**The Wallman compactification.**

In Filters IV, we have shown that we could realize the Stone-Čech compactification ß*X* of a discrete space *X* as its space of ultrafilters of subsets **U***X*. The topology on the latter had the sets *U*^{♯} as a basis, where *U*^{♯} = {*F* in **U***X *| *U* in *F*}.

In 1938, Henry Wallman gave a more general construction [1], which applies to any T_{1} space *X*, and produces the Stone-Čech compactification ß*X* of *X* (up to isomorphism) whenever X is T_{4}. I will describe it in detail. Many extensions were given since thence, notably by Orrin Frink in 1964 [2], but I will not say anything about them.

Last time, I’ve also warned you that this post would be technical. I guess you’ll agree it holds to its promises. I’ve tried to postpone the hard, Stone-duality based material, as much as I could. Still, this post is going to be packed.

Instead of considering ultrafilters of subsets, that is, maximal non-trivial filters of subsets, Wallman considers maximal non-trivial filters in the complete lattice **H***X* of closed subsets of *X*. (The **H** is for Hoare: **H***X* is the so-called Hoare powerdomain over X, up to a small detail. I will talk about it another time.) Let us write **ω***X* for the subset of maximal non-trivial filters of closed subsets of *X*. I’ll say *Wallman filter* (over X) instead of “maximal non-trivial filter of closed subsets of X”, which starts to be lengthy.

We do as in Filters IV, replacing subsets by closed subsets, ultrafilters of subsets by Wallman filters, and **U***X* by **ω***X***.**

In **U***X*, we had defined *A*^{♯} as the set of ultrafilters of subsets that contained * A*, and these were the opens (and also the closed sets) of **U***X*. We would like to adapt this to **ω***X*. Reasoning with opens is awkward, here, however, since Wallman filters are filters of closed subsets. So we shall define *C*^{♯} only when *C* is a closed subset of X, and declare these sets *C*^{♯} to be *closed*. In other words, the *complements* of these sets form a subbasis for a topology on **ω***X*. We shall call it the *Wallman topology*.

** Theorem.** **ω***X* is a compact T_{1} space.

*Proof.* As for **U***X*, we do this with filters.

- [T
_{1}] If*F*and*F’*are two distinct Wallman filters, then there is a closed subset*C*in*F*and outside*F’*. Otherwise,*F*would be included in*F’*, hence equal to it by maximality. Then*F’*is in the open complement of*C*^{♯}, and*F*is not (i.e., F is in*C*^{♯}), by definition. By a symmetric argument, from an element of*F’*that is not in*F*, we build an open neighborhood of*F*that does not contain*F’*. - [Compact] As for
**U***X*, we shall use the Kowalsky sum, or a slight variant. We now define*F*^{♭}as the set {*C*in**H***X*|*C*^{♯}is in}. Using the fact that # commutes with intersections, we see that*F**F*^{♭}is a non-trivial filter. It may fail to be maximal, but, by Zorn’s Lemma,*F*^{♭}is contained in some Wallman filter*F*^{♭’}.

The rest of the argument for compactness is as in the discrete case:*F*^{♭’}is a limit of. It suffices to show that any basic open neighborhood*F**(*complementof a basic closed subset**U***C*^{♯}) of*F*^{♭’}is in… and this is similar to what we did in Filters IV: since*F**F*^{♭’}is in, it is not in**U***C*^{♯},*so C*is not in*F*^{♭’}[definition of #], in particular C is not in the smaller filter*F*^{♭}; it follows that*C*^{♯}is not in[definition of*F**F*^{♭}], and becauseis an ultrafilter, the complement**F**of**U***C*^{♯}is in.*F*

In particular, every ultrafilterof subsets of*F***ω**has a limit. So*X***ω***X*is compact. ◻︎

**Proposition.** If *X* is T_{1}, then *X* embeds into** ω***X* through the map η that sends every point *x* to the filter of all the closed subsets that contain *x*.

The assumption that *X* is T_{1} is necessary: if *X* embeds into **ω***X*, since **ω***X* is T_{1}, *X* must be, too. Note that I am reusing the notation η for the embedding. This not quite the same as the one of Filters IV, but close.

*Proof.* We use the fact that *X* is T_{1} to show that the non-trivial filter η(*x*) is maximal: if it was not maximal, then there would be a strictly larger filter *F*, and a closed set *C* in *F* that does not contain *x*; since cl(*x*)={x} is also in *F*, and *F* is non-trivial, the intersection C ∩ {*x*} cannot be empty, contradiction.

The inverse image η^{-1}(*C*^{♯}) is *C*, so the inverse image of every closed set is closed, hence η is continuous. The map η is injective: for *x*≠*y* in *X*, {*x*} is a closed set in η(*x*) that is not in η(*y*). Finally, the direct image of *C* is Im η ∩ *C*^{♯}, so the inverse η^{-1} : Im η → *X* is continuous (the inverse images of closed subsets by η^{-1} are closed). Therefore η is an embedding. ◻︎

To go further, we need the following observation: on any distributive lattice, every maximal non-trivial filter *F* is prime, that is, if *F* contains the sup of finitely many elements *a_{i}*, then it contains some

*a*. In particular, every Wallman filter is prime: a Wallman filter that contains a finite union of closed sets ∪

_{i}_{i=1}

*must contain one of them.*

^{n}C_{i}(Here is a proof for two closed sets *C* and *C*‘. I’ll let you generalize. We first show: (*) *C* intersects every element of F, or *C’* does. Otherwise, for every *C”* in *F*, *C* ∩ *C”* and *C*‘ ∩ *C”* are both empty; so (*C* ∪ *C’*) ∩ *C”* is, too [this is where distributivity is needed]; but this is impossible since *C* ∪ *C’* and *C”* are both in *F*, hence also their intersection, but *F* is non-trivial. Using (*), we show the claim that *F* is prime. By symmetry, we may assume that *C* intersects every element of F. Add *C* to *F*, and complete so as to obtain a filter: that is, consider the filter *F’* of all supersets of sets of the form *A* ∩ *C* with *A* in *F*. *F’* is a strictly larger filter than *F* because *C* was not in *F*, and is non-trivial because* C *intersects every* A* in* F*. This is impossible since *F* is maximal.)

It follows that (∪_{i=1}* ^{n} C_{i}*)

^{♯}= ∪

_{i=1}

^{n}C_{i}^{♯}, so the sets

*C*

^{♯}form a basis (not just a subbasis) of closed sets: every closed set in the Wallman topology is an intersection of such sets. (In fact, since # also commutes with finite intersections, every closed set is a

*filtered*intersection of such sets.)

Using this, we can show that every non-empty open subset ** U** of

**ω**

*X*intersects Im η.

**must contain a non-empty basic set, obtained as the complement of some**

*U**C*

^{♯}. Since this complement is non-empty, there is a Wallman filter

*F*that is not in

*C*

^{♯}. This means that

*C*is not in

*F*. In particular,

*C*cannot be the whole of

*X*. Let

*x*be a point outside

*C*. Then C is not in η(

*x*), so η(

*x*) is not in

*C*

^{♯}, hence in

*. This implies the following:*

**U****Lemma.** If *X* is T_{1}, and modulo the embedding η : *X* → **ω***X*, *X* is a dense subset of **ω***X*. ◻︎

This does not make **ω***X* a compactification of *X* yet, as we have defined compactifications as compact **T _{2}** spaces. And we know that the spaces that have compactifications are exactly the T

_{3 1/2}spaces. So we should at least assume that. We shall in fact require

*X*to be T

_{1}and normal, that is, T

_{4}.

This will require us to give an equivalent characterization of the Wallman topology. For an arbitrary subset *A* of *X*, let us define *A** as the collection of Wallman filters that *contain* a closed subset of *A*.

This allows us to define another topology on **ω***X*: the *Wallman* topology* has the sets *U** as basic open sets, when *U* ranges over the open subsets of *X*. Let us write **ω***X** for the space of all Wallman filters on *X* with the Wallman* topology.

**Lemma.** For every normal space *X*, **ω***X** is T_{2}.

Proof. Let *F*, *F’* be two distinct Wallman ultrafilters. There is a closed subset *C* in *F* and outside *F’*, say.

There must be a closed subset *C’* in *F’* that does not intersect *C*. Indeed, otherwise, the collection *F”* of all the closed subsets that contain the intersection of *C* with an (arbitrary) element of *F’* is again a non-trivial filter: since all the closed subsets *C’* in *F’* intersect *C*, all the elements of *F”* are non-empty; that *F”* is a filter is by construction. By Zorn’s Lemma, F” is included in a maximal non-trivial filter, and since *F”* is strictly larger than *F’* (as it contains *C*), this contradicts the maximality of *F’*. This proves the claim: there is a closed subset *C’* in *F’* that does not intersect *C*.

Since *X* is normal, we can find two disjoint open neighborhoods, *U* of *C*, and *U’* of *C’*. By construction, *F* is in *U**, and *F*‘ is in *U’**. It remains to show that the intersection of *U** and *U’** is empty: any Wallman filter *F”* that is in both should contain a closed subset of *U* and a closed subset of *U’*, hence also their intersection, which is empty; this is impossible since *F”* is non-trivial. ◻︎

**Proposition.** If *X* is normal, then the subsets of the form *U**, when U ranges over the open subsets of X, form a basis of the Wallman topology on **ω***X* — in other words, **ω***X*=**ω***X**. Moreover, **ω***X* is compact T_{2}.

Proof. We first check that *U** is open in the Wallman topology. Let *F* be a Wallman filter in *U**. By definition, it contains an element *C* that is included in *U*. By normality, we can find an open set U’ and a closed set C’ such that *C* ⊆ *U’* ⊆ *C’* ⊆ *U*. The complement * U* of (complement of

*U’*)

^{♯}is an open neighborhood of F: that means that the complement of

*U’*is not in

*F*, and indeed it cannot be in

*F*because F is non-trivial and

*C*⋂ (complement of

*U’*) is empty. Also,

*is included in*

**U***U**: for every Wallman filter

*F’*in

*,*

**U***F’*does not contain the complement of

*U’*, but contains (complement of

*U’*) ⋃

*C’*=

*X*; since

*F’*is a prime filter (see above: every maximal non-trivial filter is prime), it must contain

*C’*, and therefore

*F’*is in

*U**.

We have shown that *U** is a neighborhood of any of its points *F*. Therefore *U** is open (in the Wallman topology). This shows that the Wallman* topology is coarser than the Wallman topology. By the preceding Lemma, the Wallman* topology is T_{2}, and we know already that the Wallman topology is compact. It follows immediately that the two topologies are identical! Indeed, any compact topology finer than a T_{2} topology must coincide with it: see Theorem 4.4.27 in the book. ◻︎

We are slowly working our way toward proving that if *X* is T_{4}, then **ω***X* is (isomorphic to) the Stone-Čech compactification ß*X*. For now, we know that **ω***X* is compact T_{2}, and that *X* is dense in **ω***X*. This is a good omen! But this is not enough.

This is where we shall need to explore some facets of Stone duality — some new, some old.

**The Stone duality view.**

Let us first rephrase what we have done in terms of more familiar constructions — more familiar, at least, if you have read Chapter 8 and, possibly, Section 9.5 of the book.

Any filter *F* of closed subsets of *X* defines a set *I*, of all opens whose complements are in *F*: *I* = {*U* | complement of *U* in *F*}. One checks easily that *I* is an ideal in the lattice **O***X* of open subsets of *X*, namely a directed, downward closed set of opens; and that this construction defines an order isomorphism between the lattice of filters of closed subsets of X and the ideal completion **I**(**O***X*) of the lattice of open sets of *X*.

Let us rephrase **ω***X* through this isomorphism: **ω***X* is the space of all maximal non-trivial ideals of opens subsets of *X*. By non-trivial we mean those ideals different from the top ideal, **O***X* itself. And the topology of **ω***X* has as basic open sets the complements of *U*^{♯}={*I* in **ω***X* | *U* is in *I*}, where *U* ranges over the open subsets of *X*.

We have seen a similar construction in proving Johnstone’s Theorem (Theorem 9.5.14). There we considered the topological space **pt** **I**(**O***X*), and we shall look at it with a different view. The elements of **pt** **I**(**O***X*) can be described as the prime elements of **I**(**O***X*). Remember that, in a distributive lattice, the maximal non-trivial filters are all prime. Since **ω***X* is the set of maximal non-trivial elements of **I**(**O***X*), **ω***X* occurs as a subset of **pt** **I**(**O***X*).

Oh, for this to hold, we should check that **I**(**O***X*) is indeed a distributive lattice. And I said that **ω***X* occurs as a subset of **pt** **I**(**O***X*), not as a sub*space*, and we should check that as well.

For the first item, **O***X* is a distributive pointed lattice, so we can rely on Exercise 9.5.11, and conclude that **I**(**O***X*) is an algebraic fully arithmetic lattice. In particular, it is distributive, but this says much more! See Exercise 9.5.6: this means that **I**(**O***X*) is an algebraic, distributive lattice in which the top element (**O***X*) is finite and where the greatest lower bound of any two finite elements is finite.

By Exercise 9.5.5, **pt** **I**(**O***X*) is then a spectral space — whatever *X* is (no need for normality or for any separation axiom). This was one of our first steps in proving Johnstone’s Theorem, which states that *X* occurs as a retract of **pt** **I**(**O***X*) as soon as *X* is stably compact. (We shall make use of this later.) Of course, in the cases we are interested in here, *X* will be at best T_{4}, not stably compact.

Let us proceed to the second item.

**Lemma.** **ω***X* is a subspace of **pt** **I**(**O***X*) [not just a subset].

*Proof.* Let us look at the topology on **pt** **I**(**O***X*). For each ideal *J* in **I**(**O***X*), there is an open subset *O _{J}* in

**pt**

**I**(

**O**

*X*), defined as the set of prime elements

*I*of

**I**(

**O**

*X*) such that

*J*is not included in

*I*. (In the book, we defined

**pt**

**I**(

**O**

*X*) as a space of completely prime filters of elements of

**I**(

**O**

*X*). Using this presentation, a point

*y*, namely a completely prime filter, would be in

*O*if and only if

_{J}*J*is in

*y*. We are profiting from the fact that completely prime filters are exactly the complements of the downward closures of prime elements

*I*[see Corollary 8.1.21], yielding the above formula for

*O*.)

_{J}This topology on **pt** **I**(**O***X*) is generated by the open subsets *O _{↓U}*, where

*U*is in

**O**

*X*. This is because, for every ideal

*J*,

*O*is the union of the opens

_{J}*O*with

_{↓U}*U*in

*J*. Writing

*C*for the (closed) complement of

*U*,

*O*then appears to be the complement of the set of prime elements

_{↓U}*I*of

**I**(

**O**

*X*) that contain

*U*. By a legitimate abuse of language, write

*U*

^{♯}for the latter set, {

*I*in

**pt**

**I**(

**O**

*X*) |

*U*is in

*I*}. The latter therefore form a base of closed sets for the topology of

**pt**

**I**(

**O**

*X*).

Does this ring a bell? We had written *U*^{♯}, until now, for the basic closed sets {*I* in **ω***X* | *U* is in *I*} of the topology of **ω***X*. So our former sets *U*^{♯} are the intersection of **ω***X* with our new sets *U*^{♯}. It follows that **ω***X* is a subspace, not just a subset, of the spectral space **pt** **I**(**O***X*). ◻︎

We therefore have a sequence of maps:

- η :
*X*→**ω***X*, which (in our reading of**ω***X*as a space of maximal non-trivial ideals) maps*x*in*X*to {*U*open in*X*|*x*is not in*U*} - the inclusion
**ω***X*→**pt****I**(**O***X*).

Both are embeddings as soon as X is T_{1}. In fact, their composition from *X* to **pt** **I**(**O***X*) is always an embedding (provided *X* is T_{0}), and factors through **ω***X* if and only if X is T_{1}. We shall write the composite map, *X* to **pt** **I**(**O***X*), again, as η. This should not cause any confusion.

Let us come back to our original problem. We would like to show that **ω*** X* is a Stone-Čech compactification of

*X*. For this, it is enough to show that it satisfies the universal property of Stone-Čech compactifications: every continuous map

*g*:

*X*→

*Y*, where

*Y*is compact T

_{2}, has exactly one continuous extension

*g’*from

**ω**

*to*

*X**Y*, where by extension we mean that

*g’*(η(

*x*))=g(x) for every

*x*in

*X*.

Uniqueness is easy, because Im η is dense in **ω***X*. Equating *x* with η(*x*) in **ω***X*, and therefore seeing *X* as a dense subspace of **ω***X*, we have that if *g’* exists, then for every *F* in **ω*** X*,

*F*will be a limit of some net (

*x*)

_{i}

_{i}_{ in }

*of elements of*

_{I, ⊑}*X*, so

*g’*(

*F*) will be the limit of (

*g*(

*x*))

_{i}

_{i}_{ in }

*; and this limit is unique because*

_{I, ⊑}*Y*is T

_{2}.

The real problem is showing that *g’* exists. Oh well, you might say: the previous paragraph gives us a formula for *g’*! However, it is almost unusable. We don’t even know whether *g’*(*F*) thus defined is independent of the chosen net (*x _{i}*)

_{i}_{ in }

*. Replacing nets by filters does not help much here: try it for yourself if you are not convinced.*

_{I, ⊑}However, there is a very principled way of showing that *g’* exists, through Stone duality. We show the following more general result first. This is actually a consequence of the already cited Johnstone Theorem.

**Theorem.** Every continuous map *g* : *X* → *Y*, where *Y* is stably compact, has at least one continuous extension *g’* from **pt** **I**(**O***X*) to *Y*, where by extension we mean that *g’*(η(*x*))=*g*(*x*) for every *x* in *X*.

Proof. Apply Stone duality, that is, look at the counterparts of g and η, in the world of frames. The counterpart of *g* is the frame homomorphism **O***g* : **O***Y* → **O***X*, which maps every open subset *V* of *Y* to * g^{-1}*(

*V*).

Similarly, there is a Stone dual counterpart to η. Let us compute what it does. The open subsets of **pt** **I**(**O***X*) are the subsets *O _{J}*, for each ideal

*J*in

**I**(

**O**

*X*), defined as the set of prime elements

*I*of

**I**(

**O**

*X*) such that

*J*is not included in

*I*. Then η

*(*

^{-1}*O*) is the set of points

_{J}*x*in

*X*such that

*J*is not included in η(

*x*) = {

*U*open in

*X*|

*x*is not in

*U*}, i.e., such that there is an

*U*in

*J*such that

*x*is in

*U*: this is just the union of all the elements

*U*of

*J*. In other words,

**O**η :

**O**

**pt**

**I**

*(*

**O**

*) ≅*

*X***I**

*(*

**O**

*) →*

*X***O**

*X*is just the familiar ‘union’ map, a.k.a., the sup map.

In the book, I usually write *r_{L}* for the sup map from

**I**

*(*

*) to*

*L**L*, where L is a complete lattice. For

*L*=

**O**

*Y*, where

*Y*is stably compact, which is the situation in Johnstone’s Theorem 9.5.14,

*r*is a retraction. Its associated section

_{L}*:*

*s*_{L}*L*→

**I**

*(*

*) maps every element*

*L**V*of

*L*(an open subset of

*Y*) to ↡

*V*. One checks easily that not only

*r*o

_{L}*= identity, but also*

*s*_{L}*o*

*s*_{L}*r*≤ identity. In particular,

_{L}*is left adjoint to*

*s*_{L}*r*:

_{L}*(*

*s*_{L}*V*) ⊆

*I*if and only

*V*

*⊆ r*(

_{L}*I*).

We are looking for a continuous map *g’* : **pt** **I**(**O***X*) → *Y* such that *g’* o η = *g*. Going to the Stone duals, and since all involved spaces are sober, it is equivalent to find a frame homomorphism *f* : **O***Y → ***I**(**O***X*) such that **O**η o *f* = **O***g*: there will be a unique continuous map *g’* : **pt** **I**(**O***X*) → *Y* determined from *f* by the adjunction **O** ⊣ **pt**. We have seen that **O**η is the sup map *r_{L}*, so we are looking for a frame homomorphism

*f*:

**O**

*Y →*

**I**(

**O**

*X*) such that

**O**

*g*=

*r*o

_{L}*f*. By the adjunction

*⊣*

*s*_{L}*r*,

_{L}**O**

*g*≤

*r*o

_{L}*f*if and only if

*o*

*s*_{L}**O**

*g*≤

*f*, so any solution

*f*to our problem must be above

*o*

*s*_{L}**O**

*g*. But

*o*

*s*_{L}**O**

*g*is itself a solution! since

*r*o (

_{L}*o*

*s*_{L}**O**

*g*) = (

*r*o

_{L}*) o*

*s*_{L}**O**

*g*=

**O**

*g*.

This finishes the proof. We have also shown that we could even find *g’* minimal, in the sense that *g’* is pointwise minimal among all possible solution frame homomorphisms. ◻︎

**Corollary. ** If *X* is T_{4}, then **ω***X* is (isomorphic to) the Stone-Čech compactification ß*X*.

*Proof.* We must show the universal property of Stone-Čech compactifications: every continuous map *g* : *X* → *Y*, where *Y* is compact T_{2}, has exactly one continuous extension *g’* from **ω*** X* to

*Y*, where by extension we mean that

*g’*(η(

*x*))=

*g*(

*x*) for every

*x*in

*X*.

By the previous theorem, we know there is at least one such extension from the larger space **pt** **I**(**O***X*) to *Y*. Its restriction to **ω***X* is then one possible candidate for an extension. However, *g* can have at most one continuous extension to **ω***X*, since Im η is dense in **ω***X*. ◻︎

**Wrapping up.**

If you’ve survived until this point, you may be happy to learn that the story does not end here. The fact that we only managed to show that the Wallman compactification **ω***X* is the Stone-Čech compactification ß*X* only when *X* is T_{4}, is nagging. The Stone-Čech compactification exists for all T_{3 1/2} spaces, right? Gillman and Jerrison showed [3] that a construction that is very similar to **ω***X* yields a Stone-Čech compactification of *X* for all T_{3 1/2} spaces *X*. Briefly, instead of considering maximal non-trivial filters of closed subsets, you need to consider maximal non-trivial filters of *zero sets* of *X*. (A zero set is the inverse image * f^{-1}* {0} of the one-element [closed] subset {0} of

**R**by a continuous map

*f*from

*X*to

**R**, with its usual, metric topology.) Frink [2] generalized this result to

*Z*-sets, for certain, so-called normal bases of closed sets

*Z*.

[1] Henry Wallman. *Lattices and Topological Spaces*. Annals of Mathematics, Second Series 39(1), January 1938, 112-126. Available on JSTOR.

[2] Orrin Frink. Compactifications and Semi-Normal Spaces. American Journal of Mathematics 86(3), July 1964, 602-607. Available on JSTOR.

[3] L. Gillman and M. Jerison. *Rings of Continuous Functions*. Princeton, 1960.