When I wrote my latest blog post, there were many things I thought would be useful to know about sublocales. Those eventually turned out to be useless in that context. However, I think they should be known, in a more general context.
Remember that a sublocale is a locale-theoretic notion that best represents the notion of subspace of topological space. I have based several intuitive arguments on the connection between the two notions, imagining that sublocales are some kind of abstract notion of subspace.
Not so: sublocales and subspaces can be very different. The latest post is an illustration of that, and today we shall see two other cases where caution has to be exerted, and which are perhaps simpler:
- Isbell’s density theorem;
- The difference between intersections of sublocales and intersections of subspaces.
Isbell’s density theorem
Given a frame Ω, we can form the residuation (or intuitionistic implication operator) ⇒: u ⇒ v is the largest element w such that u ⋀ w ≤ v.
When v = ⊥, we obtain intuitionistic negation ¬u. When Ω is the frame of opens of a topological space, ¬U is the interior of the complement of U — not the complement of U, which would in general fail to be open.
Let S0 be the family {¬u | u ∈ Ω}. This is a sublocale. Let us check this. S0 is closed under arbitrary infima because the intuitionistic negation of ⋁i ∈ I ui ∈ I is ⋀i ∈ I ¬ui ∈ I (beware: the dual property is wrong: the negation of an infimum need not be the corresponding sup of negations). And for every element ¬u of S0, for every v in Ω v ⇒ ¬u is equal to ¬(v ⋀ u), hence is in S0.
S0 is a very particular sublocale, as we shall see. By the way, Picado and Pultr [2] call it BL.
Every sublocale S has a closure cl(S), which is simply equal to the smallest closed sublocale c(u) containing S. Recall that c maps suprema to infima, so this is equal to the intersection of all the closed sublocales c(u) that contain S, or equivalently this is c(u) where u is the supremum of all the elements v of Ω such that S ⊆ c(v).
Recall also that c(u) is simply the upward closure ↑u. For example, c(⊥), the largest closed sublocale, is just Ω itself, the largest sublocale at all.
Call a sublocale S dense if and only if cl(S) is the largest possible sublocale, Ω itself. In other words, S is dense if and only if the only u in Ω such that S ⊆ ↑u is ⊥.
We can simplify the definition further. For every sublocale S, the infimum of all the elements of S is again in S, since S is closed under arbitrary infima. This shows that every sublocale has a least element u. Clearly S ⊆ ↑u, so if S is dense, then u=⊥. Conversely, any sublocale containing ⊥ is dense.
It follows that:
A sublocale S is dense if and only if it contains ⊥.
Immediately, we obtain that S0 is a dense sublocale: indeed ⊥ is the intuitionistic negation of ⊤, ¬⊤.
More strangely — this is Isbell’s density theorem (see III.8.3 in [2]):
Lemma. S0 is the smallest dense sublocale.
Proof. Imagine S is another dense sublocale. S contains ⊥, and since S contains v ⇒ u for every v in Ω and every u in S, then by taking u=⊥, it contains ¬v for every v in Ω. In other words, S0 is included in S. ☐
That has absolutely no equivalent in the realm of topological spaces: if there is a smallest dense subset, then it is the intersection of all dense subsets of a space; but that is in general not dense. For example, if you start from a space X where no point is isolated (i.e., the space is dense-in-itself; a point x is isolated if {x} is open), then X-{x} is dense for every point x, and the intersection of those sets when x varies is empty—certainly not dense, unless X itself is empty. The space R of real numbers with its usual T2 topology, for one, is dense-in-itself, and has no smallest dense subset.
In the pointfree version of R, however, the smallest dense sublocale S0 is the collection of interiors of closed sets of R, or equivalently, the collection of regular open subsets of R (see Exercises 8.1.7 and 8.1.8 in the book). There are plenty of them! Every open interval of R is regular open, for example. In particular S0 is very different from the smallest locale, which is just {⊤} (where ⊤=X itself).
Intersection of sublocales and sublocales of intersections
We apply that to elucidate the relationship between intersection of sublocales and intersection of subspaces.
Let us recall that the set of sublocales Sl(Ω) of a frame Ω forms a coframe under inclusion. The greatest lower bound of two sublocales S1 and S2 is simply their intersection.
In particular, the intersection of two sublocales S1 and S2 is another sublocale. One may think of that construction as the analogue of taking the intersection of two subspaces of a topological space, but one should really be cautious here; in fact the analogy breaks pretty quickly, as we shall see.
Any subspace A of a topological space X gives rise to a sublocale of O(X), given by SA = {U ∈ O(X) | U is the largest open subset of X whose intersection with A equals U ∩ A}. In other words, SA is the set of fixed points Fix(νA) of the nucleus νA mapping each U ∈ O(X) to the largest open subset V of X such that V ∩ A=U ∩ A.
SA is, naturally, order-isomorphic to O(A). We have:
Lemma. The mapping A ↦ SA is monotonic.
Proof. Assume A ⊆ B. We must show that SA ⊆ SB. Since the coframe of sublocales is isomorphic to the opposite of the frame of nuclei, it is equivalent to show that νB ≤ νA. For every open U, νB (U) is the largest open set V such that V ∩ B=U ∩ B. Taking intersections with A, and recalling that B ∩ A = A since A ⊆ B, we obtain V ∩ A=U ∩ A. Hence V = νB (U) is an open subset whose intersection with A equals U ∩ A, and is therefore certainly included in the largest such open subset, νA (U). ☐
Given two subspaces A and B of X, how do SA ∩ SB and SA ∩ B compare? The following is easy.
Lemma. SA ∩ B is included in SA ∩ SB.
Proof. It suffices to show that SA ∩ B is included both in SA and in SB, which follows from the previous Lemma. ☐
In general, SA ∩ B and SA ∩ SB differ, and can in fact differ by a very large margin. Consider the subspace A=Q of R consisting of the rational numbers, and the complementary subspace B of irrational numbers. Clearly A ∩ B is empty, so SA ∩ B is the smallest sublocale {⊤}. But A and B are both dense (in the usual sense) in R, so the sublocales SA and SB are both dense, in the sense we have just given. By Isbell’s density theorem, they both contain S0, and we have seen that it is much larger than {⊤}.
How should one view locales and sublocales?
A normal first reaction to that kind of result would be: `this cannot be the right theory, let us forget about it’. I am sometimes tempted to abide by this. However, the mathematics of locales is so beautiful that there should be some truth in it.
Here is one possible explanation of the difference between sublocales and subspaces. Instead of considering points as the primary ingredients of a space, and opens as some kind of structure that would play a useful, but secondary role, let us recognize that spaces contain both points and some glue between the points. (I don’t remember where I have seen this analogy. The idea is certainly not mine, perhaps Johnstone’s or Isbell’s.) The opens, and notions of continuity are here to express how elastic that glue is.
Topology is one extreme, where spaces are spaces of points, and, oh, in passing, they are glued together in some way.
Locale theory is another extreme, where you recognize a space as a big amount of glue, and, oh, in passing, that glue sometimes connects infinitesimal things called points.
A point of view that reconciles the two is the equivalence between sober spaces and spatial lattices: points exist, and so does glue (the topology). Moreover, glue gives structure to points, but also, conversely, points give structure to the glue, in the case of spatial lattices. Chu spaces are one way of representing both on an equal footing, and have been actively promoted by Vaughn Pratt since 1994.
- John Isbell. Product spaces in locales. Proceedings of the American Mathematical Society, 81(1), January 1981.
- Jorge Picado and Aleš Pultr. Frames and locales — topology without points. Birkhäuser, 2010.
— Jean Goubault-Larrecq (September 25th, 2017)