Projective limits of topological spaces I: oddities

I like to explain projective limits as follows.  Imagine you take a photo of some landscape with an old, low-resolution camera.  You can vaguely recognize the landscape, but the image is somehow blurred.  Hence you decide to use a second camera, with better resolution: each pixel in the first picture now corresponds, say, to a square of four pixels in the second picture.  The image is better, but not perfect, so you decide to use an even better camera, and so on.  Intuitively, if you were able to build a limit of that series of pictures, you would obtain a perfect image of the landscape in the end.  That is the projective limit of the sequence of pictures you have taken.

The definition of projective limits

Formally, we consider a family of spaces Xi indexed by some iI (those are our pictures), we assume that I is preordered and directed, and that, for all ij in I, there is a so-called bonding map pij from Xj to Xi: for every point x in Xj, pij (x) gives you the position of pixel where x would lie in the coarser picture Xi.  We require that pii  be the identity map for every i, and that pij o pjk = pik for all ijk.  The data of all spaces Xi and all bonding maps pij is called a projective system of spaces.

Categorically, a projective system is just a functor from I, seen as a (thin) category, to the category Top of topological spaces.  This construction works with any category instead of Top, and we can define projective systems of sets, of groups, etc.  We might even replace I with an arbitrary category, but the case where I is a directed preordered set is important in mathematics, and there are some results which require that restriction (notably Prohorov’s construction of a projective limit of measures).

The projective limit of a projective system is just a limit of that functor.  Explicitly, a cone over the projective systems is a space X, together with (“projection”) maps pi : X → Xfor each i, such that pij o pj = pi for all ij. And a projective limit is a universal cone, namely a cone (X, pi) such that for every cone (Y, qi), there is a unique morphism f from Y to X such that pi o f = qi for every i.  That is the usual category-theoretic definition, but if you are not that categorically minded, that will not tell you much.

Instead, let me describe one particular projective limit, the canonical projective limit X of the projective system (Xipij) (in Top).  Its elements are the tuples (xi)i∈ I where each xi is in Xi, and where xi = pij (xj) for all ij in I.  This makes X a subset of the product space Πi∈ I Xi, and we topologize it with the subspace topology.  The projection map pi : X → Xmaps every such tuple to xi.

For those accustomed to domain theory…

There is a similar construction in domain theory, where in addition to bonding maps pij,we have maps eij running in the other direction, from Xi, to Xj, and such that pij o eij = id, eij o pij ≤ id (for all ij),  eii = id and ejk o eij = eik for all ijk.  That is called an ep-system in the book (Section 9.6).

The projective limit of an ep-system is the projective limit X of the underlying projective system, forgetting about the maps eij.  That construction has plenty of nice properties.  Here is a list of interesting properties we have in that case—I will leave the task of proving them to you:

  1. For every i in I, there is a map ei : Xi → X, defined as follows.  The jth component of ei (x) is pjk(eik(x)), where k is any index above both i and j. (That k exists because I is directed.  I will let you check that this is independent of k.)  It is continuous, and po e= id, eo p≤ id.
  2. For all ij in I, eo eij = ei.
  3. For every point x in the projective limit X, the family of points ei(pi(x)) forms a monotone net (hence is directed), and its supremum is x.
  4. For every open subset U of X, the family of open subsets (ei o pi)-1 (U) is also a monotone net, and its union is exactly U.

Note that, according to intuition, the projective limit is big: through the injective maps ei, it contains every Xi.  In particular, as soon as some Xi is non-empty, then X is non-empty.

Also, the projection maps pi are all surjective.  Indeed, they are retractions.

None of that will survive for general projective limits, as we will now see.

A first oddity

We assume that every Xi is non-empty.  Is the projective limit necessarily non-empty?  The domain-theoretic intuition for the last section should tell us it must be so, but that is not true.  Here is a simple example.

Example 1. We take I=N with its usual ordering, Xn = the open interval (0, 1) in R, and for every pair mnpmn (x) = x/2n—m.  It is easy to see from the explicit construction of the projective limit as a space of tuples that the projective limit is empty: there is no tuple of numbers in (0, 1) such that each one is twice as large as the previous one in the list.

All right, but we cheated: in Example 1, the bonding maps are not surjective.  Hence, by going from Xn to Xn+1, we lose a fraction of (1—2n—m) of all the points.  In the limit, we have lost of all them: that should have been expected.

Let us refine our question: if the spaces Xi are non-empty and all the bonding maps pij are surjective, is the projective limit non-empty?

Leon Henkin was the first to answer the question [1], in the negative.  His paper is pretty complex, and involves a sophisticated construction based on ordinals.

Waterhouse later realized that there was a much simpler solution [2].  Indeed, his paper is 6 lines long!  (excluding title, author, and bibliography; I will explain it in slightly more detail).  Here it is.

Example 2. We consider a projective limit of sets, not topological spaces here, because topology is in fact irrelevant in that case.  If needed, put the discrete topology on all the sets involved in the argument.
Let A be an uncountable set, such as R for example. Let I be the lattice of finite subsets of A, ordered by inclusion. For each iI, i is a finite subset of A, and we define Xi as the set of all injective maps from i to N.  For all ij (i.e., i⊆j) in I, we define pij as mapping every f in Xj (an injective map from j to N) to its restriction f|i to the subset i.  This defines a projective system of sets with surjective bonding maps.
Given any element (fi)i∈ I of the canonical projective limit X of that system, we can “glue together” the maps fi to obtain a single map f from A to N: for every a in A, all the maps fi such that a belongs to i must map it to the same element, and this is f(a) by definition.  Then f is injective: for all ab in Af(a) = f{a,b}(a) ≠ f{a,b}(b) = f(b), since f{a,b} is injective.  However, there is no injective map from A to N, since A is uncountable.  It follows that X is empty.

The case of cofinal countable chains

It is known that such an oddity does not happen if I is countable (which the I of Example 2 is not), or more generally when I has a cofinal monotone sequence i0i1 ≤ … ≤ ik ≤ … (cofinal means that every element of I is below some ik); the latter is the case with I=R with its usual ordering, for example.  I will just give a sketch of a proof.

An easy check shows that the projective limit of the projective system (Xipij) (with ij in I) is isomorphic to the projective limit of the subsystem (Xipij) with ij in the chosen cofinal monotone sequence.  Replacing I with that monotone sequence if necessary, and reindexing, we can therefore assume that I is N itself.  Now pick a point x0 from X0, then a point x1 from X1 such that p01(x1)=x0 (remember that p01 is surjective), then a point x2 from X2 such that p12(x2)=x1 (p12 is surjective), etc.  The tuple of points xn thus obtained is an element of the projective limit.

One may “generalize” that result to the case where I has a cofinal countable set A (not necessarily totally ordered).  But in that case, I must in fact have a cofinal monotone sequence, so that is no genuine generalization: enumerate A as a0a1a2, … then define i0 as a0, i1 as some element of I above i0 and a1 (using directedness), i2 as some element of I above i1 and a2 (using directedness),  etc.

Further oddities

Non-emptiness is one thing, and compactness is another.  We may hope that a projective limit of compact spaces is compact (as usual, we understand compactness without Hausdorffness).  Again, that is not true.

To be precise, it is true that a projective limit of compact Hausdorff spaces is compact (and Hausdorff).  That had been known since Bourbaki, and we will see a generalization of that result next time (from [4]).

In the meantime, here is a counterexample to the claim that a projective limit of compact spaces is compact.  This is due to A. H. Stone [3, Example 3], and it even shows that even a projective limit of Noetherian Tspaces (remember that Noetherianness—Section 9.7 of the book—is a very strong compactness property), with bonding maps that are even bijective, can fail to be compact.

Example 3.  For every nN, let Xn be 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.  For all min N, we simply define pmn : XnXm as the identity map. Every Xn is compact, in fact Noetherian: every filtered family of closed sets is stationary (i.e., has a least element), and is also T1 (the closure of every point is the point itself). Every pmn is surjective, in fact even bijective. I will leave to you the task of checking that pmn is continuous, and that the canonical projective limit of the projective system we have just defined is isomorphic to N with the discrete topology, and that is not compact.

Next time

… we will see that projective limits of compact sober spaces are compact (and sober).  I found this in a pretty big, recent paper on so-called rigid geometry by Fujiwara and Kato [4, Theorem 2.2.20], who call it Steenrod’s theorem.

Steenrod indeed had a similar theorem [5, Theorem 2.1], but instead claims that projective limits of compact T1 (instead of sober) spaces are compact.  The latter claim is faulty, as Example 3 above shows.  The error seems to lie in the very first line of the proof of Steenrod’s Theorem 2.1, which ascertains that a certain set is closed, a fact that would be true if the bonding maps were closed maps, not just continuous maps.  (The case of closed continuous bonding maps is exactly the subject of A. H. Stone’s paper [3].)  It does not seem that this has any impact on the rest of Steenrod’s results, whose spaces will anyway be Hausdorff, from what I can see.

The proof that projective limits of compact sober spaces are compact, and which, according to Fujiwara and Kato, is due to O. Gabber, is pretty clever, but it would too much to explain it now.  Next time, then!

  1. Henkin, Leon. 1950. A Problem on Inverse Mapping Systems. Proceedings of the American Mathematical Society, 1, 224–225.
  2. Waterhouse, William C. 1972. An Empty Inverse Limit. Proceedings of the American Mathematical Society, 36(2), 618.
  3. Stone, Arthur Harold. 1979. Inverse Limits of Compact Spaces. General Topology and its Applications, 10, 203–211.
  4. Fujiwara, Kazuhiro, and Kato, Fumiharu. 2017 (Feb.). Foundations of Rigid Geometry I. arXiv 1308.4734, v5.
  5. Steenrod, Norman E. 1936. Universal Homology Groups. American Journal of Mathematics, 58(4), 661–701.

Jean Goubault-Larrecqjgl-2011