First, season’s greetings to you all!
One of the big thing about uniform spaces is that they have a notion of completeness, and a notion of completion, which generalize those which are available in metric spaces. A similar thing can be said for quasi-uniform spaces, too. However, remember that the situation was already complicated in quasi-metric spaces, where we have two notions of completeness, Smyth-completeness and Yoneda-completeness, one notion of completion (Yoneda-completion or the formal ball completion, giving isomorphic spaces), that the completion of a quasi-metric space is always Yoneda-complete and even algebraic Yoneda-complete, but that a quasi-metric space is isomorphic to its completion if and only if it is Smyth-complete. All those facts about quasi-metric spaces are in Chapter 7 of the book.
In today’s post, we will explore natural quasi-uniform generalizations of the notions of Smyth-completeness, symcompactness, and total boundedness. My plan is to show that the following two theorems from the book generalize to quasi-uniform spaces: every Smyth-complete quasi-metric space is sober, and the symcompact quasi-metric spaces are exactly the quasi-metric spaces that are Smyth-complete and totally bounded. The second one will be rather more intricate to prove than the first one, and there are new complications that do not arise in the quasi-metric case: be warned!
Smyth-completeness
A quasi-metric space X, d is Smyth-complete if and only if every Cauchy net (xi)i ∈ I, ⊑ has a limit in the symmetrized space X, dsym. And a net (xi)i ∈ I, ⊑ is Cauchy if and only if for every for every r>0, there is an i0 ∈ I such that for all i, j in I such that i0 ⊑ i ⊑ j, d(xi,xj) < r.
In other words, a net (xi)i ∈ I, ⊑ is Cauchy if and only if for every for every r>0, there is an i0 ∈ I such that for all i, j in I such that i0 ⊑ i ⊑ j, (xi,xj) is in the basic entourage [<r] of the quasi-uniformity defined by d.
Generalizing, this, we can therefore define Cauchy nets in arbitrary quasi-uniform spaces by: a net (xi)i ∈ I, ⊑ in a quasi-uniform space X, with quasi-uniformity U, is Cauchy if and only if for every entourage R in U, there is an i0 ∈ I such that for all i, j in I such that i0 ⊑ i ⊑ j, (xi,xj) is in R. Then, on quasi-metric spaces, a net is Cauchy in the quasi-metric sense if and only if it is Cauchy in the new, quasi-uniform sense.
Generalizing the notion of symmetrized space is even easier. Given a quasi-uniformity U on X, there is a dual quasi-uniformity U–1, whose entourages are the opposites R–1 (also written as Rop) of entourages R in U. We have seen that last time. There is also a symmetrized quasi-uniformity Usym, obtained as the smallest quasi-uniformity that contains both U and U–1. Explicitly, a base of entourages of Usym is given by the intersections R ∩ S–1 of an entourage R of U and the opposite of an entourage S of U. It is easy to see that another, smaller base of entourages is given by the symmetrizations R ∩ R–1 of entourages R of U. In both cases, the conditions (reflexivity), (filter3) and (relaxed transitivity) of bases of entourages are easy to check; the two bases contain the whole of U and of U–1; and any quasi-uniformity containing both U and U–1 must contain each of those bases.
The quasi-uniformity Usym is symmetric, in the sense that the opposite of any entourage of Usym is again in Usym. Hence Usym is a uniformity. When U is the quasi-uniformity of a quasi-metric d, a base of entourages of Usym is given by entourages of the form [<r] ∩ [<r]–1, r>0, which are just the basic entourages of the uniformity defined by the symmetrized metric dsym.
Hence the notion of Smyth-completeness extends to quasi-uniform spaces as follows: a quasi-uniform space X, with quasi-uniformity U, is Smyth-complete if and only if every Cauchy net in X has a limit in Xsym, where Xsym is X with the (topology induced by the) uniformity Usym.
Let us put the notion to the test.
Smyth-complete implies quasi-sober
Proposition 8.2.12 (c) of the book states that every Smyth-complete quasi-metric space is sober in its open ball topology. Let us generalize this to the setting of quasi-uniform spaces. Since the induced topology of a quasi-uniform space need not be T0, we will use a modified notion of sobriety that does not require being T0: a topological space is quasi-sober if and only if every irreducible closed subset is the closure ↓x of some (not necessarily unique) point x.
We will need the following analogue of Lemma 7.1.9 of the book. Note that I say “a” largest limit, because largest elements may fail to be unique in the absence of the T0 property. Largest is taken with respect to the specialization preordering, as usual.
Lemma. Let X be a quasi-uniform space, and (xi)i ∈ I, ⊑ be a net in X. If (xi)i ∈ I, ⊑ converges to some point x in Xsym, then x is a largest limit of (xi)i ∈ I, ⊑ in the induced topology of X.
Proof. We first check that x is a limit of (xi)i ∈ I, ⊑ in X. For every open neighborhood U of x in X, there is an entourage R such that R[x] ⊆ U. Then (R ∩ R–1)[x] is an open neighborhood of x in Xsym, and by definition of x, xi will be in it for i large enough. Since (R ∩ R–1)[x] is included in R[x], hence in U, xi is in U for i large enough.
Let now y be any limit of (xi)i ∈ I, ⊑. In order to show that x is largest, we will show that y ≤ x. Let me recall that ≤ is the specialization preordering of X in the induced topology and that, as we have seen last time (see Lemma A), ≤ is the intersection of all the entourages of X. Hence it suffices to show that for every entourage R of X, (y,x) is in R. We use the (relaxed transitivity) property of quasi-uniformities in order to find a further entourage S such that S o S ⊆ R. By definition of x as a limit in Xsym, xi is in (S ∩ S–1)[x] for i large enough. By definition of y as a limit in X, xi is in S[y] for i large enough. Hence, for i large enough, xi is both in (S ∩ S–1)[x] and in S[y]. We pick such a large enough i. Then xi is in S–1[x], so (xi,x) is in S, and xi is in S[y], so (y,xi) is in S. It follows that (y,x) is in S o S, hence in R. ☐
Proposition. Every Smyth-complete space X is quasi-sober in its induced topology.
Proof. Let C be an irreducible closed subset of X. One possible strategy would be to invoke Lemma 8.2.27 of the book, and write C as the set of points of some self-convergent net. Then we would show that a self-convergent net is Cauchy, hence has a limit in Xsym (using Smyth-completeness). By the previous lemma, this would be a largest limit x in X, and standard arguments would show that x is the largest element of C. However, it does not seem that every self-convergent net is Cauchy.
Instead, I will build a Cauchy net from C, directly. The index set I will be the collection of pairs (R, x) where R is an entourage and x is a point of C, and we compare such pairs by (R, x) ⊑ (S, y) if and only if R ⊇ S and R[x] ⊇ S[y]. This is clearly a reflexive and transitive relation.
We claim that I is directed under ⊑. First, it is of course non-empty. Next, let (R1, x1) and (R2, x2) be two elements of I. The interiors of R1[x1] and of R2[x2] (in the induced topology) both intersect C, at x1 and at x2 respectively, so their intersection V also intersects at C, say at y: this is because C is irreducible. Since V is an open neighborhood of y, there is an entourage S such that S[y] ⊆ V. By replacing S by its intersection with R1 ∩ R2, we may even assume that S ⊆ R1 ∩ R2. It follows that (R1, x1) and (R2, x2) are both below (S, y) with respect to ⊑.
Using I, we define a net (zi)i ∈ I, ⊑ by letting zi be x, where i=(R,x). For short, z(R,x)≝x.
We verify that this net is Cauchy. Let us fix a point x0 in C. For every entourage R, it is enough to show that for all indices (S, y) and (S’, y’) in I such that (R, x0) ⊑ (S, y) ⊑ (S‘, y‘), (z(S,y),z(S’,y’)) is in R. And indeed, (z(S,y),z(S’,y’)) = (y, y’) is in R: since (S, y) ⊑ (S‘, y‘), S’[y’] is included in S[y], so that y’ is in S[y], or equivalently (y, y’) is in S; and since (R, x0) ⊑ (S, y), S is included in R.
Since (zi)i ∈ I, ⊑ is Cauchy and since X is Smyth-complete, it has a limit z in Xsym. By the previous lemma, z is a largest limit of (zi)i ∈ I, ⊑ in X. We use the fact that limits of elements of the closed set C must be in C (Corollary 4.7.10 in the book), so z is in C.
We now observe that every element x of C is a limit of (zi)i ∈ I, ⊑. Let us consider any open neighborhood U of x in X. By definition of the induced topology, there is an entourage R such that R[x] ⊆ U. For every element (S, y) of I such that (R, x) ⊑ (S, y), we have S[y] ⊆ R[x] ⊆ U, in particular z(S,y) = y is in U. (As a side note, this shows that (zi)i ∈ I, ⊑ is self-convergent.)
Since z is a largest limit of (zi)i ∈ I, ⊑ in X, z is therefore larger than or equal to every element x of C. Recalling that z is in C, and that closed sets are downwards-closed, we conclude that C = ↓z. ☐
Symcompact and totally bounded quasi-uniform spaces
Let us proceed and generalize a second theorem from the book. Theorem 7.2.22 says that a quasi-metric space is symcompact if and only if it is Smyth-complete and totally bounded. We will see that this generalizes to quasi-uniform spaces.
A quasi-uniform space X is precompact if and only if, for every entourage S, there is a finite subset E of X such that S[E] ≝ {y ∈ X | ∃ x ∈ E, (x,y) ∈ S} contains the whole of X. If you specialize this to quasi-metric spaces, with S ranging over the basic entourages [<r], you will see that this means that for every r>0, there is a finite collection of open balls of radius r whose union is the whole of X, so that we recover the usual notion of precompactness for quasi-metric spaces.
A quasi-uniform space X, with quasi-uniformity U, is totally bounded if and only if the symmetrized space Xsym is precompact. Again, this generalizes the usual notion of total boundedness, which is familiar for quasi-metric spaces.
I will say that a quasi-uniform space X is symcompact if and only if Xsym, seen with its induced topology, is compact.
If X is symcompact, then it is certainly totally bounded. Indeed, for every entourage S in Usym, the interiors (in Xsym) of the sets S[x], where x ranges over X, form an open cover of Xsym. Since Xsym is compact, there is a finite subcover. In other words, there is a finite set E such that ∪x ∈ E S[x], namely S[E], contains the whole of X.
If X is symcompact, then it is also Smyth-complete. (This generalizes Lemma 7.2.21 in the book.) Indeed, consider any Cauchy net (xi)i ∈ I, ⊑ in X. Since Xsym is compact, it has a subnet (xα(j))j ∈ J, ⪯ with a limit x in Xsym. Then (xi)i ∈ I, ⊑ also converges to x in Xsym, by the following Lemma (similar to Lemma 7.2.9 in the book), allowing us to conclude.
Lemma. Let X be a quasi-uniform space, (xi)i ∈ I, ⊑ be a Cauchy net with a subnet (xα(j))j ∈ J, ⪯ that converges to a point x in Xsym. Then (xi)i ∈ I, ⊑ also converges to x in Xsym.
Proof. Let U be the quasi-uniformity of X. Let U be any open neighborhood of x in Xsym. By definition of the induced topology, there is an entourage S of Usym such that S[x] is included in U. By definition of the symmetrized uniformity Usym, S contains an entourage of the form R ∩ R–1, where R is an entourage of U.
We use the (relaxed transitivity) property of quasi-uniformities in order to find a further entourage R’ of U such that R’ o R’ is included in R.
Next, since (xα(j))j ∈ J, ⪯ converges to x in Xsym, there is an index j0 in J such that for every j ∈ J with j0⪯j, xα(j) lies in (R‘ ∩ R’–1)[x]. Since (xi)i ∈ I, ⊑ is Cauchy, there is an index i0 in I such that for all i, i’ ∈ I with i0⊑i⊑i‘, (xi,xi’) is in R’. Since I is directed, we may also assume that α(j0)⊑i0.
We claim that for every i ∈ I with i0⊑i, xi is in S[x], hence in U, which will finish to prove the Lemma. Let us recall that, by definition of a subnet, α is monotonic, and cofinal in the sense that every element of I is smaller than or equal to some element in the range of α. All right. We fix i ∈ I with i0⊑i.
- By cofinality, there is a j ∈ J such that i⊑α(j), and since J is directed and α is monotonic, we can assume that j0⪯j: if necessary, we replace j by some index above both j and j0. Now xα(j) is in (R‘ ∩ R’–1)[x], in particular in R’–1[x], or in other words (xα(j),x) is in R’. Since i0⊑i⊑α(j), the Cauchy property enables us to say that (xi,xα(j)) is in R’. It follows that (xi,x) is in R’ o R’, hence in R.
- Also, xα(j0) is in (R‘ ∩ R’–1)[x], hence in R’[x], so (x,xα(j0)) is R’. Now the Cauchy property implies that (xα(j0),xi) is in R’, and therefore (x,xi) is in R’ o R’, hence in R.
Combining the latter two arguments, (x,xi) is both in R and in R–1, hence in S. This shows that xi is in S[x], hence in U. This holds for every i above i0. Since U is an arbitrary open neighborhood of x, x is a limit of (xi)i ∈ I, ⊑ in Xsym. ☐
Symcompact = Smyth-complete+totally bounded
We have proved the ‘only if’ direction of the following quasi-uniform analogue of Theorem 7.2.22, whose proof will occupy the rest of the post. A very close result is Proposition 2 of [1].
Theorem. For every quasi-uniform space X, X is symcompact if and only if X is totally bounded and Smyth-complete.
Proof. It remains to show the ‘if’ direction. We use a similar argument as in the book, replacing hemi-metric spaces by quasi-uniform spaces. Let X be a totally bounded and Smyth-complete quasi-uniform space. We fix a net (xi)i ∈ I, ⊑ in X. We claim that it has a convergent subnet in Xsym, and this will show that Xsym is compact (Corollary 4.7.27 in the book: a space is compact if and only if every net has a convergent subnet).
For every subset A of X, let us define I[A] as the subset of those indices i in I such that xi is in A. We start by showing that: (*) for every finite cover (Ak)k ∈ K of X, at least one of the families I[Ak] must be cofinal in I. Indeed, assume the contrary. For every k ∈ K, there is an index ik in I such that no larger (or equal) index is in I[Ak]. Since I is directed, we can find an index i above every ik in I. Then i (or any index that is larger than or equal to it) is outside every I[Ak], which means that xi is outside every set Ak, and that is impossible since (Ak)k ∈ K is a cover of X.
We use (*) in order to build a subnet of (xi)i ∈ I, ⊑.
Here is a first idea how we could achieve that. For every entourage R of X, there is a finite subset ER of X such that (R ∩ R–1)[ER] contains the whole of X, by total boundedness. This means that the sets (R ∩ R–1)[z] with z ∈ ER form a finite cover of X. Therefore, we can use (*) and find a point zR of ER such that JR ≝ I[(R ∩ R–1)[zR]] is cofinal in I.
We would like to define the desired subnet by taking the collection of pairs (R, j) as its family of indices, where R ranges over the entourages of X, and j ranges over JR. The idea would be to define a sampler α that would map (R, j) to j, to show that (xα(F, j))(F, j) ∈ P, ⪯ is Cauchy, and then Smyth-completeness would imply that it has a limit in Xsym.
That does not seem to work so well. In particular, finding a way of ordering those pairs (R, j) so as to make it a directed family, and so that the map (R, j) ↦ j is monotonic, seems to be somewhat of a challenge.
Hence we need another proof strategy. This will be more complex. I will obviate the difficulty somewhat by using Rado’s selection lemma (Exercise 4.5.15 in the book). Its statement is as follows. We change the notations so as to best match our later needs: for example, we write U for the index set, because we will indeed need to index our selections by the entourages R of the space X, in its quasi-uniformity U.
Lemma (Rado). Let (ER)R ∈ U be an arbitrary family of finite sets. For every F ⊆ U, a selection function sF on F is a map with domain F such that sF(R) ∈ ER for each R in U. Given any collection of selection functions sF, one for each finite subset F of U, there is a selection function sU on the whole index set U such that for every finite subset F of U, there is a finite subset F’ of U such that F ⊆ F’ and such that sU and sF’ coincide on F.
We now apply this lemma. For every finite family F of entourages of X, and for every R ∈ F, the family {(R ∩ R–1)[z] | z ∈ ER} is a finite cover of X. Let us enumerate the entourages of F as R1, …, Rn. Then the family of all the intersections (R1 ∩ R1–1)[z1] ∩ … ∩ (Rn ∩ Rn–1)[zn], where (z1, …, zn) ranges over ER1 × … x ERn, is also a finite cover of X. By using (*) again, there is a tuple (z1, …, zn) ∈ ER1 × … x ERn such that I[(R1 ∩ R1–1)[z1] ∩ … ∩ (Rn ∩ Rn–1)[zn]] is cofinal in I. We pick one, and we define the selection function sF as mapping R1 to z1, …, Rn to zn. Good.
We build a global selection function sU, as given in Rado’s selection lemma above.
Our next task is to build a subset JF of I, for each finite set F of entourages. As above, let us enumerate the entourages of F as R1, …, Rn, let z1 be sU(R1), …, zn be sU(Rn), and finally let JF be I[(R1 ∩ R1–1)[z1] ∩ … ∩ (Rn ∩ Rn–1)[zn]]. We may write this more synthetically as: JF = I[∩R ∈ F (R ∩ R–1)[sU(R)]].
The point of this construction is to ensure that JF is antitonic in F, namely that, for every pair of finite sets F and F’ of entourages such that F ⊆ F’, then JF ⊇ JF’. The latter formula for JF should make this obvious.
We verify that JF is a cofinal subset of I, for every finite subset F of U. This is proved by using the funny, final property ensured by Rado’s selection lemma: there is a finite subset F’ of U such that F ⊆ F’ and such that sU and sF’ coincide on F. Hence JF = I[∩R ∈ F (R ∩ R–1)[sF’(R)]]. That contains I[∩R ∈ F’ (R ∩ R–1)[sF’(R)]] (note that F was replaced by F’ in the subscript to the intersection), which is cofinal in I by construction. Every subfamily of I that contains a cofinal subfamily is itself cofinal, so JF is cofinal.
Whew. That was the hard part. And we are probably lost, too. If anything, just remember that we have build a cofinal subset JF of I, for every finite subset F of U, and that JF varies antitonically as a function of F. All the complications above stemmed from the fact that we will need both properties, but we can now forget all about them. (Oh well, we will also use the actual definition of JF, but this will not happen before the third paragraph below: for now, please relax.)
Let us return to our initial goal. We wish to build a subnet of (xi)i ∈ I, ⊑ in X. Its index set P is the collection of pairs (F, j) where F is a non-empty finite family of entourages and j is in JF, ordered pairwise, namely: (F, j) ⪯ (F‘, j’) if and only if F ⊆ F’ and j ⊑ j’.
Let us verify that P is directed. It is non-empty: pick any entourage R of X, define F as {R}, and then pick any element j of the cofinal (hence non-empty) family JF. Next, directedness per se. Let (F1, j1) and (F2, j2) be two elements of P. We build an upper bound (F, j) of those two by first defining F as F1 ∪ F2. Since I is directed, there is an index i ∈ I such that j1, j2 ⊑ i. Now we have seen that JF is cofinal, so it contains an element j such that i ⊑ j. It follows that (F1, j1), (F2, j2) ⪯ (F, j).
We now check that the subnet (xα(F, j))(F, j) ∈ P, ⪯ is Cauchy. We fix an arbitrary entourage R0 of X. Let R1 be an entourage of X such that R1 o R1 is included in R0. Let also j0 be any element of J{R}. For all elements (F1, j1) and (F2, j2) of P such that ({R1}, j0) ⪯ (F1, j1) ⪯ (F2, j2), we claim that (xα(F1, j1), xα(F2, j2)) is in R0, namely, that (xj1, xj2) is in R0. Since j1 is in JF1, xj1 is in ∩R ∈ F1 (R ∩ R–1)[sU(R)], and is in particular in (R1 ∩ R1–1)[sU(R1)]. Similarly, xj2 is in (R1 ∩ R1–1)[sU(R1)]. In particular (xj1,sU(R1)) and (sU(R1),xj2) are both in R1, so (xj1, xj2) is in R1 o R1, hence in R0.
We are done! We have found a Cauchy subnet of (xi)i ∈ I, ⊑. Since X is Smyth-complete, it converges to some point in Xsym. To sum up, every net has a subnet that converges in Xsym, so Xsym is compact. ☐
Conclusion
This is already part III in a series on quasi-uniform spaces. I do not know yet what I will talk about next time, but a safe bet would be the say that this will again be on quasi-uniform spaces; most likely completions. In the quasi-metric case, formal balls are an amazing tool to approach notions of completeness, but we desperately miss an analogue of that construction in the realm of quasi-uniform spaces… what a pity.
- Hans-Peter Albert Künzi and Guillaume C. L. Brümmer. Sobrification and bicompletion of totally bounded quasi-uniform spaces, Mathematical Proceedings of the Cambridge Philosophical Society 101(2):237–247, 1987.
— Jean Goubault-Larrecq (December 20th, 2020)