g’*(η(*x*))=g(x) for every *x* in *X*.

Recall that a continuous map has the property that for every filter that converges to a point, the image filter converges to the image of the point. For every ultrafilter *F* in **U***X*, let *F**=***U**η(*F*); this is an ultrafilter of subsets of **U***X*, which converges to **F**^{♭}=♭ o **U**η(*F*)=*F*. If g’ exists, and is continuous, it follows that the image filter *g’*[* F*]=

**U**

*g’*(

*) =*

**F****U**

*g’*(

**U**η(

*F*)) =

**U**(

*g’*o η) (

*F*) =

**U**

*g*(

*F*)=

*g*[

*F*] must converge to

*g’*(

**F**

^{♭}) =

*g’*(

*F*). Since limits are unique in the T

_{2}space

*Y*, this shows that, if

*g’*exists, then it is unique. This also suggests to define

*g’*(

*F*) as the limit of

*g*[

*F*], which exists because

*Y*is compact. I’ll let you check that g’ thus defined is continuous, solving the whole question:

When

Xis a discrete space, the space of ultrafiltersUXis a Stone-Čech compactification ofX, i.e., it is isomorphic to ßX.

Oh, by the way, you probably cannot imagine any ultrafilter on X that would not be principal. This might lead you to conjecturing that **U***X*=*X* (up to η) when X is discrete. This is true when *X* is finite, but definitely wrong if *X* is infinite. Assuming that* X* is infinite, one can for example consider the cofinite filter Cof(*X*) on *X*, namely the family of all those subsets of *X* whose complement is finite. Using Zorn’s Lemma, there is an ultrafilter that contains Cof(*X*), and it cannot be principal: if it were equal to η(*x*) for some *x*, since it also contains the complement of {x} (which is in Cof(*X*)), it would contain the empty set, contradiction.

I’m stopping there for now. You’ve probably had enough for today. Next time, I’ll tell you about Wallman compactifications, a variant on this construction which also works (whatever that means) on non-discrete spaces. I claim this is best explained through Stone duality… You’ll have to understand Stone duality pretty deeply to understand my next post: sorry… but I’ll do my best to remain understandable.