Filters IV: compactifications

The Stone-Čech compactification: a reminder.

Remember from Exercise 6.7.23 that every T3 1/2 topological space X can be embedded in a compact T2 space, called its Stone-Čech compactification ßX. This has the universal property of being the free compact T2 space over X as a T3 1/2 space.

More elementarily, every subspace of a compact T2 space must be T3 1/2, and every T3 1/2 space embeds into a compact T2 space.  ßX is in a sense the largest such completion, and X is dense in ßX.

As a free object, ßX 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 ßX, 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 UX of all ultrafilters of subsets of X.  This will work out as expected, and provide another implementation of ßX, 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 ßN 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 UX 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 Ui is (⋂i Ui), so the subsets U form a base for a topology.

I claim that with this topology turns UX into a compact T2 space.  This can be checked by hand:

  • [T2] 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 UX is T2.
  • [Compact] we shall use a handy gadget: the Kowalsky sum operation.  This is a map ♭ : UUX → UX, defined by F={AX | A is in F}.  Yes, I know, this is hard to digest: F is an ultrafilter of subsets of the set UX of ultrafilters of subsets (!).
    This map ♭ has plenty of properties, but let us concentrate on one in particular: for every ultrafilter F of subsets of UXF is a limit of F.  Indeed, it suffices to show that any basic open neighborhood U of F is in F… but this is completely mechanical: since F is in U, U is in F [definition of #], so U is in F [definition of F].
    In particular, every ultrafilter F of subsets of UX has a limit.  We have seen in Filters, part I that that meant that UX was compact.

We can embed X into UX, too.  Define η : XUX 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!):

  • ♭ : UUX → UX is continuous: the inverse image of U is U♯♯;
  • U is a functor; on morphisms (functions) f : X Y, it acts by: Uf : UX UY maps every ultrafilter F in to its image filter f[F];
  • ♭ is natural in X, that is, ♭ o UUf = Uf o ♭ for every map f;
  • in fact η is also natural in XUf o η = η o f;
  • The composite ♭ o η : UX → UX is equal to the identity map;
  • The composite ♭ o Uη : UX → UX is also equal to the identity map;
  • The composites ♭ o ♭ and ♭ o U♭ : UUUX → UX 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 UX is also (isomorphic to) the Stone-Čech compactification ßX, it suffices to show that it satisfies the universal property of being the free compact T2 space over X, since all solutions to a universal property are isomorphic.  Consider any continuous map g : XY, where Y is compact T2.   We must show that it extends to a unique continuous map g’ from UX 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 UX, let F=Uη(F); this is an ultrafilter of subsets of UX, which converges to F=♭ o Uη(F)=F.  If g’ exists, and is continuous, it follows that the image filter g’[F]=Ug’ (F) = Ug’ (Uη(F)) = U(g’ o η) (F) = Ug (F)=g[F] must converge to g’(F) = g’(F).  Since limits are unique in the T2 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 X is a discrete space, the space of ultrafilters UX is a Stone-Čech compactification of X, 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 UX=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.

Jean Goubault-Larrecqjgl-2011

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