Projective limits of topological spaces III: finishing the proof of Steenrod’s theorem

Last time, we had started to prove the following theorem [3, Theorem 2.2.20]:

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.

My goal this time is to finish its proof.  That is unfortunately pretty technical.  If there were anything to remember from the proof, that would be that the following Proposition, which we had proved last time, is the cornerstone of the proof.

Proposition.  Let  (p’_jk : C_k → C_j)_j≤k∈J be a projective system of compact sober spaces.  If every C_j is non-empty, then its projective limit is non-empty as well.

The Proposition of course only deals with the final part of the Theorem.  Showing that the projective limit is sober is easy, and we deal with that point next.  Showing that it is compact is much more technical, and we will repeatedly use the Proposition to prove that.  The general idea will be to find non-empty closed (hence compact and sober) subsets C_i of each X_i, in such a way as to define a new projective limit (p_ij : C_j → C_i)_{i≤j ∈ I} of compact sober subspaces, and to use the fact that this projective limit is non-empty in order to make progress.

Sobriety

Not all subspaces of a sober space are sober: take any non-sober space X, and realize that it occurs (up to homeomorphism) as a subspace of its sobrification.  However, any subspace [g₁=g₂] of X that is built as an equalizer of two continuous maps g₁, g₂ : X → Y with X sober (Lemma 8.4.12 of the book; recall that, in an explicit form, [g₁=g₂] is the subspace of all x in X such that g₁(x)=g₂(x).)

This shows immediately that:

Lemma 1.  A projective limit X of a projective system (p_ij : X_j → X_i)_{i≤j ∈ I} of sober spaces is sober.

This is because Π_{i∈ I} X_i is sober (Theorem 8.4.8 in the book) and X occurs as the equalizer [g₁=g₂] where g₁ : Π_{i∈ I} X_i → Π_{i≤j∈ I} X_i maps (x_i)_{i∈ I} to (p_ij(x_j))_{i≤j∈ I} and g₂ maps (x_i)_{i∈ I} to (x_i)_{i≤j∈ I}.  (Please pay attention to the fact that the target space of g₁ and g₂ is a product indexed, not by i, but by all pairs of indices i, j such that i≤j.)

This also shows, for example, that every open subspace U of a sober space X is sober, being the equalizer of the characteristic map of U (with values in Sierpiński space S) with the constant 1 map.

Every closed subspace C is also sober, as the equalizer of the characteristic map of its complement with the constant 0 map.

A compactness lemma

Let X be the projective limit of a projective system of compact sober spaces (p_ij : X_j → X_i)_{i≤j ∈ I}.   Let also p_i : X → X_i be the projection maps.  We have the following, which one can think as a sort of compactness lemma for the set of indices I.

We will not have any use of it in this post, but this is a generally useful lemma, and it can be taken as a gentle example of how we can make good use of the Proposition we mentioned at the beginning of the post (the projective limit of a projective system of non-empty compact sober spaces is non-empty).

Lemma 2.  For every i in I, and every open neighborhood V of the image of X by p_i, there is an index j≥i in I such that that V already contains the image of X_j by p_ij.

Proof.  We reason by contradiction, and assume that V does not contain the image of X_j by p_ij for any j≥i.  Let J be the subset of those indices of I above i.  This is a directed set.  For each j in J, C_j=X_j–p_ij^-1(V) is then non-empty.  It is closed in a compact space, hence is itself compact.  As a compact subset, it is also a compact subspace (Exercise 4.9.11 of the book).  As a closed subspace of a sober space, it is sober.

For all j≤k in J, the restriction p’_jk of p_jk to C_k maps every element to an element of C_j.  Indeed, for every element x of C_k, if p_jk(x) were in p_ij^-1(V), then p_ij(p_jk(x))=p_ik(x) would be in V, and x would be in p_ik^-1(V): contradiction.

Therefore  (p’_jk : C_k → C_j)_j≤k∈J is a projective system of non-empty compact sober spaces.  The Proposition we mentioned at the beginning of the post shows that it has a non-empty projective limit.  We pick a tuple (x_j)_{j∈ J} in that projective limit.  We can complete it to a tuple indexed by I instead, defining x_j for j not in J as p_jk(x_k) for some arbitrary k in I above both i and j.  This defines an element x of X such that p_j(x)=x_j for every j in J, in particular p_i(x)=x_i.  It follows that x_i is in the image of p_i, hence in V, and that is impossible since x_i is in C_i. ☐

Compactness

In order to show the Theorem, we need to show that the projective limit X of a projective system of compact sober spaces (p_ij : X_j → X_i)_{i≤j ∈ I} is compact.   Let also p_i : X → X_i be the projection maps, as usual.  That is the hard part, and I hope you will not be lost in indices. There will be quite a lot of them.  In the sequel, i and j will always denote indices from the index set I, while k will index the open subsets from a directed open cover of X.

In order to show that X is compact, we consider a directed family (U_k)_{k ∈ K} of open subsets of X, whose union is equal to X, and we wish to show that some U_k is already equal to the whole of X.  (This is Proposition 4.4.7 of the book.  Being able to consider the family as directed will simplify the argument.)

For each k in K, and every i in I, there is a largest open subset U_ki of X_i such that p_i^-1(U_ki) ⊆ U_k: we just take the union of all open subsets U of X_i such that p_i^-1(U) ⊆ U_k.

We notice that as i grows, p_i^-1(U_ki) becomes larger.  (Formal argument: If i≤j, then p_i= p_ij o p_j, p_i^-1(U_ki) is equal to p_j^-1(p_ij^-1(U_ki)).  This shows that p_ij^-1(U_ki) is an open subset U of X_j such that p_j^-1(U) ⊆ U_k.  It must then be included in the largest one, U_kj.  So p_i^-1(U_ki) ⊆ p_j^-1(U_kj).)  Hence the family of sets p_i^-1(U_ki), when i varies, is directed.  Its union is included in U_k, and we claim that:

Lemma 3.  The directed union of the sets p_i^-1(U_ki), i∈I, is equal to U_k.

Proof.  Consider any point (x_i)_{i∈ I} in U_k.  Since U_k is open, and remembering that the topology on X is the subspace topology from a product topology, there is a finite subset J of I and open subsets V_i of X_i, i ∈ J, such that x_i ∈ V_i for every i ∈ J, and such that every tuple whose ith coordinate is in V_i for every i ∈ J, is in U_k.  Since I is directed, there is a an index j in I above every element of J.  For every i ∈ J, p_ij(x_j)=x_i is in V_i, so x_j is in ∩_i∈J p_ij^-1(V_i). The open set p_j^-1(∩_i∈J p_ij^-1(V_i)) consists of tuples (y_i)_{i∈ I} such that y_j is in ∩_i∈J p_ij^-1(V_i), namely such that y_i is in V_i for every i ∈ J, and is therefore included in U_k.  By the maximality property of U_kj, ∩_i∈J p_ij^-1(V_i) is included in U_kj.  Therefore x_j is in U_kj.  It follows that (x_i)_{i∈ I} is in p_j^-1(U_kj), showing the claim.  ☐

Since X is the directed union of the sets U_k, k∈K, X is also the directed union over k of the directed union over i of the sets p_i^-1(U_ki). Switching the two unions, X is also the directed union over i of the open sets V_i, where V_i is defined as the (directed) union of (U_ki)_k∈K.

Note that V_i is an open subset of the compact set X_i, and is defined as a directed union of open subsets.  This sounds good, but the latter is not (yet known to be) an open cover, so we cannot conclude (yet).

Instead, we define C_i as the complement of V_i in X_i.  This is a closed subset of X_i, and as in the proof of Lemma 2, C_i is a compact sober subspace of X_i.

For all i≤j in I, p_ij^-1(U_ki) is included in U_kj.  (Formal argument: It suffices to check that p_j^-1(p_ij^-1(U_ki)), which is equal to p_i^-1(U_ki), is included in U_k, and to invoke the maximality of U_ki.)  Taking unions over k in K, p_ij^-1(V_i) is included in V_j.  Hence the image of C_j by p_ij is included in C_i: if there were a point of C_j whose image by p_ij were not in C_i, hence in V_i, it would be in p_ij^-1(V_i) and therefore in V_j, contradiction.

All this means that (p’_ij : C_j → C_i)_{i≤j ∈ I} is a(nother) projective system of compact sober spaces, where p’_ij is the restriction of p_ij to C_j.

But that projective system has an empty projective limit!  Let us check that.
Imagine there were an element its its projective limit.  That element would be in p_i^-1(C_i) for every i in I, hence not in p_i^-1(V_i) for any i in I, but that is impossible since X is the union over all i in I over  p_i^-1(V_i).

The Proposition mentioned at the beginning of this post (projective limits of non-empty compact sober spaces are non-empty) tells us that this cannot happen if every C_i is non-empty.  Hence C_i is empty for some i in I, and this means that V_i=X_i.

Since V_i is the directed union of (U_ki)_k∈K, and X_i is compact, X_i=U_ki for some k in K.  Finally, every element of the projective limit X is in p_i^-1(X_i) (meaning that it ith coordinate must be in X_i), namely in p_i^-1(U_ki), which is included in U_k.  Hence X=U_k.  ☐

That finishes the proof of the Theorem.

Instead of ending this post here, let me conclude with an easy observation on sobrifications of projective limits.

Sobrifications of projective limits

Remember that the sobrification functor S preserves products (Theorem 8.4.8 in the book), so we may think that it would perhaps preserve all limits in Top.  That is wrong, since it does not preserve equalizers: see the warning comments after Lemma 8.4.12 in the book.

We will now argue that S does not preserve projective limits in general either.

Recall the following counterexample, due to A. H. Stone, and mentioned as Example 3 in my first post on projective limits.  For every n ∈ N, we define X_n as N with the following topology, akin to the cofinite topology: its closed subsets C are those subsets such that C ∩ ↑n is finite or equal to the whole of ↑n.  The bonding maps p_mn : X_n →X_m (m≤n) are identity maps.

We had seen that every X_n is compact, in fact Noetherian, and T₁.  The projective limit X is N with the discrete topology, and we had observed that this is not compact.

Let us look at the irreducible closed subsets C of X_n.  The only ones that are finite are the one-element sets.  It remains to look at those such that C ∩ ↑n=↑n.  Those are obtained as ↑n union finitely many elements below n–1.  Irreducibility implies there cannot be any element below n–1, hence C=↑n.  Conversely, it is easy to see that ↑n is irreducible closed in X_n.

It follows that the sobrification S(X_n) of X_n is obtained by adding a fresh element ω_n (corresponding to ↑n), above n, n+1, …, but incomparable with 0, 1, …, n–1.

Since sobrification is a functor, the projective system (p_mn : X_n → X_m)_{m≤n ∈ N} gives rise to a new projective system (S(p_mn): S(X_n) → S(X_m))_{m≤n ∈ N}.  The sobrification of a compact space is compact, since it has an isomorphic lattice of open subsets.  Hence we have obtained a  projective system of compact sober spaces, and Steenrod’s theorem tells us that its projective limit X’ is compact and sober.

It follows that X’ cannot be the sobrification of X, which is not compact.  So the sobrification functor does not preserve projective limits.

Explicitly, X is N with the discrete topology, hence is already sober.  X’, instead, is N plus an extra element ω, incomparable with all natural numbers, and is indeed different from X.

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 (October 23rd, 2018)