Overlap Algebras as Almost Discrete Locales

Boolean locales are"almost discrete", in the sense that a spatial Boolean locale is just a discrete locale (that is, it corresponds to the frame of open subsets of a discrete space, namely the powerset of a set). This basic fact, however, cannot be proven constructively, that is, over intuitionistic logic, as it requires the full law of excluded middle (LEM). In fact, discrete locales are never Boolean constructively, except for the trivial locale. So, what is an almost discrete locale constructively? Our claim is that Sambin's overlap algebras have good enough features to deserve to be called that. Namely, they include the class of discrete locales, they arise as smallest strongly dense sublocales (of overt locales), and hence they coincide with the Boolean locales if LEM holds.


Introduction
A Boolean space is a topological space whose open sets form a complete Boolean algebra (instead of a mere complete Heyting algebra).A Boolean space is almost discrete: a discrete space is just a T 0 (Kolmogorov) Boolean space or, equivalently, a sober Boolean space.In other words, a spatial Boolean locale [Joh86] is the same thing as a discrete locale.For this reason, a Boolean locale can be considered as an "almost discrete" locale.All this is fine, at least when classical logic is assumed.Intuitionistically, on the contrary, the property of being Boolean is no longer a necessary condition for discreteness.Indeed, if one wants to generalise the above discussion from the usual mathematical universe of sets to the internal universe provided by a topos E, then discrete locales (except for the trivial locale) fail to be Boolean, in general.For instance, the subobject classifier Ω in E is a discrete locale (it is the powerset of the terminal object), but it is Boolean precisely when the Law of Excluded Middle (LEM) holds in the internal language of E, that is, when E is a Boolean topos.Therefore, in a sense, the notions of Boolean-ness and discreteness appear orthogonal to each other from a constructive point of view.So what could serve as an intuitionistic definition of an almost discrete locale?(And so, in a sense, what is a more intuitionistically robust analogue for Boolean-ness?)We would like our definition of an almost discrete locale to satisfy at least the following two conditions: (i) every discrete locale has to be almost discrete, and (ii) almost discrete locales must coincide with Boolean locales classically.
The aim of this paper is to show that Sambin's notion of an overlap algebra [Samar, CS10, Cir13, CMT13, CC20] can serve as a constructive definition of an almost discrete locale.Our claim is motivated and supported by some results, the main of which is given in Section 3: overlap algebras arise precisely as the smallest strongly dense sublocales [Joh89]  of overt locales.Therefore overlap algebras give an intuitionistically robust account of Boolean-ness.And in the presence of LEM, overlap algebras and Boolean locales coincide (because, in that case, strong denseness boils down to denseness, and overtness comes for free).
Furthermore, every discrete locale is an overlap algebra, actually an atomic one.This prompts the natural question: what is a "minimal" condition which makes an overlap algebra discrete?(If we agree that the requiring atomicity seems too strong and not topological at all.)It is natural to wonder if spatiality could be such a property (that is the case in light of LEM).Spatial overlap algebras are sober spaces in which every open set is "weakly" regular in the sense that it coincides with the interior of the set of its adherent points.Spaces (not necessarily sober) enjoying this property are studied in Section 4 and they are compared with Boolean ones (the two notions coincide classically).Unfortunately, spatial overlap-algebras cannot be proved to be discrete constructively, as we show in Subsection 4.1.So understanding what the right condition is (which makes an overlap algebra discrete) is still an open question.
Section 2 contains a few preliminaries about locales and a specific limiting result about Boolean locales within an intuitionistic framework (roughly speaking, the result is that there are no nontrivial, overt, Boolean locales).
Unless otherwise stated, we work constructively in the sense that we assume neither LEM nor the Axiom of Choice.

Preliminaries about frames and locales
A frame is a complete lattice (that is, a poset with arbitrary joins and meets) satisfying the infinite distributive law x (half of the equality comes for free, of course).A frame homomorphism is a map which preserves finite meets (hence, in particular, the top element 1) and arbitrary joins (hence, in particular, the bottom element 0).Frames are the same thing as complete Heyting algebras, with x → y = {z | z ∧ x ≤ y}, although a frame homomorphism need not preserve implications.We often write −x instead of x → 0, the pseudo-complement of x.
The category Loc of locales is the opposite of the category of frames.For an arrow f : X → Y in Loc, we write Ωf : ΩY → ΩX for the corresponding frame homomorphism.
Loc has a terminal object 1 whose corresponding frame, which is usually written Ω instead of Ω1, is the powerset of the set 1 = {0}; Ω can be interpreted as the set of truth-values, and it is in bijection with the two-element set 2 = {0, 1} if and only if LEM holds.
The following are some features of Ω that are valid intuitionistically.For p, q ∈ Ω, one has p ≤ q precisely when p = 1 implies q = 1.Moreover, p ̸ = 1 if and only if p = 0 if and only if −p = 1.1 Also, for {p i | i ∈ I} ⊆ Ω, one has ( i∈I p i ) = 1 if and only if p i = 1 for some i ∈ I.
Given a locale X, the unique arrow !X : X → 1 in Loc corresponds to the frame homomorphism defined by Ω!
(which happens precisely when Ω! X preserves all meets).In particular, Pos X preserves joins.
Classically, every locale is overt, and Pos X (x) = 1 if and only if x ̸ = 0. Constructively Pos X (x) = 1 can be read as a positive (strong) way to express that x is different from 0. We call Pos X the positivity predicate of X.
Every topological space determines a locale X, where ΩX is the frame of open sets.Locales obtained in this way are called spatial.They are always overt and Pos X (x) = 1 means that the open set x is inhabited.A locale X is discrete if ΩX is the powerset of some given set; so every discrete locale is spatial and can be seen as the locale corresponding to a discrete space. 2 Different spaces can happen to determine the same spatial locale, the canonical one (up to homeomorphism) being the sober one, that is, the one in which every (inhabited) completely prime filter of opens is the collection of open neighbourhoods of a unique point.

Sublocales.
A closure operator on a poset is an endofunction c such that the conditions x ≤ cx = ccx and x ≤ y ⇒ cx ≤ cy hold identically.We write Fix(c) for the collection of all fixed points of c; since c is idempotent, Fix(c) = Im(c), the image of c.
A nucleus on a locale X is a closure operator j on ΩX that, in addition, preserves binary meets.In this case, Fix(j) is a frame where finite meets are computed in ΩX and joins are given by j-closure of joins in ΩX.A notable example of a nucleus is the map x → − − x (double negation nucleus).The locale X j corresponding to the frame ΩX j = Fix(j) is what is called a sublocale of X [Joh86] and the mapping x → jx gives a regular monomorphism X j → X in Loc. 3 For j 1 and j 2 nuclei on X, X j 1 is a sublocale of X j 2 when Fix(j 1 ) ⊆ Fix(j 2 ) or, equivalently, when j 2 ≤ j 1 pointwise. 4Therefore, the poset of all sublocales of a given locale X is the opposite of the poset N (X) of nuclei on X (with pointwise order); actually, N (X) is a frame [Joh86].The join of a family {X j i } i∈I of sublocales corresponds to the pointwise meet of nuclei x → i∈I (j i x).Meets of sublocales are better seen from another perspective.The sets of the form Fix(j), for j a nucleus on X, are precisely the subsets of ΩX which are closed under arbitrary meets and which contain x → y whenever they contain y (see [Joh86, PP06, PP12] for details). 5Therefore, an arbitrary intersection of sets of that form still has the same form, and hence i∈I Fix(j i ) corresponds to a sublocale, which is the meet of the family {X j i } i∈I .
The sublocale generated by a family of elements.Given any a ∈ ΩX, the nucleus x → (x → a) → a defines the smallest sublocale of X whose frame contains a.More generally, given any subset A ⊆ ΩX, the nucleus gives the smallest sublocale of X whose frame contains A. For x, y ∈ ΩX we thus have The nucleus "generated" by a closure operator.Given a closure operator c : ΩX → ΩX, one can consider the sublocale of X generated by Fix(c) in the sense of the previous paragraph.The corresponding nucleus is j Fix(c) (y) = b∈ΩX ((y → cb) → cb).We claim that this can be written as Note that j Fix(c) is the nucleus which best approximates c in the following sense: (i) j Fix(c) (x) ≤ c(x) for all x in ΩX and (ii) if j is another nucleus on X such that j(x) ≤ c(x) for all x ∈ ΩX, then j(x) ≤ j Fix(c) (x) for all x ∈ ΩX.

Boolean locales.
A locale X is Boolean when ΩX is a Boolean algebra, that is, x ∨ −x = 1 holds identically or, equivalently, − − x = x holds identically.
All possible examples of Boolean locales are of the form X −− , the sublocale corresponding to the double negation nucleus on some given locale X.In fact, X is Boolean if and only if X −− = X.Moreover (see [Joh86] Exercise II.2.4), a sublocale X j → X is Boolean if and only if it is generated by a singleton (in the sense of the previous section), that is, there exists a ∈ ΩX such that jx = (x → a) → a for all x.
Classically, every discrete locale is Boolean.Intuitionistically, on the contrary, discrete locales typically fail to be Boolean: an exception is the (locale whose frame is the) powerset of the empty set, that is, the trivial locale.Here we are going to show a particular limiting result about Boolean locales within a constructive framework.
Fact: Ω is the free suplattice 6 on one generator.In other words, for any complete lattice X and any a ∈ X, there is precisely one join-preserving function f : Ω → X such that f (1) = a.Indeed, every p ∈ Ω can be written as {q ∈ Ω | q = 1 = p}, and hence the only possible candidate for f is given by f (p) = {f (q) | q = 1 = p} = {r ∈ X | r = a and p = 1}.Proposition 2.1.If there exists a locale X with the following two properties (1) X is Boolean, and (2) there exists a join-preserving map F : ΩX → Ω such that F (1) = 1, then Ω is Boolean, that is, LEM holds.
Proof.The map F • Ω! X : Ω → Ω preserves joins and top, and hence it is the identity by the fact recalled above.Moreover, In view of this result, there are a number of things one cannot expect to prove within an intuitionistic setting, in general.For instance, it is impossible to construct a locale X which is Boolean, overt and with Pos X (1) = 1.Also, it is impossible to construct any point of any Boolean locale. 7his limitation appears still more striking within a topos E in which LEM is provably false: if ¬(∀p ∈ Ω)(p ∨ −p) is provable in the internal logic of E,8 then the only overt Boolean locale in E is the trivial locale (because Pos X (1) ̸ = 1 means 1=0).
A topological space is Boolean if the corresponding locale is Boolean, that is, if its open subsets form a Boolean algebra.It is a corollary of the above discussion that the existence of an inhabited and Boolean topological space is equivalent to LEM.

Overlap algebras
Let int be the interior operator on the subsets of a topological space X.An open set A is regular if it equals the interior of its closure, that is, A = int ( int A c ) c , where ( ) c is the set-theoretic complement.Since int A c is just the pseudo-complement −A in the frame of open sets, we have that A is regular when A = − − A. For this reason, an element x of a frame/locale is said regular if x = − − x; and a locale is Boolean if and only if all its elements are regular.
Given a set D of points, there are at least two (classically equivalent) ways to define the topological closure of D. One possibility is to take ( int D c ) c .The other possibility is to consider the set cl D of all adherent points of D; so x ∈ cl D if and only if x ∈ A ⇒ D ≬ A for every open A. Here, following Sambin, we write X ≬ Y to mean that X ∩ Y is inhabited (classically, X ∩ Y ̸ = ∅), that is, X and Y overlap each other.The latter definition, which is the one commonly adopted in a constructive approach, results in an intuitionistically weaker notion of closure9 in the sense that D ⊆ cl D ⊆ ( int D c ) c always holds, and hence D = ( int D c ) c implies D = cl D. Therefore, every closed subset is weakly closed; and the converse holds only classically. 10y replacing ( int A c ) c with cl A we get an intuitionistically weaker notion of regularity for open sets.We shall use the term weakly regular for an open set A such that A = int cl A.
There are spaces (notably the discrete ones) in which all open subsets are weakly regular, so that the following equation (between operators on subsets) holds: notion of strong density recalled in Subsection 3.1 below.For our purposes, however, we do not need to recall explicitly such a notion of weak closure for sublocales (thankfully, since an intrinsic characterization of weakly closed nuclei is not at hand [Joh89, p.7]). 10 Indeed, cl D = D = int D holds for all D in a discrete space (such as Ω); therefore the statement "( int D c ) c = cl D holds identically in all spaces" is precisely LEM .A space of this kind is a "positive" version of a Boolean space (perhaps the name "weakly Boolean" would be appropriate); it corresponds, as we will see, to a spatial overlap algebra.
Here we show how to express "weakly regular" in the point-free language of overt locales.As usual, we seek inspiration in the spatial case.
For X a spatial locale and a ∈ ΩX, the open set int cl a is the union of all opens x ∈ ΩX such that x ⊆ cl a.By definition, x ⊆ cl a means that z ≬ x implies z ≬ a for all z ∈ ΩX.Now x ≬ y holds precisely when Pos X (x ∧ y) = 1 (spatial locales are overt).Thus, we propose the following.Definition 3.1.Let X be an overt locale with positivity predicate Pos X .We say that a ∈ ΩX is weakly regular when In other words, a ∈ ΩX is weakly regular when, for every x ∈ ΩX, We shall show (Proposition 3.6) that the weakly regular elements of an overt locale X form a sublocale which is, in fact, the smallest strongly dense sublocale of X.11 Definition 3.2.An overlap algebra (o-algebra for short) is an overt locale in which all elements are weakly regular.More explicitly, an o-algebra is an overt locale X such that, for every x, y ∈ ΩX, if Pos X (z ∧ x) ≤ Pos X (z ∧ y) f or all z ∈ ΩX, then x ≤ y . (3.4) The following Proposition is evidence of the fact that an o-algebra is an appropriate intuitionistic version of a Boolean locale, that is, of an almost discrete locale.Further evidence will be provided by Corollary 3.7, which we consider the main result of this paper.
Proposition 3.3.Let X be a locale.
(1) If X is discrete, then X is an o-algebra. 122) If X is overt and Boolean, then X is an o-algebra.
(3) Classically, if X is an o-algebra, then X is (overt and) Boolean.
Item 1. says that every powerset is an o-algebra in a natural way: this is the prototypical example which actually motivated the introduction of o-algebras by Sambin (see [Samar]  and also [CS10, Cir13, CMT13]).Items 2. and 3. together say that o-algebras and Boolean locales coincide classically; so spatial o-algebras correspond to discrete spaces, classically, as a consequence of Proposition 4.1 below.Item 3. cannot hold intuitionistically because discrete locales are o-algebras by item 1., but they are not Boolean unless LEM holds.Item 2. is of questionable interest, of course, because of Proposition 2.1; indeed, if LEM fails, then the only overt and Boolean locale is the trivial one (that is, the powerset of the empty set).
O-algebras are often (and originally were) presented as complete lattices equipped with an extra binary relation > < (intended as an algebraic version of the relation ≬ between subsets), essentially as follows.
Proposition 3.4.The frame underlying an o-algebra is precisely a complete lattice equipped with a symmetric relation > < such that the following conditions are identically satisfied.
Proof.Assume we have a complete lattice satisfying the three conditions above.First, we show that such a lattice satisfies the infinite distributive law (2.1)so that it is a frame.By (3.5c), that is equivalent to show that z > < (x ∧ i∈I y i ) ⇒ z > < ( i∈I (x ∧ y i )) for all z.So assume the premise; by (3.5a) one gets (z ∧ x) > < ( i∈I y i ) and hence (z ∧ x) > < y i for some i ∈ I, by (3.5b).By (3.5a), that becomes z > < (x ∧ y i ) for some i ∈ I and so z > < ( i∈I (x ∧ y i )) by (3.5b).Second, we show that the corresponding locale X is overt with Pos X (x) = 1 ⇔ x > < x.We must check that x > < x ⇒ p = 1 if and only if x ≤ Ω! X (p).By (3.5c) and (3.5b), x ≤ Ω! X (p) means that, for every z, if z > < x, then both z > < 1 and p = 1.Clearly, z > < x yields z > < 1; therefore x ≤ Ω! X (p) is just equivalent to ∀z(z > < x ⇒ p = 1).By logic, this is just ∃z(z > < x) ⇒ p = 1.Now ∃z(z > < x) is tantamount to x > < x and we are done.Third, we show that (3.4) holds for Pos X .By (3.5a), (x ∧ y) > < (x ∧ y) is equivalent to x > < y.So Pos X (x ∧ y) = 1 is equivalent to x > < y and hence (3.4) follows by (3.5c).We now come to the opposite direction.Let X be an o-algebra.We define x > < y as Pos X (x ∧ y) = 1.Clearly, > < is symmetric and satisfies (3.5a).Also, (3.5b) easily follows from (2.1) and from the fact that Pos X preserves joins.Finally, (3.5c) is a consequence of (3.4) and of the fact that Pos X is monotone.
3.1.O-algebras are smallest strongly dense sublocales.There exists a well-known connection between Boolean locales and dense sublocales.A sublocale X j → X is dense if ΩX j contains the bottom element of ΩX, that is, if j(0) = 0.And X −− is the smallest dense sublocale of X.Now X −− is Boolean, and every Boolean locale is of this form (we refer the reader to [Joh86] for details).Therefore, Boolean locales arise precisely as smallest dense sublocales.Here we want to show that o-algebras enjoy an analogous characterization.A stronger notion of density was introduced in [Joh89]: X j → X is strongly dense if j(Ω! X (p)) = Ω! X (p) for all p ∈ Ω, that is, Ω! X j (p) = Ω! X (p) for all p ∈ Ω.In particular,