Projective limits of topological space II: Steenrod’s theorem

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 T₁, then the projective limit is non-empty and compact T₁.  (The T₁ condition is not mentioned explicitly, because what Steenrod includes the T₁ 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)_{i≤j ∈ I} of compact sober spaces is compact and sober. It is non-empty if every X_i is non-empty.

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)_{i≤j ∈ I} be a projective system of compact sober spaces.  If every X_i is non-empty, then its projective limit X is non-empty as well.

Recall that X consists of the tuples (x_i)_{i∈ I} where each x_i is in X_i, and where x_i = p_ij (x_j) for all i≤j in I.  X is topologized as a subspace of the product space Π_{i∈ I} X_i.  The projection map p_i : X → X_i maps every such tuple to 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_i, it is enough to find an irreducible closed subset Z_i in each X_i—subject to some conditions—instead.  That Z_i will be the closure of a unique point by sobriety, and that will be x_i.

The dcpo Φ

Let Φ be the collection of all families (C_i)_{i∈ I} where each C_i is closed and non-empty in X_i.  and such that the image of C_j by p_ij is included in C_i for all i≤j in I.  We order Φ by pointwise reverse inclusion, namely (C_i)_{i∈ I}≤(C’_i)_{i∈ I} if and only if C_i contains C’_i for every 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)_{i∈ I} is an element of Φ.

It is easy to see that given any directed family ((C^j_i)_{i∈ I})_{j∈ J} of elements of Φ (namely, for each i in J, the collection of sets (C^j_i)_{j∈ J} is filtered, since we are working with reverse inclusion), its pointwise intersection, (C_i)_{i∈ I}, where each C_i is defined as ∩_{j∈ J} C^j_i, is again an element of Φ. The key point is that C_i 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.)

By Zorn’s Lemma, Φ has a maximal element.  Let us call it (Z_i)_{i∈ I} in the sequel.

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_j] intersects C for cofinally many j≥i, then C=Z_i.

Remark.  If p_ij[Z_j] intersects C for cofinally many j≥i, then p_ij[Z_j] intersects C for every j≥i.  (Indeed, take any j≥i.  Then there is a further k≥j such that p_ik[Z_k] intersects C, by cofinality.  Hence there is a point x in Z_k such that p_ik(x_k) is in C, and then p_jk(x_k) is a point of Z_j whose image by p_ij is in 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_k, as follows.  By the Remark, for every j≥i,k, p_ij[Z_j] intersects C.  The set p_ij[Z_j] is included in X_j, but the fact that it intersects C can be expressed by saying that, equivalently, p_ij^-1(C) ∩ Z_j is non-empty.  Note that the set p_ij^-1(C) ∩ Z_j is included in X_i instead.

We can project back p_ij^-1(C) ∩ Z_j onto a closed subset of X_k, by taking the closure cl (p_kj[p_ij^-1(C) ∩ Z_j]) of its image by p_kj.  This is a familiar trick with directed families (here, 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_j]) where 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_j]) become smaller and smaller.  Since 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’_k’ to Z’_k.  (Exercise.  Think of using the fact that p_kk’ is continuous, so the image of a closure is included in the closure of the image.)  Therefore (Z’_k)_{k∈ I} is an element of Φ.

By construction of Z’_k, Z’_k is included in cl (p_kj[p_ij^-1(C) ∩ Z_j]) for at least one j, hence in cl (p_kj[Z_j]) ⊆ cl (Z_k) = Z_k. By maximality of (Z_k)_{k∈ I}, Z’_k=Z_k for every k in I.

In particular, Z’_i=Z_i.  But   Z’_i ⊆ cl (p_ij[p_ij^-1(C) ∩ Z_j]) for at least one 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

Using the Lemma, we can now proceed and show that Z_i is irreducible closed for every i.  Imagine that Z_i is included in the union of two closed subsets C’ and C” of X_i.  We wish to show that Z_i is included in C’ or in C”, namely that Z_i ∩ C’ or Z_i ∩ C” equals Z_i.  To do so, we use the Lemma with 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_j] intersects Z_i ∩ C’, and let J” be the set of indices j≥i such that p_ij[Z_j] intersects 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_j, and p_ij(x) is in Z_i.  If 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_i is the closure ↓x_i of a unique point x_i.  By the definition of Φ, p_ij maps every point of Z_j, in particular x_j, to a point of Z_i=↓x_i, so 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_i.  In particular x_i is in Z’_i, hence in cl (p_ij[p_ij^-1(C) ∩ Z_j]) ⊆ cl (p_ij[Z_j]) = cl (p_ij[↓x_j]).  Since p_ij is continuous, hence monotonic, p_ij[↓x_j] ⊆ ↓p_ij(x_j).  Therefore x_i is in cl (↓p_ij(x_j)) = ↓p_ij(x_j).  This shows that x_i≤p_ij(x_j), hence x_i=p_ij(x_j).

As this holds for all i≤j, the tuple (x_i)_{i∈ I} is in the projective limit 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.

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

— Jean Goubault-Larrecq (September 23rd, 2018)