Heyting Locally Small Spaces and Esakia Duality

Abstract

We develop the theory of Heyting locally small spaces, including Stone-like dualities such as a new version of Esakia duality and a system of concrete isomorphisms and equivalences. In such a way, we continue building tame topology, realising Grothendieck’s ideas. We use up-spectral spaces and define the standard up-spectralification of a Kolmogorov locally small space. This research gives more understanding of locally definable spaces over structures with topologies.

1. Introduction

Equivalences (including isomorphisms) and dualities between categories are important and fruitful forms of symmetry in pure mathematics, capable of explaining some phenomena in one branch of mathematics by other phenomena in another. Category theory is full of equivalences, isomorphisms and other forms of such abstract symmetry (one of suggested monographs in category theory is [1]). We continue exploring them in tame topology (see the introductions to [2,3]), an emerging branch of general topology with an algebra-friendly flavour.

More precisely, we extend Esakia Duality to Heyting locally small spaces, which we introduce. This duality was originally proved by Leo Esakia in [4] (see also [5]) for spaces that combine topology and order, which he called hybrids (we are especially interested in strict hybrids). He worked in modal logics, which are important in theoretical computer science since they model computer programs. A good monograph on modal logics is [6]. Then, we provide a system of concrete isomorphisms, equivalences and dual equivalences between our categories that give a deeper understanding of locally small spaces and notions related to them.

Esakia Duality has been studied and used after L. Esakia by such authors as: Guram and Nick Bezhanishvili [7,8] together with their collaborators [9,10], as well as S.A. Celani and R. Jansana [11] and many others.

We also see Esakia Duality as a subcase of Stone Duality [12] or Priestley Duality [13]. Many extensions of Stone Duality have been achieved in recent years. For example: the locally compact Hausdorff case in [14], removing the zero-dimensionality and the commutativity assumptions in [15], generalisations of Gelfand–Naimark–Stone Duality to completely regular spaces in [16] and its application to the characterisation of normal, Lindelöf and locally compact Hausdorff spaces in [17]. There exist extensions of Stone Duality that drop compactness completely: the paper [18] considers all zero-dimensional Hausdorff spaces. Stone Duality has been also extended in [19] to (non-distributive) orthomodular lattices (corresponding to spectral presheaves), in [20] to some non-distributive (implicative, residuated, or co-residuated) lattices and applied to the semantics of substructural logics, in [21] to a non-commutative case of left-handed skew Boolean algebras. Applications of Stone Duality have appeared in [22] (canonical extensions of lattice-ordered algebras) and [23] (the semantics of non-distributive propositional logics). Some recent applications of Stone Duality have appeared in [24] in the theory of $C^{*}$-algebras. Priestley Duality was employed in [25] in the analysis of hyperspaces that are used in domain theory and theoretical computer science where they model parallel processing and nondeterministic choice.

The main objective of this paper is to extend our results from [3] by presenting a version of Esakia Duality for Heyting locally small spaces (we try to follow the conventions of [26]). The present paper continues the research from [2,3] on some versions of Stone Duality or Priestley Duality for locally small spaces as well as a version of Esakia Duality for small spaces.

We are developing tame topology, postulated by Alexander Grothendieck in his scientific programme [27]. Such an approach to topology is intended to eliminate pathological phenomena of the usual topology. We are encouraged by the authors of [28], who concluded that the tame topology suggested by A. Grothendieck had not been defined. Notice that Grothendieck’s ideas on the notion of space interest some people concerned with physical and philosophical questions (see Cruz Morales [29]). We develop the theory of tame spaces, such as small spaces or locally small spaces, as a way to realise Grothendieck’s postulate in a purely topological context.

One important useful tool for us is the theory of up-spectral (called also almost-spectral) spaces developed by M. Hochster [30], O. Echi together with his collaborators [31,32], L. Acosta and I.M. Rubio Perilla [33]. Up-spectral spaces can be better understood in the wider context of Balbes–Dwinger spaces, see [34,35]. In particular, the theory of spectral spaces is well-presented in the monograph [26].

Smopologies have appeared implicitly in real algebraic geometry (see [36] (Definition 7.1.14) or [37] (p. 12)), in o-minimality [38,39,40], and in more general contexts of model theory [41,42,43]. Notice that the property of being Heyting appears in the theory of real spaces [37] as the axiom called there AC.

They should be helpful in such branches of mathematics as the generalisations of o-minimality, analytic geometry or algebraic geometry. While in the usual topology we have only spectral reflections (see [26] (Chapter 11)), smopologies allow transferring structural information without any losses between algebraic and topological structures using dualities or equivalences. Transferring information between the topological and the algebraic languages in new versions of Esakia Duality should give more understanding of locally definable spaces over structures with (especially definable) topologies.

Regarding the set-theoretical foundations, we use (without mentioning this) Mac Lane’s standard Zermelo–Fraenkel axioms with the Axiom of Choice and the existence of a set which is a universe as in [44] (p. 23). Such a setting allows speaking about proper classes of sets or categories while staying formally in the axiomatic system ZFC. (See “Axiomatic assumptions” in [45] for the full explanation of our axiomatic system.)

2. Generalities about Locally Small Spaces

Notation. We shall use a special notation for operations on families of sets, for example for family intersection

$$\mathcal{U}{\cap }_1\mathcal{V}=\{U\cap V:U\in \mathcal{U},V\in \mathcal{V}\},\mathcal{U}{\cap }_{1}V=\mathcal{U}{\cap }_1\left\{V\right\}$$

or for other operations

$${int}_1\left(\mathcal{A}\right)=\{int\left(A\right):A\in \mathcal{A}\},Ge{n}_1\left(\mathcal{A}\right)=\{Gen\left(A\right):A\in \mathcal{A}\},$$

$$\mho \mathcal{A}=\{\bigcup \mathcal{B}:\mathcal{B}\subseteq \mathcal{A}\},\Omega \mathcal{A}=\{\bigcap \mathcal{B}:\mathcal{B}\subseteq \mathcal{A}\}.$$

Let a family $\mathcal{A}$ of subsets of a set X be given. Define

$$ba\left(\mathcal{A}\right)=\mathrm{the} \mathrm{Boolean} \mathrm{algebra} \mathrm{generated} \mathrm{by}\mathcal{A}\mathrm{in}\mathcal{P}\left(X\right)$$

$${\mathcal{A}}^o=\{Y\subseteq X|Y{\cap }_1\mathcal{A}\subseteq \mathcal{A}\}.$$

Elements of ${\mathcal{A}}^o$ are called the sets compatible with the family $\mathcal{A}$, while elements of $ba\left(\mathcal{A}\right)$ are the sets constructible from$\mathcal{A}$.

Definition 1 

([2,46]). A locally small space is a pair $(X,{\mathcal{L}}_X)$, where X is any set and ${\mathcal{L}}_X\subseteq \mathcal{P}\left(X\right)$ satisfies the following conditions:

(LS1) 

$\varnothing \in {\mathcal{L}}_X$,

(LS2) 

if $A,B\in {\mathcal{L}}_X$, then $A\cap B, A\cup B\in {\mathcal{L}}_X$,

(LS3) 

$\bigcup {\mathcal{L}}_X=X$.

Elements of ${\mathcal{L}}_X$ are also called smops (i.e., small open subsets of X). The family ${\mathcal{L}}_X$ is called a smopology. Assume $(X,{\mathcal{L}}_X)$ and $(Y,{\mathcal{L}}_Y)$ are locally small spaces. Then, a mapping $f:X\to Y$ is:

(a) 

bounded if ${\mathcal{L}}_X$ refines $f^{-1}\left({\mathcal{L}}_Y\right)$, which means that each $A\in {\mathcal{L}}_X$ admits $B\in {\mathcal{L}}_Y$ such that $A\subseteq f^{-1}\left(B\right)$,

(b) 

continuous if $f^{-1}\left({\mathcal{L}}_Y\right){\cap }_1{\mathcal{L}}_X\subseteq {\mathcal{L}}_X$ (i.e., $f^{-1}\left({\mathcal{L}}_Y\right)\subseteq {\mathcal{L}}_X^o$),

(c) 

strongly continuous if $f^{-1}\left({\mathcal{L}}_Y\right)\subseteq {\mathcal{L}}_X$.

We have the category $\mathbf{LSS}$ of locally small spaces and bounded continuous maps, called sometimes also strictly continuous maps.

Definition 2.

If $X\in {\mathcal{L}}_X$, then $(X,{\mathcal{L}}_X)$ is called a small space. The full subcategory in $\mathbf{LSS}$ generated by the small spaces is denoted by $\mathbf{SS}$.

Remark 1. 

The isomorphisms of $\mathbf{LSS}$ are such bijections $f:X\to Y$ between locally small spaces that $f^{-1}\left({\mathcal{L}}_Y^o\right)={\mathcal{L}}_X^o$ and the families ${\mathcal{L}}_X$ and $f^{-1}\left({\mathcal{L}}_Y\right)$ refine each other. The latter condition means that the bornologies generated by those families are equal. One can easily check that isomorphisms are exactly those mappings that satisfy the condition $f^{-1}\left({\mathcal{L}}_Y\right)={\mathcal{L}}_X$. Such mappings are called strict homeomorphisms.

Definition 3. 

We have the following additional families and types of subsets of X:

1. 

The family ${\mathcal{L}}_X^{\prime }=X{\setminus }_1{\mathcal{L}}_X$ is called a co-smopology and its members are called co-smops.

2. 

The family $Con\left(X\right)=ba\left({\mathcal{L}}_X\right)$ contains the constructible sets, which are the Boolean combination of smops.

3. 

The family ${\mathcal{L}}_X^o$ contains subsets compatible with smops, which will be called (admissible) open sets, while their complements will be called (admissible) closed sets. The family of all admissible open sets ${\mathcal{L}}_X^o$ in X will be usually denoted by $AdOp\left(X\right)$, and the family of all admissible closed sets will be denoted by $AdCl\left(X\right)$ or ${\mathcal{L}}_X^c$.

4. 

A subset of a smop that is a constructible set will be called a small constructible set. The family of all small constructible sets will be denoted by $sCon\left(X\right)$.

5. 

A subset of X whose traces on smops are constructible is a locally constructible set. The family of all locally constructible sets will be denoted by $LCon\left(X\right)$.

Remark 2. 

Notice that for a small space (i.e., if $X\in {\mathcal{L}}_X$) we have

$${\mathcal{L}}_X=AdOp\left(X\right),{\mathcal{L}}_X^{\prime }=AdCl\left(X\right)andsCon\left(X\right)=LCon\left(X\right)=Con\left(X\right).$$

In this case, ${\mathcal{L}}_X^{\prime }$ is also a smopology and the small space $(X,{\mathcal{L}}_X^{\prime })$ is called the inverse space of $(X,{\mathcal{L}}_X)$ and denoted by $X_{inv}$ (see [3]).

Fact 1 

([46]). We have the functor of smallification $sm:\mathbf{LSS}\to \mathbf{SS}$ given by $sm:(X,{\mathcal{L}}_X)\to (X,{\mathcal{L}}_X^o)$ and $sm\left(f\right)=f$, which is a concrete reflector (see [1]).

Definition 4. 

We have the following important topologies in a locally small space:

1. 

The topology $\tau \left({\mathcal{L}}_X\right)$ generated by the smops will be called the original topology and denoted ${\tau }_{orig}$. The closure, the interior and the exterior operations in the original topology will be denoted by $\overline{·}$ (or by $cl(·)$), $int(·), ext(·)$, respectively.

2. 

The topology ${\tau }_{inv}=\tau \left(AdCl\left(X\right)\right)$ generated by the admissible closed sets will be called the inverse topology. The closure and the interior operations in the inverse topology will be denoted by ${\overline{·}}^{inv}, in{t}_{inv}(·)$, respectively.

3. 

The topology ${\tau }_{tinv}=\tau \left({\mathcal{L}}_X^{\prime }\right)$, generated by the co-smops, which can be called the tiny inverse topology.

4. 

The topology $\tau \left({\mathcal{L}}_X{\setminus }_1{\mathcal{L}}_X\right)$ generated by the differences of smops will be called the constructible topology. The closure and the interior operations in the constructible topology will be denoted by ${\overline{·}}^{con}, in{t}_{con}(·)$, respectively.

Example 1 

(Khalimsky line). The Khalimsky topology on a digital line $\mathbb{Z}$ is given by the sub-basis $\left\{\right\{2n-1,2n,2n+1\}:n\in \mathbb{Z}\}$. We can see this line as a locally small space by defining the smopology ${\mathcal{L}}_{\mathbb{Z}}$ to be the family of finite unions of the sets from the sub-basis and odd singletons. Then, $AdOp(\mathbb{Z},{\mathcal{L}}_{\mathbb{Z}})$ is the family of arbitrary unions of the sets from the sub-basis and odd singletons, $AdCl(\mathbb{Z},{\mathcal{L}}_{\mathbb{Z}})$ contains those subsets of $\mathbb{Z}$ that contain an even number if they contain one of its neighbours, $Con(\mathbb{Z},{\mathcal{L}}_{\mathbb{Z}})=Fin\left(\mathbb{Z}\right)\cup Cofin\left(\mathbb{Z}\right)$, $sCon(\mathbb{Z},{\mathcal{L}}_{\mathbb{Z}})=Fin\left(\mathbb{Z}\right)$, $LCon(\mathbb{Z},{\mathcal{L}}_{\mathbb{Z}})=\mathcal{P}\left(\mathbb{Z}\right)$. Moreover, ${\tau }_{tinv}(\mathbb{Z},{\mathcal{L}}_{\mathbb{Z}})=(Cofin\left(\mathbb{Z}\right)\cup \{\varnothing \})\cap AdCl(\mathbb{Z},{\mathcal{L}}_{\mathbb{Z}})={\mathcal{L}}_{\mathbb{Z}}^{\prime }\cup \{\varnothing \}$. Here, $Fin\left(X\right)$ and $Cofin\left(X\right)$, respectively, denote the family of finite and cofinite, respectively, subsets of X.

Example 2 

(Khalimsky plane). The Khalimsky topology on a digital plain may be defined by giving the smallest neighbourhood of every point of ${\mathbb{Z}}^2$. For an even point $(2m, 2n)$, its smallest neighbourhood $\{2m-1, 2m, 2m+1\}\times \{2n-1, 2n, 2n+1\}$ has nine points, for an odd point $(2m+1, 2n+1)$ its smallest neighbourhood is a singleton, and for an even-odd point $(2m, 2n+1)$ its smallest neighbourhood $\{2m-1, 2m, 2m+1\}\times \{2n+1\}$ has three points. Similarly for an odd-even point. The Khalimsky plane can be seen as a locally small space, where the smopology ${\mathcal{L}}_{{\mathbb{Z}}^2}$ contains exactly the finite unions of such smallest neighbourhoods. Then, $AdOp({\mathbb{Z}}^2,{\mathcal{L}}_{{\mathbb{Z}}^2})$ is the family of arbitrary unions of such smallest neighbourhoods, $Con({\mathbb{Z}}^2,{\mathcal{L}}_{{\mathbb{Z}}^2})=Fin\left({\mathbb{Z}}^2\right)\cup Cofin\left({\mathbb{Z}}^2\right)$, $sCon({\mathbb{Z}}^2,{\mathcal{L}}_{{\mathbb{Z}}^2})=Fin\left({\mathbb{Z}}^2\right)$, $LCon({\mathbb{Z}}^2,{\mathcal{L}}_{{\mathbb{Z}}^2})=\mathcal{P}\left({\mathbb{Z}}^2\right)$. We again have ${\tau }_{tinv}({\mathbb{Z}}^2,{\mathcal{L}}_{{\mathbb{Z}}^2})=(Cofin\left({\mathbb{Z}}^2\right)\cup \{\varnothing \})\cap AdCl({\mathbb{Z}}^2,{\mathcal{L}}_{{\mathbb{Z}}^2})={\mathcal{L}}_{{\mathbb{Z}}^2}^{\prime }\cup \{\varnothing \}$.

Remark 3. 

While the family $AdOp\left(X\right)$ usually properly contains ${\mathcal{L}}_X$, they generate the same original topology $\tau \left({\mathcal{L}}_X\right)=\tau \left(AdOp\left(X\right)\right)$.

Definition 5. 

Weakly open sets are the sets belonging to the original topology, and their complements (=intersections of admissible closed sets) are the weakly closed sets. We can express this in symbols as follows:

$${\tau }_{orig}=\mho {\mathcal{L}}_X=\mho AdOp\left(X\right),X{\setminus }_1{\tau }_{orig}=\Omega {\mathcal{L}}_X^{\prime }=\Omega AdCl\left(X\right).$$

Similarly, we have constructibly weakly open and constructibly weakly closed sets, inversely weakly open and inversely weakly closed sets, etc. A mapping that is continuous in the topologies of weakly open sets is called weakly continuous.

Proposition 1. 

For a locally small space $(X,{\mathcal{L}}_X)$, we have

$$\overline{A}=\bigcup _{V\in {\mathcal{L}}_X}\overline{A\cap V},int\left(A\right)=\bigcup _{V\in {\mathcal{L}}_X}int(A\cap V).$$

The topological operations on the right-hand side may be taken in X or in V.

Proof. 

We always have ${\bigcup }_{V\in {\mathcal{L}}_X}\overline{A\cap V}\subseteq \overline{A}$. If $x\in \overline{A}$, then there exists $V\in {\mathcal{L}}_X$ such that $x\in \overline{A}\cap V$. For any $W\in {\mathcal{L}}_X$ if $x\in W$ then $W\cap V\in {\mathcal{L}}_X$ and $W\cap V\cap A\ne \varnothing$. That is why $x\in \overline{A\cap V}\cap V$, so x belongs to the set on the right-hand side of the first claim. For the non-trivial inclusion $int\left(A\right)\subseteq {\bigcup }_{V\in {\mathcal{L}}_X}int(A\cap V)$ in the second claim, assume $x\in int\left(A\right)\cap V$ for some $V\in {\mathcal{L}}_X$. Then, $x\in W\subseteq A$ for some $W\in {\mathcal{L}}_X$. Hence, $x\in W\cap V\subseteq A\cap V$ and $x\in int(A\cap V)$. □

Example 3. 

The tiny inverse topology is usually strictly smaller than the inverse topology. In symbols: we usually have ${\tau }_{tinv}=\mho {\mathcal{L}}_X^{\prime }\subset {\tau }_{inv}=\mho AdCl\left(X\right)$.

Indeed, consider an infinite set X and define ${\mathcal{L}}_X=Fin\left(X\right)$. Then, $\tau \left({\mathcal{L}}_X^{\prime }\right)=Cofin\left(X\right)\cup \{\varnothing \}$, but the inverse topology equals $\tau \left(AdCl\right(X\left)\right)=AdCl\left(X\right)=\mathcal{P}\left(X\right)$.

Example 4. 

The family $ba\left(AdOp\right(X\left)\right)$ may be a proper subfamily of $LCon\left(X\right)$. Indeed, take as X the disjoint union ${⨆}_{n\in \mathbb{N}}{\mathbb{R}}_{om}^{2n}$ of $2n$-th powers of the o-minimal real line ${\mathbb{R}}_{om}$ [46] indexed by $n\in \mathbb{N}$. Then, the smops of ${\mathbb{R}}_{om}^{2n}$ are the finite unions of Cartesian products of open intervals. Smops in the disjoint union are the finite unions of smops in particular Cartesian powers ${\mathbb{R}}_{om}^{2n}$. The locally constructible set $A={\bigcup }_{n\in \mathbb{N}}{A}_n$ where $A_n=\{x\in {\mathbb{R}}^{2n}:\#\{k:x_k=0\}is even\}$ is not a Boolean combination of admissible open sets. To see this, consider the closure algebra $(LCon\left(X\right),\overline{·})$ in the sense of Definition 2.2.1 of [5]. If A were a Boolean combination of m admissible open sets, then A would be a union of at most $2^m$ locally closed sets in this closure algebra, hence, by Proposition 2.5.27 of [5], we would have ${\rho }^{2^m}\left(A\right)=\varnothing$ where $\rho \left(Z\right)=\partial (\partial Z)$ and $\partial Z=\overline{Z}\setminus Z$. For $n>2^m$, we have ${\rho }^n\left(A_n\right)={\left\{0\right\}}^{2n}\ne \varnothing$, so $A_n$ is not a Boolean combination of m smops in ${\mathbb{R}}_{om}^{2n}$. Consequently, $ba\left(AdOp\right(X\left)\right)\subset LCon\left(X\right)$.

Example 5. 

Usually the Boolean algebra $Con\left(X\right)=ba\left({\mathcal{L}}_X\right)$ is also strictly smaller than the Boolean algebra $ba\left(AdOp\right(X\left)\right)$. Indeed, for the space from Example 3, we have $Con\left(X\right)=Fin\left(X\right)\cup Cofin\left(X\right)$ and $ba\left(AdOp\right(X\left)\right)=\mathcal{P}\left(X\right)$.

Proposition 2. 

We have the following equality between topologies:

$$\tau \left({\mathcal{L}}_X{\setminus }_1{\mathcal{L}}_X\right)=\tau \left(ba\left({\mathcal{L}}_X\right)\right)=\tau \left(ba\left(AdOp\left(X\right)\right)\right)=\tau \left(sCon\left(X\right)\right)=\tau \left(LCon\left(X\right)\right).$$

Hence, we have the following descriptions of the constructible topology:

$$\mho LCon\left(X\right)=\mho sCon\left(X\right)=\mho ba\left(AdOp\left(X\right)\right)=\mho ba\left({\mathcal{L}}_X\right)=\mho \left({\mathcal{L}}_X{\setminus }_1{\mathcal{L}}_X\right).$$

Proof. 

The inclusions ${\mathcal{L}}_X{\setminus }_1{\mathcal{L}}_X\subseteq ba\left({\mathcal{L}}_X\right)\subseteq ba\left(AdOp\left(X\right)\right)\subseteq LCon\left(X\right)$ are obvious. Each locally constructible set is a union of a family of smop differences. □

Definition 6. 

For a locally small space $(X,{\mathcal{L}}_X)$ and a subset $Y\subseteq X$, we have the induced subspace  $(Y,{\mathcal{L}}_X{\cap }_{1}Y)$ of $(X,{\mathcal{L}}_X)$. This subspace is called: open if $Y\in AdOp\left(X\right)$, closed if $Y\in AdCl\left(X\right)$, decent if Y is constructibly dense, and small if Y is a subset of a smop.

Lemma 1. 

Assume that X is a locally small space.

1. 

If $V\in AdOp\left(X\right)$, then for the induced subspace $(V,{\mathcal{L}}_X{\cap }_{1}V)$ we have $AdOp\left(V\right)=AdOp\left(X\right){\cap }_{1}V$.

2. 

If $F\in AdCl\left(X\right)$, then for the induced subspace $(F,{\mathcal{L}}_X{\cap }_{1}F)$ we have $AdOp\left(F\right)=AdOp\left(X\right){\cap }_{1}F$.

3. 

If S is a small subset of X, then for the induced subspace $(S,{\mathcal{L}}_X{\cap }_{1}S)$ we have $AdOp\left(S\right)=AdOp\left(X\right){\cap }_{1}S$.

4. 

If $Y\subseteq X$ is decent, then for the induced subspace $(Y,{\mathcal{L}}_X{\cap }_{1}Y)$ we have $AdOp\left(Y\right)=AdOp\left(X\right){\cap }_{1}Y$.

Proof. 

• We check $AdOp\left(X\right){\cap }_{1}V\subseteq AdOp\left(V\right)$ first. For each $W\in AdOp\left(X\right)$, we have $W\cap V\in AdOp\left(X\right)\cap \mathcal{P}\left(V\right)$. For any $A\in {\mathcal{L}}_X{\cap }_{1}V$, we have $W\cap V\cap A\in {\mathcal{L}}_X\cap \mathcal{P}\left(V\right)={\mathcal{L}}_X{\cap }_{1}V$. This proves $W\cap V\in AdOp\left(V\right)$. For the other inclusion, assume $U\in AdOp\left(V\right)$. For each $L\in {\mathcal{L}}_X$, we have $U\cap L=U\cap L\cap V\in {\mathcal{L}}_X{\cap }_{1}V\subseteq {\mathcal{L}}_X$. This means $U\in AdOp\left(X\right)\cap \mathcal{P}\left(V\right)$. Since $U=U\cap V\in AdOp\left(X\right){\cap }_{1}V$, we have $AdOp\left(V\right)\subseteq AdOp\left(X\right){\cap }_{1}V$.
• Assume $W\in AdOp\left(X\right)$. For any $L\in {\mathcal{L}}_X$, the set $W\cap (L\cap F)=(W\cap L)\cap F\in {\mathcal{L}}_X{\cap }_{1}F$. Since $W\cap F\in AdOp\left(F\right)$, the inclusions $AdOp\left(X\right){\cap }_{1}F\subseteq AdOp\left(F\right)$ and $AdCl\left(X\right){\cap }_{1}F\subseteq AdCl\left(F\right)$ are clear. We now prove $AdCl\left(F\right)\subseteq AdCl\left(X\right){\cap }_{1}F$. For $M\in AdCl\left(F\right)$, we have $(F\setminus M){\cap }_1\left({\mathcal{L}}_X{\cap }_{1}F\right)\subseteq {\mathcal{L}}_X{\cap }_{1}F$. Take $L\in {\mathcal{L}}_X$. Then, $(X\setminus M)\cap L=\left(\right(X\setminus F)\cap L)\cup \left(\right(F\setminus M)\cap L)$. But $(F\setminus M)\cap L=F\cap K_L$ for some $K_L\in {\mathcal{L}}_X$ and we may assume $K_L\subseteq L$. We obtain $(X\setminus M)\cap L=((X\setminus F)\cap L)\cup K_L\in {\mathcal{L}}_X$. This proves $X\setminus M\in AdOp\left(X\right)$, so $M=M\cap F\in AdCl\left(X\right){\cap }_{1}F$.
• We know that $S\subseteq L$ for some $L\in {\mathcal{L}}_X$. Since $(S,{\mathcal{L}}_X{\cap }_{1}S)$ is a small space, we have $AdOp\left(S\right)={\mathcal{L}}_S={\mathcal{L}}_X{\cap }_{1}S=\left({\mathcal{L}}_X{\cap }_{1}L\right){\cap }_{1}S=\left(AdOp\left(X\right){\cap }_{1}L\right){\cap }_{1}S=AdOp\left(X\right){\cap }_{1}S$.
• For the non-trivial inclusion $AdOp\left(Y\right)\subseteq AdOp\left(X\right){\cap }_{1}Y$, assume $W\in AdOp\left(Y\right)$. Then, for each $L\in {\mathcal{L}}_X$ there exists $W_L\in {\mathcal{L}}_X{\cap }_{1}L$ such that $W\cap L=W_L\cap Y$. We check if the set $\tilde{W}={\bigcup }_{L\in {\mathcal{L}}_X}{W}_L$ is admissible open in X. For any $M\in {\mathcal{L}}_X$, we conclude $\tilde{W}\cap M={\bigcup }_{L\in {\mathcal{L}}_X}{W}_L\cap M$ is equal to $W_M\in {\mathcal{L}}_X$ since ${\mathcal{L}}_X\ni V\mapsto V\cap Y\in {\mathcal{L}}_Y$ is an isomorphism of lattices and, consequently, for each $L\in {\mathcal{L}}_X$ we have $M\cap W_L\subseteq W_M$. Finally, $W=\tilde{W}\cap Y$.

□

3. Specialisation

Now we extend the facts from Section 4 of [3] about the relation of specialisation to the case of locally small spaces. While we have not defined the inverse smopology in this case, we have the inverse topology as well as the constructible topology, which do not change if we pass to the smallification $(X,{\mathcal{L}}_X^o)$ of the locally small space $(X,{\mathcal{L}}_X)$. That is why a series of facts passes to the locally small case.

Recall that for a topological space $(X,{\tau }_X)$ the point x specialises to y (we write $x⇝y$) if each neighbourhood containing y also contains x. In this situation, x is a generalisation of y, and y is a specialisation of x. For $a\in X$ and subsets $A\subseteq X$, we write:

$Gen\left(a\right)=\{x\in X:x⇝a\}$, $Gen\left(A\right)=\{x\in X:x\in Gen(a) \mathrm{for} \mathrm{some} a\in A\}$,

$Spez\left(a\right)=\{x\in X:a⇝x\}$, $Spez\left(A\right)=\{x\in X:x\in Spez(a) \mathrm{for} \mathrm{some} a\in A\}$.

A specialisation relation is always a preorder (called also a quasi-order) on X and does not depend on the ambient topological space (if $Y\subseteq X$ and $x,y\in Y$, then $x⇝y$ in X iff $x⇝y$ in Y).

Fact 2. 

For a locally small space $(X,{\mathcal{L}}_X)$ and $x,y\in X$, the following conditions are equivalent:

1. 

x specialises to y ($x⇝y$) in the original topology on X,

2. 

$y\in \overline{\left\{x\right\}}$,

3. 

$y\notin ext\left\{x\right\}$,

4. 

each smop containing y also contains x,

5. 

each admissible open set containing y also contains x,

6. 

each co-smop containing x also contains y,

7. 

$y{⇝}_{inv}x$ (read: y specialises to x in ${\tau }_{inv}$.)

Fact 3. 

For each locally small space $(X,{\mathcal{L}}_X)$ and $A\subseteq X$, we have: $Spez\left(A\right)\subseteq \overline{A}$ and $Gen\left(A\right)\subseteq {\overline{A}}^{inv}$.

Fact 4. 

In a pre-Boolean locally small space (Definition 16), $x⇝y$ iff $x=y$.

Fact 5. 

In any locally small space $(X,{\mathcal{L}}_X)$, we have the following:

1. 

If $A\subseteq X$ is weakly closed ($A\in \Omega AdCl\left(X\right)$), then $Spez\left(A\right)=A$.

2. 

If $A\subseteq X$ is weakly open ($A\in \mho {\mathcal{L}}_X$), then $Gen\left(A\right)=A$.

Recall that a subset $Q\subseteq X$ is called saturated if it is an intersection of weakly open sets, i.e., $Q\in \Omega \mho {\mathcal{L}}_X$. Moreover, a locally small space $(X,{\mathcal{L}}_X)$ is called $T_0$ (or Kolmogorov) if the family ${\mathcal{L}}_X$ separates points ([26] (Reminder 1.1.4)), which means that for $x,y\in X$ the following condition is satisfied:

$$\mathrm{if} x\in A\iff y\in A\mathrm{for} \mathrm{each}A\in {\mathcal{L}}_X, \mathrm{then} x=y.$$

By ${\mathbf{LSS}}_0$, we denote the full subcategory in $\mathbf{LSS}$ generated by $T_0$ objects, while by ${\mathbf{SS}}_0$ we denote the full subcategory in $\mathbf{LSS}$ generated by small $T_0$ objects.

Fact 6. 

If $(X,{\mathcal{L}}_X)$ is $T_0$, then the specialisation relation ⇝ is a partial order.

Fact 7. 

If $(X,{\mathcal{L}}_X)$ is $T_0$ and $Q\subseteq X$, then the following are equivalent:

1. 

Q is saturated,

2. 

$Gen\left(Q\right)=Q$.

4. Up-Spectral Locally Small Spaces

Definition 7. 

For a topological space $(X,{\tau }_X)$, we have the following notation:

1. 

$CO\left(X\right)=CO(X,{\tau }_X)$ is the family of all compact open subsets of X,

2. 

$ICO\left(X\right)=ICO(X,{\tau }_X)$ is the family of all open subsets of X such that any intersection with a compact open set is compact,

3. 

co-$ICO\left(X\right)$ is the family of complements of sets from $ICO\left(X\right)$,

4. 

$ClOp\left(X\right)=ClOp(X,{\tau }_X)$ is the family of clopen subsets of X.

Moreover, $(X,{\tau }_X)$ is called

1. 

semi-spectral if $CO\left(X\right){\cap }_{1}CO\left(X\right)\subseteq CO\left(X\right)$,

2. 

coherent if $CO\left(X\right)$ forms a basis of the topology and X is semi-spectral.

Definition 8 

([2,32,33]). A topological space $(X,{\tau }_X)$ is up-spectral if it is ($T_0$) sober and coherent but not necessarily compact.

Fact 8 

(compare [26] (Corollary 1.5.5) ). In an up-spectral topological space, the following holds for any $A\subseteq X$:

1. 

$\overline{A}=Spez\left({\overline{A}}^{con}\right)$ where ${\tau }_{con}=\mho \left(CO\left(X\right){\setminus }_{1}CO\left(X\right)\right)$,

2. 

${\overline{A}}^{inv}=Gen\left({\overline{A}}^{con}\right)$ where ${\tau }_{inv}=\mho co-ICO\left(X\right)$.

The importance of this class of spaces can be seen in the following theorem.

Theorem 1 

([30,31,33], characterisation of up-spectral spaces). For any topological space $(X,{\tau }_X)$, the following conditions are equivalent:

1. 

X is almost-spectral,

2. 

X is open dense in a spectral space,

3. 

X is open in a spectral space,

4. 

X is a sober BD-space,

5. 

X is homeomorphic to the prime spectrum of a distributive lattice with minimum,

6. 

X is up-spectral,

7. 

X admits a trivial one-point spectralification,

8. 

X is semi-spectral and locally spectral,

9. 

X is the underlying topological space of a scheme,

10. 

X is the underlying topological space of an open sub-scheme in an affine scheme.

Proof. 

This is Theorems 7 and 8 in [33], Theorem 2.1 in [31] and Proposition 16 in [30]. □

Definition 9 

([2]). A mapping $g:(X,{\tau }_X)\to (Y,{\tau }_Y)$ between up-spectral spaces is called spectral if the following conditions are satisfied:

1. 

g is bounded: $g\left(CO\right(X\left)\right)$ refines $CO\left(Y\right)$,

2. 

g is s-continuous: $g^{-1}\left(ICO\left(Y\right)\right)\subseteq ICO\left(X\right)$.

Following [2], we shall denote by $\mathbf{uSpec}$ the category of up-spectral topological spaces and spectral mappings between them.

Definition 10. 

An up-spectral locally small space is a locally small space $(X,{\mathcal{L}}_X)$ where $(X,\mho {\mathcal{L}}_X)$ is an up-spectral topological space and ${\mathcal{L}}_X=CO\left(X\right)$. We obtain the category $\mathbf{uSpLSS}$ of up-spectral locally small spaces and bounded continuous mappings.

Remark 4. 

For objects of $\mathbf{uSpLSS}$, we have (see [32] and Chapter 1 of [26])

$$AdOp\left(X\right)=ICO\left(X\right),AdCl\left(X\right)=co-ICO\left(X\right),LCon\left(X\right)=ClOp(X,{\tau }_{con}).$$

Definition 11. 

An up-Priestley space is a system $(X,{\sigma }_X,{\le }_X)$ where $(X,{\sigma }_X)$ is a Boolean (i.e., zero-dimensional Hausdorff locally compact, [47]) topological space and ${\le }_X$ is a partial order on X satisfying the Priestley separation axiom

$$if x{\nleqq }_{X}y, then \exists V\in CO\left(X\right)such that V=\uparrow V,x\in V,y\notin V.$$

A Priestley mapping between up-Priestley spaces is a continuous non-decreasing mapping. We have the category $\mathbf{uPri}$ of up-Priestley spaces and Priestley mappings.

Definition 12. 

A Priestley locally small space is a system $(X,{\mathcal{L}}_X^{\sigma },{\le }_X)$ where $(X,{\mathcal{L}}_X^{\sigma })$ is a Boolean locally small space (Definition 16) and ${\le }_X$ is a partial order on X satisfying the Priestley separation axiom

$$if x{\nleqq }_{X}y, then \exists V\in {\mathcal{L}}_X^{\sigma }such that V=\uparrow V,x\in V,y\notin V.$$

A Priestley morphism between Priestley locally small spaces is a strictly (equivalently: weakly) continuous non-decreasing mapping. We have the category $\mathbf{PLSS}$ of Priestley locally small spaces and Priestley morphisms.

Theorem 2. 

All the following categories are concretely isomorphic:

$$\mathbf{uSpec},\mathbf{uPri},\mathbf{PLSS},\mathbf{uSpLSS}.$$

Proof. 

We have four concrete functors:

• The functor $k_1:\mathbf{uSpec}\to \mathbf{uPri},k_1(X,{\tau }_X)=(X,{\tau }_{con},{⇝}_{inv})$.
  For an up-spectral space $(X,{\tau }_X)$, the constructible topology ${\tau }_{con}$ is Boolean and the generalisation relation ${⇝}_{inv}$ (as well as the specialisation relation ⇝) clearly satisfies the Priestley separation axiom.
  Assume $f:X\to Y$ is bounded s-continuous. Then, it is continuous in the constructible topologies as well as in the original topologies, so non-decreasing in the generalisation relation.
• The functor $k_2:\mathbf{uPri}\to \mathbf{PLSS},k_2(X,{\sigma }_X,\le )=(X,CO(X,{\sigma }_X),\le )$.
  For an object $(X,{\sigma }_X,\le )$ of $\mathbf{uPri}$, the space $(X,CO(X,{\sigma }_X))$ is clearly locally small ($CO(X,{\sigma }_X)$ is a smopology) and Boolean since $AdOp\left(X\right)=ClOp(X,{\sigma }_X)=AdCl\left(X\right)$. Obviously, ≤ satisfies the Priestley separation axiom.
  Assume $f:X\to Y$ is non-decreasing and continuous. Then,

  $$f^{-1}\left(ICO\left(Y\right)\right)=f^{-1}\left(ClOp\left(Y\right)\right)\subseteq ClOp\left(X\right)=ICO\left(X\right).$$
  Hence, f is continuous between the locally small spaces.
  Since the image of a Hausdorff compact set is compact, it is also a subset of a compact open set by local compactness. That is why $f\left(CO\right(X\left)\right)$ is a refinement of $CO\left(Y\right)$. Hence, f is bounded between the locally small spaces.
• The functor $k_3:\mathbf{PLSS}\to \mathbf{uSpLSS},k_3(X,{\mathcal{L}}_X^{\sigma },\le )=(X,U{p}_{\le }\cap {\mathcal{L}}_X^{\sigma })$.
  The family $U{p}_{\le }\cap {\mathcal{L}}_X^{\sigma }$ is the compact-open basis of an up-spectral topology ${\tau }_X$ (compare the characterisation of spectral topologies in Theorem 1.5.11 of [26]). This compact-open basis in the up-spectral topology is, in particular, a smopology. Hence, $(X,U{p}_{\le }\cap {\mathcal{L}}_X^{\sigma })$ is an up-spectral locally small space.
  Assume $f:X\to Y$ is non-decreasing and bounded continuous in the constructible smopologies ${\mathcal{L}}_X^{\sigma },{\mathcal{L}}_Y^{\sigma }$. Each compact-open in ${\tau }_X$ is compact-open in ${\sigma }_X$ so its image under f is contained in a small constructible in ${\tau }_Y$, a subset of a compact-open in ${\tau }_Y$. We also have $f^{-1}\left(ICO(Y,{\tau }_Y)\right)\subseteq ICO(X,{\tau }_X)$ since the preimage of a clopen upset is a clopen upset. Hence, f is bounded continuous as the mapping between the up-spectral locally small spaces.
• The functor $k_4:\mathbf{uSpLSS}\to \mathbf{uSpec},k_4(X,{\mathcal{L}}_X)=(X,\mho {\mathcal{L}}_X)$.
  For an up-spectral locally small space $(X,{\mathcal{L}}_X)$, the topological space $(X,\mho {\mathcal{L}}_X)$ is up-spectral by Definition 10.
  Assume $f:X\to Y$ is bounded continuous. Then, f is spectral as a mapping between the up-spectral topological spaces by Definition 9.

Moreover, $k_4{k}_3{k}_2{k}_1$, $k_1{k}_4{k}_3{k}_2$, $k_2{k}_1{k}_4{k}_3$, $k_3{k}_2{k}_1{k}_4$ are the identity functors since

$${\tau }_X=\mho (U{p}_{{⇝}_{inv}}\cap CO(X,{\sigma }_X)),{\sigma }_X=\tau {(U{p}_{\le }\cap CO(X,{\sigma }_X))}_{con},$$

$${\mathcal{L}}_X^{\sigma }=CO(X,\tau {(U{p}_{\le }\cap {\mathcal{L}}_X^{\sigma })}_{con}),{\mathcal{L}}_X=U{p}_{{⇝}_{inv}}\cap CO(X,\tau {\left({\mathcal{L}}_X\right)}_{con})$$

and $\le ={⇝}_{inv}$ is the generalisation relation of the up-spectral topology ${\tau }_X$. □

We give an analogue of (1.3) in [26] and Example 3 in [3].

Proposition 3. 

For an up-spectral topological space $(X,{\tau }_X)$, the corresponding up-spectral locally small space $(X,{\mathcal{L}}_X)$ with ${\mathcal{L}}_X=CO\left(X\right)$ has the following properties:

1. 

$sCon\left(X\right)\cap \mho {\mathcal{L}}_X={\mathcal{L}}_X$,

2. 

$LCon\left(X\right)\cap \mho {\mathcal{L}}_X=AdOp\left(X\right)$.

Proof. 

Small constructible sets are clopen in the constructible topology, so compact. If they are additionally open in the original topology, then they are smops. The second property follows. □

Remark 5. 

From [3], Examples 4–7, we know that:

1. 

A weakly open constructible set in a locally small space may not be a smop.

2. 

A $T_0$ locally small space may have all smops (topologically) compact but not be sober.

3. 

A $T_0$ locally small space may be (topologically) sober but not compact.

4. 

A $T_0$ locally small space may be sober and compact with not all smops being compact.

Proposition 4. 

If X is a $T_0$ sober locally small space with all smops compact, then X is an up-spectral locally small space.

Proof. 

Since ${\mathcal{L}}_X\subseteq CO\left(X\right)$, we have $CO\left(X\right)={\mathcal{L}}_X$. The other conditions for up-spectrality are obvious. □

Proposition 5. 

If a locally small space has all smops compact, then:

• the ideals in ${\mathcal{L}}_X$ are in a bijective correspondence with the weakly open sets, so also with the weakly closed sets,
• the prime ideals of ${\mathcal{L}}_X$ are in a bijective correspondence with the non-empty, irreducible, weakly closed sets.

Proof. 

The proof is similar to that of Proposition 3 of [3]. □

Recall that if not all smops are compact, then by Example 8 from [3] it may happen that $I\subset i\left(s\right(I\left)\right)$, using notation from the above mentioned proof.

5. Stone Duality for Up-Spectral Spaces

Corollary 1 

(Stone Duality for spectral spaces). The categories $\mathbf{Spec}$, $\mathbf{SpSS}$, $\mathbf{Pri}$, $\mathbf{PSS}$ are dually equivalent to $\mathbf{Lat}$.

Proof. 

Follows from Theorem 2 and the classical Stone Duality. □

Definition 13. 

A bornology in a bounded lattice $(L,\vee ,\wedge ,0,1)$ is an ideal $B\subseteq L$ such that

$$\bigvee B=1.$$

Recall that each distributive bounded lattice L corresponds to the spectral space $(\mathcal{PF}\left(L\right),\tau \left(\tilde{L}\right))$ where $\tilde{L}=\{\tilde{a}:a\in L\}$. Similarly, we define $\tilde{B}=\{\tilde{a}:a\in B\}$. A bornology B will be called special if ${(\tilde{B}{\cap }_1\bigcup \tilde{B})}^o=\tilde{L}{\cap }_1\bigcup \tilde{B}$ as subsets of $\mathcal{P}(\bigcup \tilde{B})$ and $\bigcup \tilde{B}$ is constructibly dense in $\mathcal{PF}\left(L\right)$.

Example 6. 

Let $\tilde{\mathbb{R}}$ be the real spectrum ${\mathrm{Spec}}_{\mathrm{r}}\mathbb{R}\left[Y\right]$ of the ring $\mathbb{R}\left[Y\right]$, see [36]. Consider the following families of compact open sets in this spectral space:

$$CO\left(\tilde{\mathbb{R}}\right)=finite unions of intervals of type [r^{+}, s^{-}] where r<s\in \mathbb{R}$$

$$or [-\infty ,s^{-}] or [r^{+},+\infty ],$$

$$\mathcal{B}=finite unions of intervals [r^{+}, s^{-}] where r<s\in \mathbb{R}.$$

Then, $\mathcal{B}$ is a bornology in the distributive bounded lattice $CO\left(\tilde{\mathbb{R}}\right)$. Notice that $CO\left(\tilde{\mathbb{R}}\right)\subset {\mathcal{B}}^o$ = $locally finite unions of intervals [r^{+}, s^{-}], r<s\in \mathbb{R} and singletons \{-\infty \}, \{+\infty \}.$

Define $X_s=\bigcup \tilde{\mathcal{B}}=\tilde{\mathbb{R}}\setminus \{-\infty ,+\infty \}.$ Still

$$CO\left(\tilde{\mathbb{R}}\right){\cap }_1{X}_s\subset \left({\mathcal{B}}^o\right){\cap }_1{X}_s={\left(\mathcal{B}{\cap }_1{X}_s\right)}^o$$

$$=the family of all locally finite unions of intervals [r^{+}, s^{-}], r<s\in \mathbb{R},in X_s.$$

Hence, the bornology $\mathcal{B}$ is not special.

Definition 14 

([2]). The category $\mathbf{SpecB}$ has

1. 

as objects: systems $((X,{\tau }_X),C{O}_s\left(X\right),X_s)$ where $(X,{\tau }_X)$ is a spectral topological space, $C{O}_s\left(X\right)$ is a special bornology in $CO\left(X\right)$ and $X_s=\bigcup C{O}_s\left(X\right)$,

2. 

as morphisms: spectral mappings $g:X\to Y$ satisfying the condition of boundedness: $C{O}_s\left(X\right)$ is a refinement of $g^{-1}\left(C{O}_s\left(Y\right)\right)$.

Remark 6. 

The condition $g\left(X_s\right)\subseteq Y_s$ follows from the condition of boundedness in the above definition.

Definition 15. 

1. 

An object of $\mathbf{LatB}$ is a system $(L,L_s)$ with $L=(L,\vee ,\wedge ,0,1)$ a bounded distributive lattice and $L_s$ a special bornology in L.

2. 

A morphism of $\mathbf{LatB}$ from $(L,L_s)$ to $(M,M_s)$ is such a homomorphism of bounded lattices $h:L\to M$ that is dominating: $\forall a\in M_s\exists b\in L_{s}a\le h\left(b\right)$.

Corollary 2 

(Stone Duality for up-spectral spaces). The concretely isomorphic categories $\mathbf{uSpec}$, $\mathbf{uSpLSS}$, $\mathbf{uPri}$, $\mathbf{PLSS}$ are equivalent to $\mathbf{SpecB}$ and dually equivalent to $\mathbf{LatB}$.

Proof. 

Follows from Theorem 2 and a special case ($X_d=X_s$) of the Stone Duality for locally small spaces ([2], Theorem 1). □

6. Boolean Locally Small Spaces

Proposition 6. 

For a locally small space, the following are equivalent:

1. 

$AdOp\left(X\right)=AdCl\left(X\right)$,

2. 

$AdOp\left(X\right)=LCon\left(X\right)$,

3. 

$AdCl\left(X\right)=LCon\left(X\right)$,

4. 

$AdOp\left(X\right)$ is a Boolean subalgebra of $\mathcal{P}\left(X\right)$.

Proof. 

Obviously, (2) or (3) are equivalent and imply (1). We prove that (1) implies (2). If $A\subseteq X$ is locally constructible, then it is locally a finite union of finite intersections of smops due to (1). That is why it is admissible open. Moreover, (2) implies (4) implies (1) is clear. □

Definition 16. 

1. 

If a locally small space satisfies the above conditions from Proposition 6, then we will call it a pre-Boolean locally small space.

2. 

A Hausdorff pre-Boolean locally small space with all smops compact will be called a Boolean locally small space.

3. 

The category of Boolean locally small spaces and bounded (weakly) continuous maps will be denoted by $\mathbf{BLSS}$.

Definition 17. 

We have the following categories:

1. 

$\mathbf{BoolSp}$ is the category of zero-dimensional Hausdorff locally compact topological spaces and continuous mappings (see Dimov [47]),

2. 

$\mathbf{BAB}$ is the category of Boolean algebras with special bornologies and dominating Boolean homomorphisms.

Proposition 7. 

$\mathbf{BLSS}$ and $\mathbf{BoolSp}$ are concretely isomorphic categories. In particular, Boolean locally small spaces are up-spectral.

Proof. 

For a Boolean topological space $(X,\sigma )$, the family ${\mathcal{L}}^{\sigma }=CO(X,\sigma )$ is a smopology for which ${\mathcal{L}}^{\sigma }\subseteq ClOp(X,\sigma )\subseteq AdOp(X,{\mathcal{L}}^{\sigma })$. On the other hand, $AdCl\left(X\right)\subseteq AdOp\left(X\right)\subseteq ClOp\left(X\right)$, hence $AdOp\left(X\right)=AdCl\left(X\right)=ClOp\left(X\right)$ and $(X,{\mathcal{L}}^{\sigma })$ is a Boolean locally small space.

For a Boolean locally small space $(X,{\mathcal{L}}^{\sigma })$, each smop is compact, so ${\mathcal{L}}^{\sigma }\subseteq CO(X,\mho {\mathcal{L}}^{\sigma })$ and $AdOp\left(X\right)=AdCl\left(X\right)\subseteq ClOp\left(X\right)$. The topology $\mho {\mathcal{L}}^{\sigma }$ is Hausdorff, locally compact and has a clopen basis $AdOp\left(X\right)=AdCl\left(X\right)$, hence it is a Boolean topology.

The morphisms are the (weakly) continuous mappings in both categories.

In particular, each smop in a Boolean locally small space is spectral. The whole space is semispectral and locally spectral, so up-spectral [2]. □

Theorem 3. 

$\mathbf{BAB}$ is dually equivalent to $\mathbf{BoolSp}$, so also to $\mathbf{BLSS}$.

Proof. 

The Dimov’s version of Stone Duality [47] says that $\mathbf{BoolSp}$ is dually equivalent to his category $ZLBA$. One needs to notice that $ZLBA$ is our category $\mathbf{BAB}$ since in Boolean algebras the special bornologies are exactly the distinguished dense ideals (i.e., the dense ideals I satisfying the condition: for each simple ideal J in I the supremum of J in the Boolean algebra exists). We check this in four steps:

• Each bornology in a Boolean algebra is a dense ideal.
  Let A be a Boolean algebra and B a bornology in A. Take $a\in A\setminus \left\{0\right\}$. Then $a^{\prime }\ne 1={\bigvee }_{b\in B}b$. There exists $b\in B$ such that $b\nleqq a^{\prime }$, so $a\nleqq b^{\prime }$. We have $a=a\wedge (b\vee b^{\prime })=(a\wedge b)\vee (a\wedge b^{\prime })$. Since $a\wedge b^{\prime }<a$, we have $a\wedge b\ne 0$ and B is dense.
• Each distinguished dense ideal is a bornology.
  If ${\bigvee }_{A}I=a\ne 1$, then $a^{\prime }\ne 0$. Take $i\in I$ such that $0\ne i\le a^{\prime }$. Then, $i=(i\vee a)\wedge (i\vee a^{\prime })=a\wedge a^{\prime }=0$. Contradiction proves that ${\bigvee }_{A}I=1$.
• Special bornologies are distinguished.
  If B is a special bornology, then each clopen set in the constructibly dense open $\bigcup \tilde{B}$ is of the form ${\tilde{a}}^s=\tilde{a}\cap \bigcup \tilde{B}$ where $a\in A$. It follows from Proposition 2.6 of [47] that B is distinguished.
• Each distinguished dense ideal is special.
  If I is a distinguished dense ideal in a Boolean algebra A, then the dense set $\bigcup \tilde{I}$ is constructibly dense. The condition ${(\tilde{I}{\cap }_1\bigcup \tilde{I})}^o=AdOp(\bigcup \tilde{I})=LCon(\bigcup \tilde{I})=ClOp(\bigcup \tilde{I})=\tilde{A}{\cap }_1\bigcup \tilde{I}$ is satisfied by Proposition 2.6 of [47].

In both cases, the morphisms are the dominating Boolean homomorphisms. □

7. The Standard Up-Spectralification

We extend the facts from Section 5 of [3] about the standard spectralification of $T_0$ small spaces to the locally small case.

Definition 18. 

An embedding of locally small spaces is an injective map $e:X\to Y$ such that $e\left({\mathcal{L}}_X\right)={\mathcal{L}}_Y{\cap }_{1}e\left(X\right)$.

Definition 19 

([2]). An up-spectralification of a locally small space X is the pair $(e,Y)$ where: Y is an up-spectral locally small space and $e:X\to Y$ is an embedding between locally small spaces with the image $e\left(X\right)$ dense in the constructible topology of Y.

Definition 20. 

The standard up-spectralification of a $T_0$ locally small space $(X,{\mathcal{L}}_X)$ is the locally small space (using notations from Theorem 1 of [2])

$$X^{usp}=(\bigcup \widetilde{{\mathcal{L}}_X},\widetilde{{\mathcal{L}}_X})$$

with the embedding $X\ni x\mapsto \widehat{x}\in \widehat{X}\subseteq \bigcup \widetilde{{\mathcal{L}}_X}$.

Remark 7. 

Identifying X with $\widehat{X}$, we have

1. 

$CO\left(X^{usp}\right){\cap }_{1}X={\mathcal{L}}_{X^{usp}}{\cap }_{1}X={\mathcal{L}}_X$,

2. 

$ICO\left(X^{usp}\right){\cap }_{1}X=AdOp\left(X^{usp}\right){\cap }_{1}X=AdOp\left(X\right)$,

3. 

$ClOp\left(X_{con}^{usp}\right){\cap }_{1}X=LCon\left(X^{usp}\right){\cap }_{1}X=LCon\left(X\right)=ClOp\left(X_{con}\right)$.

Since X is constructibly dense in $X^{usp}$, we have the following bijection:

$$LCon\left(X\right)\ni A\mapsto A^{usp}\in LCon\left(X^{usp}\right),$$

where $A^{usp}$ is the only member of $LCon\left(X^{usp}\right)$ such that $A^{usp}\cap X=A$. One can see that $A^{usp}={\overline{A}}^{con}$ taken in $X^{usp}$ and $\overline{A^{usp}}\cap X={\overline{A}}^X.$ By bijectivity, we have

(i) 

$V\in AdOp\left(X\right)$ iff $V^{usp}\in AdOp\left(X^{usp}\right)$,

(ii) 

$F\in AdCl\left(X\right)$ iff $F^{usp}\in AdCl\left(X^{usp}\right)$.

Fact 9. 

If X is small, then $X^{usp}=X^{sp}$ is an object of $\mathbf{SpSS}$.

The following fact is purely topological.

Fact 10. 

If C is a non-empty, weakly closed set in the original topology of a locally small space X, then the following conditions are equivalent:

1. 

C is irreducible in X,

2. 

the closure $\overline{C}$ in the standard up-spectralification $X^{usp}$ is irreducible.

Remark 8. 

We have the functor $usp:{\mathbf{LSS}}_0\to \mathbf{uSpLSS}$ given by formulas

$$usp(X,{\mathcal{L}}_X)=(\bigcup \widetilde{{\mathcal{L}}_X},\widetilde{{\mathcal{L}}_X}),usp\left(f\right)={\left({\left({\mathcal{L}}^{o}f\right)}^{•}\right)}_s$$

where ${\left({\left({\mathcal{L}}^{o}f\right)}^{•}\right)}_s$ is the restriction of ${\left({\mathcal{L}}^{o}f\right)}^{•}$ to the standard up-spectralifications of the domain and the codomain of f.

8. Heyting Locally Small Spaces

Definition 21. 

1. 

A locally small space will be called pre-semi-Heyting if any of the following equivalent conditions are satisfied:

(a) 

the interior in the original topology of any admissible closed set is an admissible open set, i.e., $in{t}_{1}AdCl\left(X\right)\subseteq AdOp\left(X\right)$,

(b) 

the closure in the original topology of any admissible open set is an admissible closed set, i.e., $c{l}_{1}AdOp\left(X\right)\subseteq AdCl\left(X\right)$.

2. 

A locally small space will be called pre-Heyting if any of the following equivalent conditions are satisfied:

(a) 

the interior in the original topology of any locally constructible set is an admissible open set, i.e., we have $in{t}_{1}LCon\left(X\right)\subseteq AdOp\left(X\right)$,

(b) 

the closure in the original topology of any locally constructible set is an admissible closed set, i.e., we have $c{l}_{1}LCon\left(X\right)\subseteq AdCl\left(X\right)$.

3. 

A locally small space will be called Heyting if it is $T_0$ and pre-Heyting.

Inspired by [3,26], we have the following propositions.

Proposition 8. 

Assume that X is a pre-semi-Heyting locally small space. Then:

1. 

the following maps are well defined:

(a) 

the open regularisation map $N:AdOp\left(X\right)\to AdOp\left(X\right)$ given by $N\left(A\right)=int\overline{A}$,

(b) 

the closed regularisation map $\overline{N}:AdCl\left(X\right)\to AdCl\left(X\right)$ given by $\overline{N}\left(F\right)=\overline{intF}$,

2. 

for each $V\in AdOp\left(X\right)$, the subspace $(V,{\mathcal{L}}_X{\cap }_{1}V)$ is pre-semi-Heyting.

Proof. 

• Obvious by definition.
• For $A,V\in AdOp\left(X\right)$, we have that the set ${\overline{A\cap V}}^V=V\cap \left(\overline{A\cap V}\right)\in V{\cap }_{1}AdCl\left(X\right)=AdCl\left(V\right)$ is a local co-smop in V. Notice that $(V,\tau \left({\mathcal{L}}_X{\cap }_{1}V\right))$ is a topological subspace of $(X,\tau \left({\mathcal{L}}_X\right))$.

□

Proposition 9 

(characterisation of pre-Heyting spaces). For a locally small space $X=(X,{\mathcal{L}}_X)$, the following conditions are equivalent:

1. 

X is pre-Heyting,

2. 

for each $B\in {\mathcal{L}}_X{\setminus }_1{\mathcal{L}}_X$, $\overline{B}\in AdCl\left(X\right)$,

3. 

for each $A\in LCon\left(X\right)$, the subspace $(A,{\mathcal{L}}_X{\cap }_{1}A)$ is pre-Heyting,

4. 

for each $A\in LCon\left(X\right)$, the subspace $(A,{\mathcal{L}}_X{\cap }_{1}A)$ is pre-semi-Heyting,

5. 

for each $F\in AdCl\left(X\right)$, the subspace $(F,{\mathcal{L}}_X{\cap }_{1}F)$ is pre-semi-Heyting.

Proof. 

$\left(1\right)\iff \left(2\right)$ Of course, (2) follows from (1). Assume $C\in LCon\left(X\right)$. Then, for each $A\in {\mathcal{L}}_X$, we have $\overline{C}\cap A=\overline{C\cap A}\cap A$. Since $C\cap A\in Con\left(A\right)$, this set is a finite union of sets of the form $B_1\setminus B_2$ with $B_1,B_2\in {\mathcal{L}}_A$. Thus, $\overline{C\cap A}\cap A\in AdCl\left(A\right)$ and $\overline{C}\in AdCl\left(X\right)$.

$\left(1\right)\Rightarrow \left(3\right)$ For $A\in LCon\left(X\right)$ and $D\in LCon\left(A\right)=LCon\left(X\right){\cap }_{1}A\subseteq LCon\left(X\right)$, we have $\overline{D}\in AdCl\left(X\right)$. Now, ${\overline{D}}^A=\overline{D}\cap A\in AdCl\left(A\right)$.

$\left(3\right)\Rightarrow \left(4\right)$ Trivial.

$\left(4\right)\Rightarrow \left(5\right)$ Trivial.

$\left(5\right)\Rightarrow \left(2\right)$ Each element of ${\mathcal{L}}_X{\setminus }_1{\mathcal{L}}_X$ is of the form $A\cap F$ with $A\in {\mathcal{L}}_X,F\in AdCl\left(X\right)$. Since $\overline{A\cap F}=\overline{A\cap F}\cap F\in AdCl\left(X\right){\cap }_{1}F$ by (5), we have $\overline{A\cap F}\in AdCl\left(X\right){\cap }_{1}AdCl\left(X\right)\subseteq AdCl\left(X\right)$. □

Remark 9. 

There exist pre-semi-Heyting spectral small spaces that are not pre-Heyting. See Example 8.3.11(iii) in [26].

Since the smallification does not change the original topology, we have

Fact 11. 

If the locally small space $(X,{\mathcal{L}}_X)$ is (pre-)Heyting, then its smallification $(X,{\mathcal{L}}_X^o)$ is (pre-)Heyting as a small space or a locally small space.

Definition 22 

(Heyting maps). A map between pre-Heyting locally small spaces $f:X\to Y$ will be called a Heyting (bounded continuous) map if it is bounded continuous and the interior operation (equivalently: the closure operation) commutes with the preimage on the locally constructible sets, that is, any of the equivalent conditions is satisfied:

1. 

$f^{-1}\left(int\left(C\right)\right)=int\left(f^{-1}\left(C\right)\right)$ for $C\in LCon\left(Y\right)$,

2. 

$f^{-1}\left(\overline{C}\right)=\overline{f^{-1}\left(C\right)}$ for $C\in LCon\left(Y\right)$.

Definition 23. 

By $\mathbf{HLSS}$, we denote the category of Heyting locally small spaces and Heyting bounded continuous maps.

Since strict homeomorphisms are homeomorphisms, we have

Fact 12. 

Strict homeomorphisms between Heyting locally small spaces are isomorphisms of $\mathbf{HLSS}$.

9. Heyting Up-Spectral Spaces

Definition 24 

([26] (Section 8.3)). A topological up-spectral space $(X,{\tau }_X)$ will be called Heyting if the closure of any constructibly clopen (i.e., locally constructible) set is a co-ICO set. A map between Heyting up-spectral spaces $g:X\to Y$ is called a Heyting spectral map if g is spectral and any of the equivalent conditions hold:

1. 

$f^{-1}\left(\overline{C}\right)=\overline{f^{-1}\left(C\right)}$ for $C\in LCon\left(Y\right)=ClOp(Y,{\tau }_{con})$,

2. 

$f^{-1}\left(int\left(C\right)\right)=int\left(f^{-1}\left(C\right)\right)$ for $C\in LCon\left(Y\right)=ClOp(Y,{\tau }_{con})$.

We have the category $\mathbf{HuSpec}$ of Heyting up-spectral (topological) spaces and Heyting spectral mappings.

Corollary 3. 

Each homeomorphism between Heyting up-spectral spaces is an isomorphism in $\mathbf{HuSpec}$.

Definition 25. 

We have the category $\mathbf{HuSpLSS}$ of Heyting up-spectral locally small spaces and Heyting strictly continuous maps.

Proposition 10. 

If the standard up-spectralification $X^{usp}$ of a locally small space X is Heyting, then X is Heyting.

Proof. 

Assume $\forall A\in LCon\left(X^{usp}\right)\overline{A}\in AdCl\left(X^{usp}\right)$. Then, by taking traces with X, we have $\forall B\in LCon\left(X\right){\overline{B}}^X\in AdCl\left(X\right)$, see Remark 7. □

Proposition 11. 

If X is a Heyting locally small space (object of $\mathbf{HLSS}$), then:

1. 

$AdOp\left(X\right)$ is a Heyting algebra and $(\widetilde{{\mathcal{L}}_X^o},\widetilde{{\mathcal{L}}_X},\widehat{X})$ is an object of $\mathbf{LatBD}$,

2. 

$X^{usp}$ is a Heyting up-spectral locally small space (object of $\mathbf{HuSpLSS}$).

Proof. 

• For $U,V\in AdOp\left(X\right)$ define $U\to V=int(V\cup (X\setminus U\left)\right)$. Then, by the Heyting assumption on the locally small space X, we have $U\to V\in AdOp\left(X\right)$ and this set is the largest element $Z\in AdOp\left(X\right)$ with the property $V\cap Z\subseteq W$. That $(\widetilde{{\mathcal{L}}_X^o},\widetilde{{\mathcal{L}}_X},\widehat{X})$ is an object of $\mathbf{LatBD}$ was proved in Step 3 of the proof of Theorem 1 of [2].
• Since X is Heyting and the lattices $AdOp\left(X\right)$ and $ICO\left(X^{usp}\right)$ are isomorphic Heyting algebras, $X^{usp}$ is Heyting by Remark 7.

□

Remark 10. 

The functor $usp:{\mathbf{LSS}}_0\to \mathbf{uSpLSS}$ from Remark 8 has a restriction that will be denoted by $usp:\mathbf{HLSS}\to \mathbf{HuSpLSS}$.

Definition 26. 

An up-Esakia space $(X,{\sigma }_X,{\le }_X)$ is an up-Priestley space such that

$$for all C\in ClOp(X,{\sigma }_X)we have \downarrow C\in ClOp(X,{\sigma }_X).$$

An Esakia mapping between up-Esakia spaces is a Priestley mapping $f:X\to Y$ such that $f(\uparrow x)=\uparrow f\left(x\right)$. We have the category $\mathbf{uEsa}$ of up-Esakia spaces and Esakia mappings.

Definition 27. 

An Esakia locally small space is such a Priestley locally small space $(X,{\mathcal{L}}_X^{\sigma },{\le }_X)$ that satisfies the condition

$$for all A\in LCon(X,{\mathcal{L}}_X^{\sigma })we have \downarrow A\in LCon(X,{\mathcal{L}}_X^{\sigma }).$$

An Esakia morphism between Esakia locally small spaces is a Priestley morphism $f:X\to Y$ that is a p-morphism, i.e., it satisfies the condition $f(\uparrow x)=\uparrow f\left(x\right)$. We have the category $\mathbf{ELSS}$ of Esakia locally small spaces and Esakia morphisms.

Corollary 4. 

The categories $\mathbf{HuSpec},\mathbf{HuSpLSS},\mathbf{uEsa},\mathbf{ELSS}$ are concretely isomorphic.

Proof. 

Follows from Theorem 2. □

Definition 28. 

The category $\mathbf{HSpecB}$ has:

1. 

systems $((X,{\tau }_X),C{O}_s\left(X\right),X_s)$ where $(X,{\tau }_X)$ is a Heyting spectral topological space, $C{O}_s\left(X\right)$ is a special bornology in $CO\left(X\right)$ and $X_s=\bigcup C{O}_s\left(X\right)$ (called the decent lump) as objects,

2. 

Heyting spectral mappings between Heyting spectral spaces satisfying the condition of boundedness and respecting the decent lump as morphisms.

Definition 29. 

The category $\mathbf{HAB}$ has

1. 

systems $(L,L_s)$ with $L=(L,\vee ,\wedge ,\to ,0,1)$ a Heyting algebra, $L_s$ a special bornology in L as objects,

2. 

homomorphisms of Heyting algebras $h:L\to M$ that are dominating (i.e., $\forall a\in M_s\exists b\in L_{s}a\le h\left(b\right)$) as morphisms from $(L,L_s)$ to $(M,M_s)$.

10. Categories of Spaces with Decent Subsets

While the category SpecD was defined in [2] and the category ESSD was defined in [3], we introduce similarly:

Definition 30. 

We have the following categories:

1. 

the category $\mathbf{PriD}$ whose objects are Priestley spaces with distinguished dense sets (called decent sets) and whose morphisms are Priestley mappings respecting the decent sets,

2. 

the category $\mathbf{EsaD}$ whose objects are Esakia spaces with distinguished dense sets (called decent sets) and whose morphisms are Esakia mappings respecting the decent sets,

3. 

the category $\mathbf{PSSD}$ whose objects are Priestley small spaces with decent sets and whose morphisms are Priestley morphisms respecting the decent sets.

We have two consequences of Theorem 2.

Corollary 5. 

$\mathbf{SpecD},\mathbf{PriD},\mathbf{PSSD}$ are concretely isomorphic.

Corollary 6. 

$\mathbf{HSpecD},\mathbf{EsaD},\mathbf{ESSD}$ are concretely isomorphic.

Two versions of dualities follow.

Corollary 7 

(Stone Duality for small spaces). The categories ${\mathbf{SS}}_0$, $\mathbf{SpecD}$, $\mathbf{PriD}$, $\mathbf{PSSD}$ are dually equivalent to $\mathbf{LatD}$.

Corollary 8 

(Esakia Duality for small spaces). The categories $\mathbf{HSS}$, $\mathbf{HSpecD}$, $\mathbf{EsaD}$, $\mathbf{ESSD}$, are dually equivalent to $\mathbf{HAD}$.

Definition 31. 

We have the following categories:

1. 

the category $\mathbf{uSpecD}$ whose objects are up-spectral spaces with distinguished decent sets that are dense in the constructible topology and whose morphisms are spectral mappings respecting the decent sets,

2. 

the category $\mathbf{uPriD}$ whose objects are up-Priestley spaces with distinguished dense sets (called decent sets) and whose morphisms are Priestley mappings respecting the decent sets,

3. 

the category $\mathbf{uEsaD}$ whose objects are up-Esakia spaces with distinguished dense sets (called decent sets) and whose morphisms are Esakia mappings respecting the decent sets,

4. 

the category $\mathbf{PLSSD}$ whose objects are Priestley locally small spaces with decent sets and whose morphisms are Priestley morphisms respecting the decent sets,

5. 

the category $\mathbf{ELSSD}$ whose objects are Esakia locally small spaces with decent sets and whose morphisms are Esakia morphisms respecting the decent sets.

Corollary 9 

(Stone Duality for locally small spaces). The categories ${\mathbf{LSS}}_0$, $\mathbf{uSpecD}$, $\mathbf{uPriD}$, $\mathbf{PLSSD}$, and $\mathbf{SpecBD}$ are dually equivalent to $\mathbf{LatBD}$.

Proof. 

Follows from Theorem 1 in [2] and Theorem 2. □

Definition 32. 

1. 

An object of $\mathbf{HuSpecD}$ is a system $((X,{\tau }_X),X_d)$ where $(X,{\tau }_X)$ is a Heyting up-spectral space and $X_d\subseteq X$ is a constructibly dense subset.

2. 

A morphism from $((X,{\tau }_X),X_d)$ to $((Y,{\tau }_Y),Y_d)$ in $\mathbf{HuSpecD}$ is such a Heyting spectral mapping between Heyting up-spectral spaces $g:(X,{\tau }_X)\to (Y,{\tau }_Y)$ that $g\left(X_d\right)\subseteq Y_d$.

Definition 33. 

1. 

An object of $\mathbf{HSpecBD}$ is a system $((X,{\tau }_X),C{O}_s\left(X\right),X_d)$ where $(X,{\tau }_X)$ is a Heyting spectral space, $C{O}_s\left(X\right)$ is a bornology in the Heyting algebra $CO\left(X\right)$ and $X_d\subseteq \bigcup C{O}_s\left(X\right)$ (a decent lump of X) is constructibly dense and such that $CO{\left(X\right)}_d={\left(C{O}_s{\left(X\right)}_d\right)}^o\subseteq \mathcal{P}\left(X_d\right)$, see [2].

2. 

A morphism from $((X,{\tau }_X),C{O}_s\left(X\right),X_d)$ to $((Y,{\tau }_Y),C{O}_s\left(Y\right),Y_d)$ in $\mathbf{HSpecBD}$ is such a Heyting spectral mapping $g:(X,{\tau }_X)\to (Y,{\tau }_Y)$ between Heyting spectral spaces that satisfies the condition of boundedness $\forall A\in C{O}_s\left(X\right)\exists B\in C{O}_s\left(Y\right)g\left(A\right)\subseteq B,$ and respects the decent lump: $g\left(X_d\right)\subseteq Y_d$.

Similarly to Lemma 3, we have

Lemma 2. 

The categories $\mathbf{HuSpecD}$ and $\mathbf{HSpecBD}$ are equivalent.

Proof. 

Theorem 5 of [2] gives us equivalence between $\mathbf{uSpec}$ and $\mathbf{uSpLSS}$. Similarly, $\mathbf{uSpecD}$ is equivalent to $\mathbf{uSpLSSD}$, which is clearly equivalent to ${\mathbf{LSS}}_0$. Now Theorem 1 of [2] gives us equivalence between ${\mathbf{LSS}}_0$ and $\mathbf{SpecBD}$. Restricting to Heyting objects and Heyting morphisms, we obtain the equivalence between $\mathbf{HuSpecD}$ and $\mathbf{HSpecBD}$. □

Definition 34. 

1. 

An object of $\mathbf{HABD}$ is a system $(L,L_s,{\mathbf{D}}_L)$ with $L=(L,\vee ,\wedge ,\to ,0,1)$ a Heyting algebra, $L_s$ a bornology in L and ${\mathbf{D}}_L\subseteq \bigcup \widetilde{L_s}\subseteq \mathcal{PF}\left(L\right)$ a constructibly dense set satisfying $\tilde{L}{\cap }_1{\mathbf{D}}_L={\left(\widetilde{L_s}{\cap }_1{\mathbf{D}}_L\right)}^o\subseteq \mathcal{P}\left({\mathbf{D}}_L\right)$.

2. 

A morphism of $\mathbf{HABD}$ from $(L,L_s,{\mathbf{D}}_L)$ to $(M,M_s,{\mathbf{D}}_M)$ is such a homomorphism of Heyting algebras $h:L\to M$ that is dominating (i.e., $\forall a\in M_s\exists b\in L_{s}a\le h\left(b\right)$) and $h^{•}\left({\mathbf{D}}_M\right)\subseteq {\mathbf{D}}_L$, see [2].

Fact 13 

([3], Theorem 2). If $h:A\to B$ is a Heyting algebra homomorphism, then $h^{•}:\mathcal{PF}\left(B\right)\to \mathcal{PF}\left(A\right)$ is a Heyting spectral map between Heyting spectral spaces.

Proposition 12. 

If $f:X\to Y$ is a Heyting bounded continuous map between Heyting locally small spaces, then ${\mathcal{L}}^{o}f:{\mathcal{L}}_Y^o\to {\mathcal{L}}_X$ is a homomorphism of Heyting algebras.

Proof. 

We only check up the condition with the intuitionistic implication. For $V,W\in {\mathcal{L}}_Y^o$, we have $\left({\mathcal{L}}^{o}f\right)(W\to V)=f^{-1}\left(ext(W\setminus V)\right)=ext(f^{-1}\left(W\right)\setminus f^{-1}\left(V\right))=\left({\mathcal{L}}^{o}f\right)\left(W\right)\to \left({\mathcal{L}}^{o}f\right)\left(V\right)$. □

11. Esakia Duality for Locally Small Spaces

Corollary 10. 

Concretely isomorphic categories $\mathbf{HuSpecD},\mathbf{uEsaD},\mathbf{ELSSD}$ are equivalent to $\mathbf{HLSS}$.

Proof. 

Follows from Theorem 2, using methods similar to those in the proof of Theorem 4. □

Theorem 4 

(Esakia Duality for locally small spaces). The categories $\mathbf{HLSS}$, $\mathbf{HuSpecD}$, $\mathbf{uEsaD}$, $\mathbf{ELSSD}$ and $\mathbf{HSpecBD}$ are dually equivalent to $\mathbf{HABD}$.

Proof. 

By the corollary above, we need only to prove that $\mathbf{HLSS}$ and $\mathbf{HSpecBD}$ are dually equivalent to $\mathbf{HABD}$. Since our proof is about the restrictions of functors from the proof of Theorem 1 in [2], we concentrate on objects and morphisms being Heyting. We shall use the notations from that proof.

Step 1: 

The restricted functor $\overline{R}:\mathbf{HSpecBD}\to \mathbf{HLSS}$.

Assume $((X,{\tau }_X),C{O}_s\left(X\right),X_d)$ is an object of $\mathbf{HSpecBD}$. Then, $C{O}_s\left(X\right)$ is a basis of the induced topology on $X_s$ and the original topology of $(X_d,C{O}_s{\left(X\right)}_d)$ is the topology induced from $(X,{\tau }_X)$. We are to check the Heyting closure condition for the locally small space $(X_d,C{O}_s{\left(X\right)}_d)$. Assume $B\in LCon(X_d,C{O}_s{\left(X\right)}_d)$. For $V\in C{O}_s\left(X\right)$, we have $B\cap V\in sCon(X_d,C{O}_s{\left(X\right)}_d)\subseteq Con(X_d,CO{\left(X\right)}_d)$, so $int(B\cap V)\in CO{\left(X\right)}_d{\cap }_{1}C{O}_s{\left(X\right)}_d=C{O}_s{\left(X\right)}_d$. Since $C{O}_s{\left(X\right)}_d$ is a basis of the induced topology of $X_d$ stable under finite intersections, $int\left(B\right)={\bigcup }_{V\in C{O}_s\left(X\right)}int(B\cap V)\in AdOp(X_d,C{O}_s{\left(X\right)}_d)$.

Assume $f:X\to Y$ is a morphism of $\mathbf{HSpecBD}$. Since

$$\forall C\in Con(Y,CO\left(Y\right))f^{-1}\left(\overline{C}\right)=\overline{f^{-1}\left(C\right)},$$

we also have

$$\forall D\in Con(Y_d,CO{\left(Y\right)}_d)f_d^{-1}\left({\overline{D}}^d\right)={\overline{f_d^{-1}\left(D\right)}}^d.$$

Indeed, for any $D=C\cap Y_d\in Con(Y_d,CO{\left(Y\right)}_d)$, we have ${\overline{D}}^d=\overline{C\cap Y_d}\cap Y_d=\overline{C}\cap Y_d$. Applying this to the Heyting mapping condition, we have $f_d^{-1}\left({\overline{D}}^d\right)=f_d^{-1}(\overline{C}\cap Y_d)=f^{-1}\left(\overline{C}\right)\cap X_d=\overline{f^{-1}\left(C\right)}\cap X_d=\overline{f^{-1}\left(C\right)\cap X_d}\cap X_d=\overline{f^{-1}\left(D\right)\cap X_d}\cap X_d={\overline{f_d^{-1}\left(D\right)}}^d$.

Take now $K\in LCon(Y_d,C{O}_s{\left(Y\right)}_d)$. We can express its topological closure in $Y_d$ (omitting ${}^d$) as $\overline{K}=\bigcup \{\overline{K\cap V}:V\in C{O}_s{\left(Y\right)}_d\}$. That is why $f_d^{-1}\left(\overline{K}\right)=\bigcup \{f_d^{-1}\left(\overline{K\cap V}\right):V\in C{O}_s{\left(Y\right)}_d\}$. Each $K\cap V$ belongs to $Con(Y_d,C{O}_s{\left(Y\right)}_d)\subseteq Con(Y_d,CO{\left(Y\right)}_d)$. Hence, $f_d^{-1}\left(\overline{K\cap V}\right)=\overline{f_d^{-1}(K\cap V)}$. We obtain $\bigcup \{f_d^{-1}\left(\overline{K\cap V}\right):V\in C{O}_s{\left(Y\right)}_d\}=\bigcup \{\overline{f_d^{-1}(K\cap V)}:V\in C{O}_s{\left(Y\right)}_d\}=\bigcup \{\overline{f_d^{-1}\left(K\right)\cap W}:W\in C{O}_s{\left(X\right)}_d\}=\overline{f_d^{-1}\left(K\right)}$ and $f_d$ is a Heyting bounded continuous mapping.

Step 2: 

The restricted functor $\overline{S}:{\mathbf{HABD}}^{op}\to \mathbf{HSpecBD}$.

By the classical Esakia Duality (see Remark 10 in [3]), the topological space $\mathcal{PF}\left(A\right)$ is Heyting spectral for a Heyting algebra A. Moreover, $h^{•}:\mathcal{PF}\left(M\right)\to \mathcal{PF}\left(L\right)$ a Heyting spectral map for a morphism $h:L\to M$ of $\mathbf{HABD}$ (Fact 13).

Step 3: 

The restricted functor $\overline{A}:\mathbf{HLSS}\to {\mathbf{HABD}}^{op}$.

For a Heyting locally small space $(X,{\mathcal{L}}_X)$, the intuitionistic implication in the bounded lattice ${\mathcal{L}}_X^o$ can be introduced by the formula $U\to V=ext(U\setminus V)$, making it a Heyting algebra by Proposition 11. For a morphism f in $\mathbf{HLSS}$, the mapping ${\mathcal{L}}^{o}f$ is a homomorphism of Heyting algebras by Proposition 12.

Step 4: 

The functor $\overline{R}\overline{S}\overline{A}$ is naturally isomorphic to $I{d}_{\mathbf{HLSS}}$.

The mapping ${\eta }_X:(\widehat{X},{\widetilde{{\mathcal{L}}_X}}^d)\to (X,{\mathcal{L}}_X)$ is a strict homeomorphism by Proposition 2 of [2] and an isomorphism of $\mathbf{HLSS}$ by Fact 12.

Step 5: 

The functor $\overline{S}\overline{A}\overline{R}$ is naturally isomorphic to $I{d}_{\mathbf{HSpecBD}}$.

Each ${\theta }_X$ is an isomorphism in $\mathbf{SpecBD}$ as well as an isomorphism of $\mathbf{HSpec}$ by Fact 11 in [3], so an isomorphism of $\mathbf{HSpecBD}$.

Step 6: 

The functor $\overline{A}\overline{R}\overline{S}$ is naturally isomorphic to $I{d}_{{\mathbf{HABD}}^{op}}$.

Each ${\kappa }_L:L\to {\tilde{L}}^d$ is an order isomorphism hence, by [35] (Exercise IX.4.3), an isomorphism of Heyting algebras. □

Corollary 11. 

Restrictions of Heyting bounded continuous maps between Heyting up-spectral locally small spaces to decent locally small subspaces are Heyting bounded continuous.

Proof. 

Follows from Step 1 in the proof of Theorem 4. □

Example 7. 

1. 

For an o-minimal structure $\mathcal{M}$ (see [39]) and a locally definable space X over $\mathcal{M}$ (see [42]), the pair $(X,LDefOp(X\left)\right)$, where $LDefOp\left(X\right)$ is the family of locally definable open subsets of X, is an object of $\mathbf{HLSS}$. (Compare Example 12 in [3].)

2. 

Each open morphism $f:X\to Y$ of $\mathbf{LDS}\left(\mathcal{M}\right)$ is Heyting bounded continuous as a map from $(X,LDefOp(X\left)\right)$ to $(Y,LDefOp(Y\left)\right)$, so a morphism of $\mathbf{HLSS}$. (Compare Example 14 in [3].)

Corollary 12 

(see Corollary 2 of [3]). Open morphisms in $\mathbf{LDS}\left(\mathcal{M}\right)$, where $\mathcal{M}$ is an o-minimal strucuture, are restrictions to constructibly dense subsets of Heyting spectral maps between Heyting up-spectral spaces.

Lemma 3. 

The categories $\mathbf{HuSpec}$ and $\mathbf{HSpecB}$ are equivalent and dually equivalent to $\mathbf{HAB}$.

Proof. 

That $\mathbf{HuSpec}$ and $\mathbf{HSpecB}$ are equivalent follows from Theorem 5 in [2] by restricting to Heyting objects and Heyting morphisms.

That $\mathbf{HSpecB}$ is dually equivalent to $\mathbf{HAB}$ follows from Theorem 4 by restricting to the case $X_d=X_s$. □

Corollary 13 

(Esakia Duality for up-spectral spaces). The categories $\mathbf{HuSpec}$, $\mathbf{HuSpLSS}$, $\mathbf{uEsa}$, $\mathbf{ELSS}$, $\mathbf{HSpecB}$ are dually equivalent to $\mathbf{HAB}$.

Proof. 

Follows from Lemma 3 and Corollary 4. □

Corollary 14 

(Esakia Duality for spectral spaces). The categories $\mathbf{HSpec}$, $\mathbf{HSpSS}$, $\mathbf{Esa}$, and $\mathbf{ESS}$ are dually equivalent to $\mathbf{HA}$.

12. Discussion

Although almost 90 years have passed since Stone’s papers [12,48,49], nearly 50 years since Esakia’s work [4], and the literature of their various extensions and variants is enormous, it is the author’s novelty to study these examples of symmetry between categories to develop tame topology.

The closest to our approach are debatably the works of G. Dimov and E. Ivanova-Dimova [18,47,50], which, however, are written in a strictly algebraic language and deal only with Boolean algebras (with distinguished ideals), and do not consider a more general case of distributive lattices. The topological spaces they consider are Boolean, while we use up-spectral spaces.

Even when, for example, the papers [10,25,51] employ many topological notions, the language used in them is mainly algebraic or categorial, and the authors do not have tame topology in their focus. It would be interesting to examine all similar Stone-type dualities (including variants of Esakia duality) and other equivalences or dualities known in topology from the point of view of tame topology.

13. Conclusions

We have proved many facts about concrete isomorphisms, equivalences and dual equivalences between categories, summarised in the following two tables. (The main new theorem is Theorem 4, followed by Theorem 2.) In each row of the first table, the first four categories are concretely isomorphic, equivalent to the category in the fifth column and dually equivalent to the category in the last column.

$\mathbf{SpSS}$	$\mathbf{Spec}$	$\mathbf{Pri}$	$\mathbf{PSS}$		$\mathbf{Lat}$
$\mathbf{HSpSS}$	$\mathbf{HSpec}$	$\mathbf{Esa}$	$\mathbf{ESS}$		$\mathbf{HA}$
$\mathbf{uSpLSS}$	$\mathbf{uSpec}$	$\mathbf{uPri}$	$\mathbf{PLSS}$	$\mathbf{SpecB}$	$\mathbf{LatB}$
$\mathbf{HuSpLSS}$	$\mathbf{HuSpec}$	$\mathbf{uEsa}$	$\mathbf{ELSS}$	$\mathbf{HSpecB}$	$\mathbf{HAB}$

In each row of the second table, the categories in the second, third and fourth columns are concretely isomorphic, equivalent to the category in the first and fifth columns and dually equivalent to the category in the last column.

${\mathbf{SS}}_0$	$\mathbf{SpecD}$	$\mathbf{PriD}$	$\mathbf{PSSD}$		$\mathbf{LatD}$
$\mathbf{HSS}$	$\mathbf{HSpecD}$	$\mathbf{EsaD}$	$\mathbf{ESSD}$		$\mathbf{HAD}$
${\mathbf{LSS}}_0$	$\mathbf{uSpecD}$	$\mathbf{uPriD}$	$\mathbf{PLSSD}$	$\mathbf{SpecBD}$	$\mathbf{LatBD}$
$\mathbf{HLSS}$	$\mathbf{HuSpecD}$	$\mathbf{uEsaD}$	$\mathbf{ELSSD}$	$\mathbf{HSpecBD}$	$\mathbf{HABD}$

This makes another step, after [2,3], in developing tame topology. To do this, we have introduced new notions and notations as well as proved many auxiliary facts about locally small spaces and presented many examples illustrating the introduced notions. Among the things worth mentioning are the notion of a special bornology (Definition 13) in a distributive bounded lattice and Example 4 using the theory of closure algebras. Moreover, we have shown (Theorem 3) how Dimov’s version (from [47]) of Stone Duality for Boolean spaces with continuous mappings agrees with our versions of Stone Duality. We have also introduced the standard up-spectralification (Definition 20) of a Kolmogorov locally small space and extended many features of the theory of Heyting small spaces from our previous paper [3] to the locally small context.