I posted this question on MSE a few days ago, but got no response (despite a bounty). I hope it will get more answers here, but I'm afraid it might not be appropriate as I'm not sure it's actually research-level. Please do tell me if it's not appropriate and if possible tell me how to modify the question so that it may become appropriate (if such modifications exist)


I recently saw a video of a presentation of Andrej Bauer [here](https://www.youtube.com/watch?v=21qPOReu4FI&ab_channel=InstituteforAdvancedStudy) about constructive mathematics; and there are two examples of locales he mentions that strike me : he explains quickly what the space of random reals might be, by saying that it's the locale of reals that are in every measure $1$ subset of $[0,1]$ (for instance): as he says, of course there are no such reals, but that should not stop us from considering the space of these reals, which may have interesting topological properties even if it has no points.


Similarly in a constructive setting (or classical setting without AC) some rings may have no prime/maximal ideals, and so their spectrum as usually conceived is uninteresting. But that should not stop us from studying the space of prime/maximal ideals with the Zariski topology, even if it has no points.


My questions are related to these examples specifically and to generalizations:


Is the first example of random reals in any way connected to the random reals one mentions in forcing ? e.g. is forcing to add some random reals in any way connected to considering the topos of sheaves on the locale of random reals ?


Has the second example been extensively studied ? What sort of properties can we get from the study of this "Zariski locale" ?


Is there some form of general theory of locales as spaces of imaginary points ? For instance is this how one usually sees locales intuitively; or better is there some actual theory (more than a heuristic) of constructing pointless (or with few points) spaces of objects that we'd like to exist but don't actually exist ? This is very vague so I'll give a further example of what one might envision: if two first-order structures $A$ and $B$ aren't isomorphic but $A\cong\_p B$, we might want to study the space of isomorphisms of $A$ and $B$, which would ideally be a pointless locale. One could say something similar about generic filters of a poset when one is trying to do some forcing : from the point of view of the small model, these generic filters don't exist: we could envision a space of generic filters. In these four cases we have some objects that don't exist (random reals, maximal ideals, isomorphisms) but that we can define and that in some very vague sense ought to exist, and so we construct the space of these objects; but it turns out that this space can have no points at all: is there a general theory of this sort of thing ?


These questions are very vague so I hope they're appropriate. I'll appreciate answers with references, but I'd also very much like answers that themselves provide some intuition (though a bit more technical than what I've expressed in the question), and some thoughts.



Locales as spaces of ideal/imaginary points
I can only answer some of your questions.


Yes, the Zariski locale is extensively studied. It's one of the ways of setting up scheme theory in a constructive context: Don't define schemes as locally ringed spaces, but as locally ringed locales. The locally ringed locale $\mathrm{Spec}(A)$ always enjoys the universal property we expect of it, namely that morphisms $X \to \mathrm{Spec}(A)$ of locally ringed locales are in one-to-one correspondence with ring homomorphisms $A \to \mathcal{O}\_X(X)$. (Only) if the Boolean Prime Ideal Theorem is available (a slightly weaker form of the axiom of choice), one can show that the Zariski locale has *enough points*. In this case it's isomorphic to the locale induced from the classical topological space of prime ideals (equipped with the Zariski topology).


(Note that the preceding paragraph assumes that you define the Zariski locale of a ring $A$ to be the locale of *prime filters* of $A$, not the locale of *prime ideals*. (A prime filter is a direct axiomatization of what's classically simply the complement of a prime ideal.) The locale of prime ideals also exists, but does not coincide with the true Zariski locale; classically, it is isomorphic to the topological space of prime ideals equipped with the flat topology.)


Yes, there is a general theory of locales as spaces of imaginary points. Briefly, to any *propositional geometric theory* $T$ (roughly speaking, a collection of axioms of a certain form, such as the axioms of a prime ideal or of a prime filter), there is a *classifing locale* $\mathrm{Set}[T]$. The points of this locale are exactly the $\mathrm{Set}$-based models of $T$ (that is, the actual prime ideals or the actual prime filters). It can happen that there are no such models, yet still the theory $T$ is consistent. In this case the classifying local doesn't have any points, yet still is not the trivial locale.


Any locale $X$ is the classifying locale of some propositional geometric theory, a theory which deserves the name "theory of points of $X$".


The theory of classifying locales indeed allows you to construct spaces (locales) of things which aren't expected to exist, just by writing down the axioms of the hypothetical objects. A particularly tantalizing example is the locale of surjections from $\mathbb{N}$ to $\mathbb{R}$. There are no such surjections, of course, therefore this locale doesn't have any points. But it's still nontrivial. The topos of sheaves over this locale contains an epimorphism from $\underline{\mathbb{N}}$, the constant sheaf with stalks $\mathbb{N}$, to $\underline{\mathbb{R}}$; this epimorphism could be named the "walking surjection from $\mathbb{N}$ to $\mathbb{R}$", as any such surjection in any topos is a pullback of this one.


(The reals starred in the preceding paragraph only for dramatic purposes. The previous paragraph stays correct if $\mathbb{R}$ is substituted by any inhabited set. The described construction has been used, as one of a series of reduction steps, in Joyal and Tierney's celebrated monograph *An Extension of the Galois Theory of Grothendieck*.)


An excellent entry point to the business of locales as spaces of ideal points is [a very accessible expository note](https://www.cs.bham.ac.uk/~sjv/LocTopSpaces.pdf) by Steve Vickers. (When you've finished with this one, be sure to check out his further surveys, all available on his webpage, for instance [this one](https://www.cs.bham.ac.uk/~sjv/GeoAspects.pdf).)


