Stone duality for Kolmogorov locally small spaces

We prove three new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of compact (not necessarily Hausdorff) open sets as objects and spectral mappings respecting the decent lumps and satisfying a boundedness condition as morphisms as well as it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. Some theory of strongly locally spectral spaces is developed.


Introduction
Stone Duality is one of the most important dualities in mathematics. It is very widely known for Boolean algebras and a little less known for bounded distributive lattices. In fact, M. H. Stone's two fundamental papers [20,21] described duality between generalized Boolean algebras (or Boolean rings) and Hausdorff locally compact Boolean spaces, where usual Boolean algebras (or unital Boolean rings) correspond to Hausdorff compact Boolean spaces. He achieved a beautiful theory of ideals in Boolean rings and a beautiful theory of representations of Boolean rings in powersets. The case of distributive lattices was considered by M. H. Stone in [22]. Many versions of this duality exist (see, for example, [7] or [9] for further literature), including versions of Priestley Duality proved by H. Priestley in [18] with many consequences developed in [19]. Stone Duality for bounded distributive lattices, while considered already in a much broader context in [8], has been presented in detail in a recent monograph [5] by M. Dickmann, N. Schwartz and M. Tressl.
In this paper, three new versions of Stone Duality are proved: for small spaces, for locally small spaces with usual morphisms (bounded continuous mappings) and for locally small spaces with bounded strongly continuous mappings as morphisms. In each of the cases, the Kolmogorov separation axiom (T 0 ) is assumed.
Locally small spaces may be understood to be a special kind of generalized topological spaces in the sense of Delfs and Knebusch ( [16]), which in turn are a special form of Grothendieck topologies (see [2,13]) or G-topologies of [4]. Locally small spaces were used in o-minimal homotopy theory ( [2,15]). A simpler language for locally small spaces was introduced and used in [14] and [16], compare also [17]. We continue developing the theory of locally small spaces in this simple language, analogical to the language of Lugojan's generalized topology ( [12]) or Császár's generalized topology ( [1]), where a family of subsets of the underlying set is satisfying some, but not all, conditions for a topology.
The main result of the paper reads as follows: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with distinguished decent lumps and with bornologies in the lattices of compact (not necessarily Hausdorff) open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms and is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices satisfying a domination condition and respecting those decent lumps as morphisms.
Small spaces are a special case of locally small spaces, with some compactness flavour. While we meet small spaces as these underlying definable spaces over structures with topologies, we meet locally small spaces as those underlying analogical locally definable spaces ( [14,16]). We show that a Kolmogorov small space is essentially a patch dense subset of a spectral space. More precisely: the category of Kolmogorov small spaces and continuous mappings is equivalent to the category of spectral spaces with distinguished patch dense subsets and spectral mappings respecting those patch dense subsets and is dually equivalent to the category of bounded distributive lattices with distinguished patch dense sets of prime filters and homomorphisms of bounded lattices respecting those patch dense sets. This means that spectralifications of a Kolmogorov topological space may be constructed by choosing bounded sublattice bases of the topology.
We have another version of Stone Duality: for Kolmogorov locally small spaces with bounded strongly continuous mappings. This category is equivalent to the category of strongly locally spectral spaces with distinguished patch dense subsets as objects and strongly spectral mappings respecting those patch dense subsets as morphisms and is dually equivalent to the category of distributive lattices with zeros and distinguished patch dense sets of prime filters as objects and lattice homomorphisms respecting zeros and those patch dense sets and satisfying a condition of domination as morphisms. Moreover, some theory of strongly locally spectral spaces is developed.
The paper is organized in the following way: Section 2 introduces categories SS 0 and LSS 0 , Section 3 deals with SpecD and SpecBD, Section 4 introduces LatD and LatBD. Section 5 gives the main theorem for LSS 0 and a version for SS 0 . Section 6 deals with spectralifications of Kolmogorov spaces. Section 7 introduces the categories slSpec and slSpec s . Section 8 deals with ZLat and establishes a dual equivalency between slSpec s and ZLat. Section 9 concludes with Stone Duality for LSS s 0 . Examples throughout the paper illustrate the topic.
Regarding the set-theoretic axiomatics for this paper, we follow Saunders Mac Lane's version of Zermelo-Fraenkel axioms with the axiom of choice plus the existence of a set which is a universe ( [11, p. 23]).
We shall freely use the notation for family intersection and family difference, compatible with [13,14,16,17]: 2. The Categories SS 0 and LSS 0 where X is any set and L X ⊆ P(X) satisfies the following conditions: Definition 2.2 ([16, Definition 2.21]). A small space is such a locally small space (X, L X ) that X ∈ L X . Definition 2.3. A locally small space (X, L X ) will be called T 0 (or Kolmogorov ) if the family L X separates points ([5, Remainder 1.1.4]), which means that for x, y ∈ X the following condition is satisfied: Definition 2.4 ([16, Definition 2.9]). If (X, L X ) is a locally small space, then the topology L wo X = τ (L X ), generated by L X in P(X), is called the family of weakly open sets in (X, L X ).
Fact 2.5. For a small space (X, L X ), the following conditions are equivalent:
(2) The functor Co : Spec → Lat op is given by: Definition 3.5. An object of SpecD is a pair ((X, τ X ), X d ) where (X, τ X ) is a spectral space and X d is a subset of X satisfying: is a spectral mapping g : X → Y between spectral spaces (X, τ X ) and (Y, τ Y ) that respects the decent subset, that is: , both the spaces PF(CO(X)) and PF(CO(X) d ) with their spectral topologies are homeomorphic to (X, τ X ). A point x ∈ X corresponds tô and tô . Let (X, τ X ) be a spectral space. Then the patch topology (or the constructible topology) on X is the topology with the family CO(X) \ 1 CO(X) as a basis. Proposition 3.9. For a spectral space (X, τ X ) and X d ⊆ X, the following conditions are equivalent: Proof. If the set X d is decent in (X, τ X ) and U is a non-empty patch open set in (X, τ X ), then we may assume Then CO( R) is the family of finite unions of basic sets and the topological space ( R, τ (B)) is spectral. The set R of real numbers is decent in this spectral space, so (( R, τ (B)), R) is an object of SpecD. (The operation · mentioned in this example is an isomorphism between the Boolean algebra of semialgebraic sets in R and the Boolean algebra of constructible sets in R, see [3, Proposition 7.2.3].) Any semialgebraic mapping g : R → R (i.e., g has a semialgebraic graph) extends (uniquely) to a maping g : R → R satisfying the condition g −1 ( T ) = g −1 (T ) for any semialgebraic T ⊆ R, as in [3, Proposition 7.2.8], which means that g : is a bornology in the bounded lattice CO(X) and X d satisfies the following conditions: in SpecBD is such a spectral mapping between spectral spaces g :

The Categories LatD and LatBD
Then D L is called a decent set of prime filters on L.
Morphisms of LatD are such homomorphisms of bounded lattices h :  (1) D L ⊆ L s ⊆ PF(L), Such D L will be called a decent lump of prime filters on L.
Moreover, the Boolean algebras P(X) and P(X) are isomorphic, where the sublattice L o X ∩ 1X corresponds to L o X and the sublattice L X ∩ 1X corresponds to L X . That is why L X ∼ = L X ∩ 1X and L o We define the restriction functorR : SpecBD → LSS 0 by formulas where g d : X d → Y d is the restriction of g : X → Y in the domain and in the codomain to the decent lumps. It is clear that CO s (X) d is a sublattice of P(X d ) with zero that covers X d . Now CO(X) separates points of X, since it is a basis of the topology τ X . Hence both CO(X) d and CO s (X) d separate points of X d . For a morphism g : X → Y in SpecBD, we have That g d is a bounded mapping between locally small spaces follows from g satisfying the condition of boundedness. Since the rest of the conditions are obvious,R is indeed a functor.
Step 2: Defining functorS : LatBD op → SpecBD. For a morphism h : We define the algebraization functorĀ : and, for a strictly continuous mapping f : , with the domination condition being the boundedness of the strictly continuous mapping f . Since the rest of the conditions are obvious,Ā is indeed a functor.
Step 4: The functorRSĀ is naturally isomorphic to Id LSS 0 .
We haveRSĀ(X, Define a natural transformation η fromRSĀ to Id LSS 0 by forx ∈X and, by the obvious isomorphism between P(X) and P(X) (compare Proposition 5.1), each η X is an isomorphism in LSS 0 , so η is truely a natural isomorphism.
Step 5: The functorSĀR is naturally isomorphic to Id SpecBD .
We haveSĀR((X, τ X ), CO s (X), with the topology τ ( CO(X) d ) on PF((CO s (X) d ) o ), since, by Definition 3.12, Define a natural transformation θ fromSĀR to Id SpecBD by Hence θ is truely a natural isomorphism.
Step 6: The functorĀRS is naturally isomorphic to Id LatBD op .
We getĀRS(L, L s , Define a natural transformation κ op fromĀRS to Id LatBD op by putting We are to check that κ op Proof. In the proof of Theorem 5.2, we restrict to the case L o X = L X , L s = L and CO s (X) = CO(X).

Spectralifications
Definition 6.1. A topological space (Y, τ Y ) will be called a spectralification of a topological space (X, τ X ) if: Example 6.2. The space ( R, τ (B)) from Example 3.10 is a spectralification of the real line (with the natural topology), homeomorphic to (PF(L om ), τ ( L om )). Example 6.3. Consider L rom from Example 2.6. The points of PF(L rom ) are: We can see that (PF(L rom ), τ ( L rom )) is another spectralification of the real line, homeomorphic to the space of types over Q of the theory T h(R, <) with the spectral topology (compare [5, Section 14.2]), obtained without using the language of model theory.
Example 6.4. For the patch topology on the "same" set of points as in the previous example, one takes as the new L X the Boolean algebra B rom generated by L rom . This changes the topology on R: the rational points become isolated. The space (PF(B rom ), τ ( B rom )) is a Hausdorff spectralification of this modified real line, and is identified with the (usual in model theory) space of types over Q of the theory T h(R, <). Theorem 6.5. For a topological space, being T 0 is equivalent to admitting a spectralification.
Proof. For (X, τ X ) a T 0 topological space, choose a basis L X of τ X that is a bounded sublattice. By Step 4 of the proof of Theorem 5.2, (X, τ X ) embeds into the spectral space (PF(L X ), τ ( L X )) whose patch topology has a basis L X \ 1 L X . Each nonempty member of this basis A \ B admits x ∈ A \ B. Thenx ∈ A \ B, so the imageX of the embedding is patch dense in (PF(L X ), τ ( L X )). Since a subspace of a T 0 space is T 0 , only T 0 topological spaces can have spectralifications.
Proposition 7.3. For any strongly locally spectral space (X, τ X ), we have Proof. Obviously, SO(X) ⊆ CO(X). Let A ∈ CO(X). Then A is covered by a finite family W 1 , ..., W n of spectral open sets. Since a finite union of spectral spaces glued together along compact open subsets is spectral, the set W 1 ∪...∪W n is spectral and its compact open subset A belongs to SO(X).
Proposition 7.4. In a strongly locally spectral space (X, τ X ), we have Proof. If V ∈ CO(X) o , then V is a union of compact open sets since CO(X) covers X. Hence V ∈ τ X and V satisfies the definition of a member of ICO(X).
If V ∈ ICO(X) and A is any member of CO(X), then V ∩ A ∈ CO(X). This means V ∈ CO(X) o . Definition 7.6. A subset X d ⊆ X in a strongly locally spectral space (X, τ X ) will be called decent if any of the two equivalent conditions is satisfied: Definition 7.7. The patch topology of a strongly locally spectral space (X, τ X ) is the topology on X with a basis CO(X) \ 1 CO(X).
Proposition 7.8. In a strongly locally spectral space (X, τ X ) the decent subsets are exactly the patch dense subsets.
Proposition 7.11. If g : (X, τ X ) → (Y τ Y ) is a mapping between strongly locally spectral spaces, then the following conditions are equivalent: (1) g is spectral, (2) g is bounded and locally spectral (i.e., for any A ∈ CO(X), B ∈ CO(Y ) such that g(A) ⊆ B, the restriction g B A : A → B is a spectral mapping between spectral spaces).
Proof. (1) =⇒ (2) If g is spectral and A, B are as in the statement, then , so g is locally spectral. (2) =⇒ (1) If g is bounded and locally spectral, then, for D ∈ ICO(Y ) and A ∈ CO(X), we have This means g is s-continuous.
Definition 7.12. By slSpec we shall denote the category of strongly locally spectral spaces and spectral mappings between them.
Definition 7.14. By slSpec s we shall denote the category of strongly locally spectral spaces and strongly spectral mappings between them.

The Category ZLat
Definition 8.2. By ZLat we denote the category of distributive lattices with zeros and dominating homomorphisms of lattices respecting zeros.  closed under finite intersections and satisfying the condition (⋆) for a closed set F and a subfamily C ⊆ CO(PI(L)) centered on F (this means: for any finite family C 1 , ..., C n of members of C the set F ∩C 1 ∩...∩C n is nonempty), the intersection F ∩ C is nonempty. (1) (CO(X), ∪, ∩, ∅) is a distributive lattice with zero, (2) Ψ : I(CO(X)) ∋ I → I ∈ τ X is an isomorphism of lattices, where I(CO(X)) is the lattice of all ideals in CO(X), (3) for each p ∈ PI(CO(X)) there exists a unique x p ∈ X such that p = ext{x p }, where the topology in PI(CO(X)) is defined as in Theorem 8.4.
The following proposition gives an explanation to Theorem 8.5.
Then F ∩ C is patch compact and members of C ∩ 1 C are patch closed in C.
Since finite subfamilies of C ∩ 1 C meet F ∩ C, the set F ∩ C is nonempty.  Proof.
Define a natural transformation β from Id slSpec s to Sp Co by β X (x) =x for any object (X, τ X ) of slSpec s , wherex = {V ∈ CO(X) | x ∈ V }. (We have X = {x | x ∈ X} = PF(CO(X)) by the dual of Theorem 8.5). For a morphism g : (X, τ X ) → (Y, τ Y ) of slSpec s , we have (β Y • g)(x) = g(x) = (Lg) • ( x) = ((Lg) • • β X )(x) for x ∈ X. Now β X is an isomorphism, since β X (CO(X)) = CO(X), where CO(X) = {Ã | A ∈ CO(X)} andÃ = {x | x ∈ A}. Hence β is a natural isomorphism. 9. Stone Duality for LSS s 0 Definition 9.1. The category slSpecD s has as objects pairs ((X, τ X ), X d ) where (X, τ X ) is a strongly locally spectral space and X d is a distinguished decent subset of X and as morphisms strongly spectral mappings respecting the decent subsets.
Definition 9.2. The category ZLatD has pairs (L, D L ) where L is a distributive lattice with zero and D L is a distinguished decent set of prime filters in PF(L) as objects and homomorphisms of lattices with zeros respecting the decent sets of prime filters and satisfying the condition of domination as morphisms.
Definition 9.3. The category LSS s 0 is a subcategory of LSS 0 with the same objects and bounded strongly continuous mappings as morphisms.
Example 9.4. Let π : R lom ⊔ R lom → R lom be the natural projection from the disjoint union of two copies of the real locally o-minimal line to the real locally o-minimal line. This finite covering mapping is an example of a bounded strongly continuous mapping but not an isomorphism of LSS s 0 . Theorem 9.5. The categories LSS s 0 , ZLatD op and slSpecD s are equivalent. Proof. Similar to the proof of Theorem 5.2, using Theorem 8.8 instead of the classical Stone Duality, with no necessity to mention explicitely the ambient bounded lattice L o X of L X , with an object L of ZLat playing the role of L s and CO(X) d playing the role of CO s (X) d in Theorem 5.2, restricting to the appropriate classes of morphisms.
Example 9.6. The mapping (Lπ) • : PF(L lom ⊕ L lom ) → PF(L lom ), where π : R lom ⊔ R lom → R lom is as in Example 9.4, is the natural projection from the disjoint union of two copies of PF(L lom ) to PF(L lom ). It is a strongly spectral mapping between strongly locally spectral spaces. Moreover,R(L lom ) is a patch dense set in PF(L lom ) and R ⊔ R(L lom ⊕ L lom ) is a patch dense set in PF(L lom ⊕ L lom ). The morphism (Lπ) • of slSpecD s corresponds to the morphism π of LSS s 0 and may be understood as an extention of π.