Last time, I explained some of the strange things that happen with projective limits of topological spaces: they can be empty, even if all the spaces in the given projective system are non-empty and all bonding maps are surjective, and they can fail to be compact, even if all the spaces in the projective system are compact.

Steenrod [1, Theorem 2.1] claimed that if all the spaces in the projective system are compact and T_{1}, then the projective limit is non-empty and compact T_{1}. (The T_{1} condition is not mentioned explicitly, because what Steenrod includes the T_{1} separation axiom in his definition of topological spaces. Also, note that he calls bicompact what we call compact.)

As I said last time, this is wrong. But Fujiwara and Kato prove [3, Theorem 2.2.20] the following—and call it Steenrod’s theorem:

**Theorem.** The projective limit of a projective system (*p _{ij }*:

*X*→

_{j}*X*)

_{i}*of compact sober spaces is compact and sober. It is non-empty if every*

_{i≤j ∈ I}*X*is non-empty.

_{i}Note that we do not need the bonding maps *p _{ij}* to be surjective for that to hold.

My goal is to explain the proof (credited to O. Gabber by Fujiwara and Kato). We will do it in (at least) two passes. This month, we will prove the following apparently very special case. We will explain the rest next month

**Proposition.** Let (*p _{ij }*:

*X*→

_{j}*X*)

_{i}*be a projective system of compact sober spaces. If every*

_{i≤j ∈ I}*X*is non-empty, then its projective limit

_{i}*X*is non-empty as well.

Recall that *X* consists of the tuples (*x _{i}*)

*where each*

_{i∈ I}*x*is in

_{i}*X*, and where

_{i}*x*=

_{i}*p*(

_{ij}*x*) for all

_{j}*i*≤

*j*in

*I*.

*X*is topologized as a subspace of the product space Π

_{i∈ I}*X*. The projection map

_{i}*p*:

_{i}*X*→

*X*maps every such tuple to

_{i }*x*.

_{i}The proof is pretty subtle, and will occupy the rest of this post. Because of sobriety, instead of picking one element *x _{i}* from each

*X*, it is enough to find an irreducible closed subset

_{i}*Z*in each

_{i}*X*—subject to some conditions—instead. That

_{i}*Z*will be the closure of a unique point by sobriety, and that will be

_{i}*x*.

_{i}## The dcpo Φ

Let Φ be the collection of all families (*C _{i}*)

*where each*

_{i∈ I}*C*is closed and non-empty in

_{i}*X*. and such that the image of

_{i}*C*by

_{j}*p*is included in

_{ij }*C*for all

_{i}*i*≤

*j*in

*I*. We order Φ by pointwise reverse inclusion, namely (

*C*)

_{i}*≤(*

_{i∈ I}*C’*)

_{i}*if and only if*

_{i∈ I}*C*contains

_{i}*C’*for every

_{i}*i*in

*I*. The idea is to find a maximal element of Φ, and to show that it fits.

In order to show that Φ has a maximal element, we plan to use Zorn’s Lemma. We first check that Φ is non-empty: (*X _{i}*)

*is an element of Φ.*

_{i∈ I}It is easy to see that given any directed family ((*C ^{j}_{i}*)

*)*

_{i∈ I}*of elements of Φ (namely, for each*

_{j∈ J}*i*in

*J*, the collection of sets (

*C*)

^{j}_{i}*is filtered, since we are working with*

_{j∈ J}*reverse*inclusion), its pointwise intersection, (

*C*)

_{i}*, where each*

_{i∈ I}*C*is defined as ∩

_{i}

_{j∈ J}*C*, is again an element of Φ. The key point is that

^{j}_{i}*C*is non-empty, because every filtered intersection of non-empty closed subsets of a compact space is non-empty (Exercise 4.4.11 in the book.)

_{i}By Zorn’s Lemma, Φ has a maximal element. Let us call it (*Z _{i}*)

*in the sequel.*

_{i∈ I}## The key lemma

Given any property *P* of indices *i* in *I*, we will say “for cofinally many *j*, *P*(*j*)” or “*P*(*j*) holds for cofinally many *j*” to say that for every *k* in *I*, there is a *j*≥*k* in *I* such that *P*(*j*) holds. (In other words, the set of indices *j* such that *P*(*j*) holds is cofinal in *I*.)

If the index set were **N** instead of *I*, that would be equivalent to saying that *P*(*j*) holds for infinitely many values of *j*, if that can be of some help.

We will also say “*P*(*j*) holds for cofinally many *j≥i*” (or the obvious variants) to say that for every *k* in *I* with *k*≥*i*, there is a *j*≥*k* in *I* such that *P*(*j*) holds. If the index set were **N**, that would mean that *P*(*j*) holds for infinitely many *j≥i*—and the `≥*i*‘ part would be useless.

**Lemma.** For every *i* in *I*, and every closed subset *C* of *Z _{i}*, if

*p*[

_{ij}*Z*] intersects

_{j}*C*for cofinally many

*j*≥

*i*, then

*C*=

*Z*.

_{i}Remark. If *p _{ij}*[

*Z*] intersects

_{j}*C*for cofinally many

*j*≥

*i*, then

*p*[

_{ij}*Z*] intersects

_{j}*C*for

*every*

*j*≥

*i*. (Indeed, take any

*j*≥

*i*. Then there is a further

*k*≥

*j*such that

*p*[

_{ik}*Z*] intersects

_{k}*C*, by cofinality. Hence there is a point

*x*in

*Z*such that

_{k}*p*(

_{ik}*x*) is in

_{k}*C*, and then

*p*(

_{jk}*x*) is a point of

_{k}*Z*whose image by

_{j}*p*is in

_{ij}*C*.) But we will really need the “cofinally many” part later.

*Proof.* We build a new family of closed subset *Z’ _{k}* of each

*X*, as follows. By the Remark, for every

_{k}*j*≥

*i,k*,

*p*[

_{ij}*Z*] intersects

_{j}*C*. The set

*p*[

_{ij}*Z*] is included in

_{j}*X*, but the fact that it intersects

_{j}*C*can be expressed by saying that, equivalently,

*p*

_{ij}^{-1}(

*C*) ∩

*Z*is non-empty. Note that the set

_{j}*p*

_{ij}^{-1}(

*C*) ∩

*Z*is included in

_{j}*X*instead.

_{i}We can project back *p _{ij}*

^{-1}(

*C*) ∩

*Z*onto a closed subset of

_{j}*X*, by taking the closure cl (

_{k}*p*[

_{kj}*p*

_{ij}^{-1}(

*C*) ∩

*Z*]) of its image by

_{j}*p*. This is a familiar trick with directed families (here,

_{kj}*I*): to go from index

*i*to index

*k*, where

*k*is possibly incomparable to

*i*, we find an index

*j*above both

*i*and

*k*, then we go from

*i*to

*j*, and then back from

*j*to

*k*.

One checks easily that the family *F* of closed sets cl (*p _{kj}*[

*p*

_{ij}^{-1}(

*C*) ∩

*Z*]) where

_{j}*j*ranges over the indices that are above both

*i*and

*k*is filtered: as

*j*grows, the sets cl (

*p*[

_{kj}*p*

_{ij}^{-1}(

*C*) ∩

*Z*]) become smaller and smaller. Since

_{j}*F*is a filtered family of non-empty closed sets in a compact space, its intersection is closed and non-empty: this is what we choose to be

*Z’*.

_{k}It is also elementary to check that if *k*≤*k’*, then *p _{kk’}* maps every element of

*Z’*to

_{k’}*Z’*. (Exercise. Think of using the fact that

_{k}*p*is continuous, so the image of a closure is included in the closure of the image.) Therefore (

_{kk’}*Z’*)

_{k}*is an element of Φ.*

_{k∈ I}By construction of *Z’ _{k}*,

*Z’*is included in cl (

_{k }*p*[

_{kj}*p*

_{ij}^{-1}(

*C*) ∩

*Z*]) for at least one

_{j}*j*, hence in cl (

*p*[

_{kj}*Z*]) ⊆ cl (

_{j}*Z*) =

_{k}*Z*. By maximality of (

_{k}*Z*)

_{k}*,*

_{k∈ I}*Z’*=

_{k}*Z*for every

_{k}*k*in

*I*.

In particular, *Z’ _{i}*=

*Z*. But

_{i}*Z’*⊆ cl (

_{i }*p*[

_{ij}*p*

_{ij}^{-1}(

*C*) ∩

*Z*]) for at least one

_{j}*j*, and that is included in cl (

*p*[

_{ij}*p*

_{ij}^{-1}(

*C*)]) ⊆

*C*. Therefore

*Z’*⊆

_{i }*C*. Equality follows since

*C*is a subset of

*Z’*. ☐

_{i}*Z*_{i} is irreducible closed

_{i}

Using the Lemma, we can now proceed and show that *Z _{i}* is irreducible closed for every

*i*. Imagine that

*Z*is included in the union of two closed subsets

_{i}*C’*and

*C”*of

*X*. We wish to show that

_{i}*Z*is included in

_{i}*C’*or in

*C”*, namely that

*Z*∩

_{i}*C’*or

*Z*∩

_{i}*C”*equals

*Z*. To do so, we use the Lemma with

_{i}*C*equal to

*Z*∩

_{i}*C’*or to

*Z*∩

_{i}*C”*.

Explicitly, let *J’* be the set of indices *j*≥*i* such that *p _{ij}*[

*Z*] intersects

_{j}*Z*

_{i}∩*C’*, and let

*J”*be the set of indices

*j*≥

*i*such that

*p*[

_{ij}*Z*] intersects

_{j}*Z*

_{i}∩*C”*: we must show that

*J’*or

*J”*is cofinal in

*I*.

To do so, we claim that *J’* ∪ *J”* is cofinal in *I*. In fact every *j*≥*i* is in *J’* ∪ *J”*. Indeed, for every *j*≥*i*, *Z _{j}* is non-empty, so there is a point

*x*in

*Z*, and

_{j}*p*(

_{ij}*x*) is in

*Z*. If

_{i}*p*(

_{ij}*x*) is in

*C’*then

*j*is in

*J’*, otherwise

*p*(

_{ij}*x*) is in

*C”*and

*j*is in

*J”*.

Now, since *J’* ∪ *J”* is cofinal in *I*, *J’* or *J”* must be cofinal in *I*: otherwise after a certain rank no element of *I* would be in *J’*, and no element of *I* would be in *J”*. This concludes the proof: summing up, if *J’* is cofinal in *I*, then we apply the Lemma with *C*=*Z _{i} ∩*

*C’*, so that

*Z*⊆

_{i}*C’*, and if

*J”*is cofinal in

*I*, then we apply the Lemma with

*C*=

*Z*

_{i}∩*C”*, so that

*Z*⊆

_{i}*C”*. ☐

## Finishing the proof of the Proposition

Since every *X _{i}* is sober, the irreducible closed subset

*Z*is the closure ↓

_{i}*x*of a unique point

_{i}*x*. By the definition of Φ,

_{i}*p*maps every point of

_{ij}*Z*, in particular

_{j}*x*, to a point of

_{j}*Z*=↓

_{i}*x*, so

_{i}*p*(

_{ij}*x*)≤

_{j}*x*.

_{i}In order to show the converse inequality, we recall from the proof of the Lemma that *Z’ _{i}*=

*Z*. In particular

_{i}*x*is in

_{i}*Z’*, hence in cl (

_{i}*p*[

_{ij}*p*

_{ij}^{-1}(

*C*) ∩

*Z*]) ⊆ cl (

_{j}*p*[

_{ij}*Z*]) = cl (

_{j}*p*[↓

_{ij}*x*

*]). Since*

_{j}*p*is continuous, hence monotonic,

_{ij}*p*[↓

_{ij}*x*

*] ⊆ ↓*

_{j}*p*(

_{ij}*x*

*). Therefore*

_{j}*x*is in cl (↓

_{i}*p*(

_{ij}*x*

*)) = ↓*

_{j}*p*(

_{ij}*x*

*). This shows that*

_{j}*x*

*≤*

_{i}*p*(

_{ij}*x*

*), hence*

_{j}*x*

*=*

_{i}*p*(

_{ij}*x*

*).*

_{j}As this holds for all *i*≤*j*, the tuple (*x _{i}*)

*is in the projective limit*

_{i∈ I}*X*. So

*X*is non-empty. This concludes the proof of the Proposition. ☐

That is enough for this month. We have only proved that the projective limit of non-empty compact sober spaces is non-empty. We have done the most complicated part! Next month, we will see that this entails that projective limits of compact sober spaces are compact. This requires much less effort.

- Steenrod, Norman E. 1936. Universal Homology Groups. American Journal of Mathematics, 58(4), 661–701.
- Stone, Arthur Harold. 1979. Inverse Limits of Compact Spaces. General Topology and its Applications, 10, 203–211.
- Fujiwara, Kazuhiro, and Kato, Fumiharu. 2017 (Feb.). Foundations of Rigid Geometry I. arXiv 1308.4734, v5.