Heyting Locally Small Spaces and Esakia Duality

Preprint

Article

Heyting Locally Small Spaces and Esakia Duality

Altmetrics

Downloads

Views

Comments

A peer-reviewed article of this preprint also exists.

Artur Piękosz^*

This version is not peer-reviewed

Submitted:

01 June 2023

Posted:

02 June 2023

You are already at the latest version

Alerts

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.

Keywords:

Subject: Computer Science and Mathematics - Geometry and Topology

1. Introduction

Equivalences (including isomorphisms) and dualities between categories are seen as important and fruitful forms of symmetry in pure mathematics. We extend Esakia Duality to Heyting locally small spaces, which we introduce. This duality was originally proved by Leo Esakia in [1] (see also [2]) for spaces that combine topology and order, which he called hybrids (we are especially interested in strict hybrids). 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 ([3] and [4]) together with their collaborators ([5,6]) as well as S.A. Celani and R. Jansana ([7]) and many others.

We also see Esakia Duality as a subcase of Stone Duality ([8]) or Priestley Duality ([9]). Many extensions of Stone Duality have been achieved in recent years. For example: the locally compact Hausdorff case in [10], removing the zero-dimensionality and the commutativity assumptions in [11], generalisations of Gelfand–Naimark–Stone Duality to completely regular spaces in [12] and its application to the characterisation of normal, Lindelöf and locally compact Hausdorff spaces in [13]. There exist extensions of Stone Duality that drop compactness completely: the paper [14] considers all zero-dimensional Hausdorff spaces. Stone Duality has been also extended in [15] to (non-distributive) orthomodular lattices (corresponding to spectral presheaves), in [16] to some non-distributive (implicative, residuated, or co-residuated) lattices and applied to the semantics of substructural logics, in [17] to a non-commutative case of left-handed skew Boolean algebras. Applications of Stone Duality have appeared in [18] (canonical extensions of lattice-ordered algebras) and [19] (the semantics of non-distributive propositional logics). Some recent applications of Stone Duality have appeared in [20] in the theory of

C^{*}

-algebras.

The main objective of this paper is to extend our results from [21] by presenting a version of Esakia Duality for Heyting locally small spaces (we try to follow the conventions of [22]). The present paper continues the research from [21,23] 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 [24]. Such an approach to topology is intended to eliminate pathological phenomena of usual topology. We are encouraged by the authors of [25] 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 [26]). 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 ([27]) O. Echi together with his collaborators ([28,29]), L. Acosta and I.M. Rubio Perilla ([30]). Up-spectral spaces can be better understood in the wider context of Balbes-Dwinger spaces, see [31] and [32].

Smopologies have appeared implicitly in real algebraic geometry (see [33] (Definition 7.1.14) or [34] (p. 12)), in o-minimality ([35,36,37]), and in more general contexts of model theory ([38,39,40]). 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 [22] (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 [41] (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 [42] 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

U \cap_{1} V = {U \cap V : U \in U, V \in V}, U \cap_{1} V = U \cap_{1} {V}

or for other operations

{i n t}_{1} (A) = {i n t (A) : A \in A}, G e n_{1} (A) = {G e n (A) : A \in A},

℧ A = {⋃ B : B \subseteq A}, Ω A = {⋂ B : B \subseteq A} .

Let a family

A

of subsets of a set X be given. Define

b a (A) = t h e B o o l e a n a l g e b r a g e n e r a t e d b y A i n P (X)

A^{o} = {Y \subseteq X | Y \cap_{1} A \subseteq A} .

Elements of

A^{o}

are called the sets compatible with the family

A

, while elements of

b a (A)

are the sets constructible from

A

Definition 1

([23,43]). Alocally small spaceis a pair

(X, L_{X})

, where X is any set and

L_{X} \subseteq P (X)

satisfies the following conditions:

(LS1)

\emptyset \in L_{X}

(LS2) if

A, B \in L_{X}

, then

A \cap B, A \cup B \in L_{X}

(LS3)

⋃ L_{X} = X

Elements of

L_{X}

are also calledsmops(i.e.,small opensubsets of X). The family

L_{X}

is called asmopology. Assume

(X, L_{X})

and

(Y, L_{Y})

are locally small spaces. Then a mapping

f : X \to Y

is:

(a): boundedif $L_{X}$ refines $f^{- 1} (L_{Y})$ , which means that each $A \in L_{X}$ admits $B \in L_{Y}$ such that $A \subseteq f^{- 1} (B)$ ,
(b): continuousif $f^{- 1} (L_{Y}) \cap_{1} L_{X} \subseteq L_{X}$ (i.e., $f^{- 1} (L_{Y}) \subseteq L_{X}^{o}$ ),
(c): strongly continuousif $f^{- 1} (L_{Y}) \subseteq L_{X}$ .

We have the category

LSS

of locally small spaces and bounded continuous maps, called sometimes alsostrictly continuousmaps.

Definition 2.

X \in L_{X}

, then

(X, L_{X})

is called asmall space. The full subcategory in

LSS

generated by the small spaces is denoted by

SS

Remark 1.

The isomorphisms of

LSS

are such bijections

f : X \to Y

between locally small spaces that

f^{- 1} (L_{Y}^{o}) = L_{X}^{o}

and the families

L_{X}

and

f^{- 1} (L_{Y})

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} (L_{Y}) = L_{X}

. Such mappings are calledstrict homeomorphisms.

Definition 3.

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

The family $L_{X}^{'} = X ∖_{1} L_{X}$ is called aco-smopologyand its members are calledco-smops.
The family $C o n (X) = b a (L_{X})$ contains theconstructiblesets, which are the Boolean combination of smops.
The family $L_{X}^{o}$ contains subsets compatible with smops, which will be called(admissible) open sets, and their complements that will be called(admissible) closed sets. The family of all admissible open sets $L_{X}^{o}$ in X will be usually denoted by $A d O p (X)$ , and the family of all admissible closed sets will be denoted by $A d C l (X)$ or $L_{X}^{c}$ .
A subset of a smop that is a constructible set will be called asmall constructible set. The family of all small constructible sets will be denoted by $s C o n (X)$ .
A subset of X whose traces on smops are constructible is alocally constructible set. The family of all locally constructible sets will be denoted by $L C o n (X)$ .

Remark 2.

Notice that for a small space (i.e., if

X \in L_{X}

) we have

L_{X} = A d O p (X) a n d L_{X}^{'} = A d C l (X) .

In this case

L_{X}^{'}

is also a smopology and the small space

(X, L_{X}^{'})

is called theinverse spaceof

(X, L_{X})

and denoted by

X_{i n v}

([21]).

Fact 1

([43]). We have the functor ofsmallification

s m : LSS \to SS

given by

s m : (X, L_{X}) \to (X, L_{X}^{o})

and

s m (f) = f

, which is a concrete reflector (see [44]).

Definition 4.

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

1): The topology $τ (L_{X})$ generated by the smops will be called theoriginal topologyand denoted $τ_{o r i g}$ . The closure, the interior and the exterior operations in the original topology will be denoted by $\bar{\cdot}$ (or by $c l (\cdot)$ ), $i n t (\cdot), e x t (\cdot)$ , respectively.
2): The topology $τ_{i n v} = τ (A d C l (X))$ generated by the admissible closed sets will be called theinverse topology. The closure and the interior operations in the inverse topology will be denoted by ${\bar{\cdot}}^{i n v}, i n t_{i n v} (\cdot)$ , respectively.
3): The topology $τ_{t i n v} = τ (L_{X}^{'})$ , generated by the co-smops, which can be called thetiny inverse topology.
4): The topology $τ (L_{X} ∖_{1} L_{X})$ generated by the differences of smops will be called theconstructible topology. The closure and the interior operations in the constructible topology will be denoted by ${\bar{\cdot}}^{c o n}, i n t_{c o n} (\cdot)$ , respectively.

Remark 3.

While the family

A d O p (X)

usually properly contains

L_{X}

, they generate the same original topology

τ (L_{X}) = τ (A d O p (X))

Definition 5.

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

τ_{o r i g} = ℧ L_{X} = ℧ A d O p (X), X ∖_{1} τ_{o r i g} = Ω L_{X}^{'} = Ω A d C l (X) .

Similarly, we haveconstructibly weakly openandconstructibly weakly closedsets,inversely weakly openandinversely weakly closedsets, etc. A mapping that is continuous in the topologies of weakly open sets is calledweakly continuous.

Proposition 1.

For a locally small space

(X, L_{X})

, we have

\bar{A} = ⋃_{V \in L_{X}} \bar{A \cap V}, i n t (A) = ⋃_{V \in L_{X}} i n t (A \cap V) .

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

Proof.

We always have

⋃_{V \in L_{X}} \bar{A \cap V} \subseteq \bar{A}

. If

x \in \bar{A}

, then there exists

V \in L_{X}

such that

x \in \bar{A} \cap V

. For any

W \in L_{X}

x \in W

then

W \cap V \in L_{X}

and

W \cap V \cap A \neq \emptyset

. That is why

x \in \bar{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

i n t (A) \subseteq ⋃_{V \in L_{X}} i n t (A \cap V)

in the second claim, assume

x \in i n t (A) \cap V

for some

V \in L_{X}

. Then

x \in W \subseteq A

for some

W \in L_{X}

. Hence

x \in W \cap V \subseteq A \cap V

and

x \in i n t (A \cap V)

. □

Example 1.

The tiny inverse topology is usually strictly smaller than the inverse topology. In symbols: we usually have

τ_{t i n v} = ℧ L_{X}^{'} \subset τ_{i n v} = ℧ A d C l (X)

Indeed, consider an infinite set X and define

L_{X} = Fin (X)

. Then

τ (L_{X}^{'}) = Cofin (X) \cup {\emptyset}

but the inverse topology equals

τ (A d C l (X)) = A d C l (X) = P (X)

Example 2.

The family

b a (A d O p (X))

may be a proper subfamily of

L C o n (X)

. Indeed, take as X the disjoint union

⨆_{n \in N} R_{o m}^{2 n}

2 n

-th powers of the o-minimal real line

R_{o m}

([43]) indexed by

n \in N

. Then the smops of

R_{o m}^{2 n}

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

R_{o m}^{2 n}

. The locally constructible set

A = ⋃_{n \in N} A_{n}

where

A_{n} = {x \in R^{2 n} : # {k : x_{k} = 0} i s e v e n}

is not a Boolean combination of admissible open sets. To see this, consider the closure algebra

(L C o n (X), \bar{\cdot})

in the sense of Definition 2.2.1 of [2]. 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 [2], we would have

ρ^{2^{m}} (A) = \emptyset

where

ρ (Z) = \partial (\partial Z)

and

\partial Z = \bar{Z} ∖ Z

. For

n > 2^{m}

, we have

ρ^{n} (A_{n}) = {0}^{2 n} \neq \emptyset

, so

A_{n}

is not a Boolean combination of m smops in

R_{o m}^{2 n}

. Consequently,

b a (A d O p (X)) \subset L C o n (X)

Example 3.

Usually also the Boolean algebra

C o n (X) = b a (L_{X})

is strictly smaller than the Boolean algebra

b a (A d O p (X))

. Indeed, for the space from Example 1, we have

C o n (X) = Fin (X) \cup Cofin (X)

and

b a (A d O p (X)) = P (X)

Proposition 2.

We have the following equality between topologies:

τ (L_{X} ∖_{1} L_{X}) = τ (b a (L_{X})) = τ (b a (A d O p (X))) = τ (s C o n (X)) = τ (L C o n (X)) .

Hence we have the following descriptions of the constructible topology:

℧ L C o n (X) = ℧ s C o n (X) = ℧ b a (A d O p (X)) = ℧ b a (L_{X}) = ℧ (L_{X} ∖_{1} L_{X}) .

Proof.

The inclusions

L_{X} ∖_{1} L_{X} \subseteq b a (L_{X}) \subseteq b a (A d O p (X)) \subseteq L C o n (X)

are obvious. Each locally constructible set is a union of a family of smop differences. □

Definition 6.

For a locally small space

(X, L_{X})

and a subset

Y \subseteq X

, we have the inducedsubspace

(Y, L_{X} \cap_{1} Y)

(X, L_{X})

. This subspace is called:openif

Y \in A d O p (X)

,closedif

Y \in A d C l (X)

,decentif Y is constructibly dense,smallif Y is a subset of a smop.

Lemma 1.

Assume that X is a locally small space.

If $V \in A d O p (X)$ , then for the induced subspace $(V, L_{X} \cap_{1} V)$ we have $A d O p (V) = A d O p (X) \cap_{1} V$ .
If $F \in A d C l (X)$ , then for the induced subspace $(F, L_{X} \cap_{1} F)$ we have $A d O p (F) = A d O p (X) \cap_{1} F$ .
If S is a small subset of X, then for the induced subspace $(S, L_{X} \cap_{1} S)$ we have $A d O p (S) = A d O p (X) \cap_{1} S$ .
If $Y \subseteq X$ is decent, then for the induced subspace $(Y, L_{X} \cap_{1} Y)$ we have $A d O p (Y) = A d O p (X) \cap_{1} Y$ .

Proof.

We check $A d O p (X) \cap_{1} V \subseteq A d O p (V)$ first. For each $W \in A d O p (X)$ we have $W \cap V \in A d O p (X) \cap P (V)$ . For any $A \in L_{X} \cap_{1} V$ , we have $W \cap V \cap A \in L_{X} \cap P (V) = L_{X} \cap_{1} V$ . This proves $W \cap V \in A d O p (V)$ . For the other inclusion assume $U \in A d O p (V)$ . For each $L \in L_{X}$ , we have $U \cap L = U \cap L \cap V \in L_{X} \cap_{1} V \subseteq L_{X}$ . This means $U \in A d O p (X) \cap P (V)$ . Since $U = U \cap V \in A d O p (X) \cap_{1} V$ , we have $A d O p (V) \subseteq A d O p (X) \cap_{1} V$ .
Assume $W \in A d O p (X)$ . For any $L \in L_{X}$ , the set $W \cap (L \cap F) = (W \cap L) \cap F \in L_{X} \cap_{1} F$ . Since $W \cap F \in A d O p (F)$ , the inclusions $A d O p (X) \cap_{1} F \subseteq A d O p (F)$ and $A d C l (X) \cap_{1} F \subseteq A d C l (F)$ are clear. We now prove $A d C l (F) \subseteq A d C l (X) \cap_{1} F$ . For $M \in A d C l (F)$ , we have $(F ∖ M) \cap_{1} (L_{X} \cap_{1} F) \subseteq L_{X} \cap_{1} F$ . Take $L \in L_{X}$ . Then $(X ∖ M) \cap L = ((X ∖ F) \cap L) \cup ((F ∖ M) \cap L)$ . But $(F ∖ M) \cap L = F \cap K_{L}$ for some $K_{L} \in L_{X}$ and we may assume $K_{L} \subseteq L$ . We get $(X ∖ M) \cap L = ((X ∖ F) \cap L) \cup K_{L} \in L_{X}$ . This proves $X ∖ M \in A d O p (X)$ , so $M = M \cap F \in A d C l (X) \cap_{1} F$ .
We know that $S \subseteq L$ for some $L \in L_{X}$ . Since $(S, L_{X} \cap_{1} S)$ is a small space, we have $A d O p (S) = L_{S} = L_{X} \cap_{1} S = (L_{X} \cap_{1} L) \cap_{1} S = (A d O p (X) \cap_{1} L) \cap_{1} S = A d O p (X) \cap_{1} S$ .
For the non-trivial inclusion $A d O p (Y) \subseteq A d O p (X) \cap_{1} Y$ , assume $W \in A d O p (Y)$ . Then for each $L \in L_{X}$ there exists $W_{L} \in L_{X} \cap_{1} L$ such that $W \cap L = W_{L} \cap Y$ . We check if the set $\tilde{W} = ⋃_{L \in L_{X}} W_{L}$ is admissible open in X. For any $M \in L_{X}$ , we conclude $\tilde{W} \cap M = ⋃_{L \in L_{X}} W_{L} \cap M$ is equal to $W_{M} \in L_{X}$ since $L_{X} ∋ V \mapsto V \cap Y \in L_{Y}$ is an isomorphism of lattices and, consequently, for each $L \in L_{X}$ 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 [21] 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, L_{X}^{o})

of the locally small space

(X, L_{X})

. That is why a series of facts passes to the locally small case.

Recall that for a topological space

(X, τ_{X})

the point xspecialises toy (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 subsets

A \subseteq X

, we write:

G e n (A) = {x \in X : x \in G e n (a) f o r s o m e a \in A}

S p e z (A) = {x \in X : x \in S p e z (a) f o r s o m e 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, L_{X})

and

x, y \in X

, the following conditions are equivalent for the original topology on X:

x specialises to y ( $x ⇝ y$ ),
$y \in \bar{{x}}$ ,
$y \notin e x t {x}$ ,
each smop containing y also contains x,
each admissible open set containing y also contains x,
each co-smop containing x also contains y,
$y ⇝_{i n v} x$ (read: y specialises to x in $τ_{i n v}$ .)

Fact 3.

For each locally small space

(X, L_{X})

and

A \subseteq X

, we have:

S p e z (A) \subseteq \bar{A}

and

G e n (A) \subseteq {\bar{A}}^{i n v}

Fact 4.

In a pre-Boolean locally small space,

x ⇝ y

iff

x = y

Fact 5.

In any locally small space

(X, L_{X})

, we have the following:

(1) If

A \subseteq X

is weakly closed (

A \in Ω A d C l (X)

), then

S p e z (A) = A

(2) If

A \subseteq X

is weakly open (

A \in ℧ L_{X}

), then

G e n (A) = A

Recall that a subset

Q \subseteq X

is called saturated if it is an intersection of weakly open sets, i.e.,

Q \in Ω ℧ L_{X}

. Moreover, a locally small space

(X, L_{X})

is called

T_{0}

(or Kolmogorov) if the family

L_{X}

separates points ([22], Reminder 1.1.4), which means that for

x, y \in X

the following condition is satisfied:

i f x \in A \Leftrightarrow y \in A f o r e a c h A \in L_{X}, t h e n x = y .

{LSS}_{0}

we denote the full subcategory in

LSS

generated by

T_{0}

objects, while by

{SS}_{0}

we denote the full subcategory in

LSS

generated by small

T_{0}

objects.

Fact 6.

(X, L_{X})

T_{0}

, then the specialisation relation ⇝ is a partial order.

Fact 7.

(X, L_{X})

T_{0}

and

Q \subseteq X

, then the following are equivalent:

Q is saturated,
$G e n (Q) = Q$ .

4. Up-spectral Locally Small Spaces

Definition 7.

For a topological space

(X, τ_{X})

, we have the following notation:

$C O (X) = C O (X, τ_{X})$ is the family of all compact open subsets of X,
$I C O (X) = I C O (X, τ_{X})$ is the family of all open subsets of X such that any intersection with a compact open set is compact,
co- $I C O (X)$ is the family of complements of sets from $I C O (X)$ ,
$C l O p (X) = C l O p (X, τ_{X})$ is the family of clopen subsets of X.

Moreover,

(X, τ_{X})

is called

semi-spectralif $C O (X) \cap_{1} C O (X) \subseteq C O (X)$ ,
coherentif $C O (X)$ forms a basis of the topology and X is semi-spectral.

Definition 8

([23,29,30]). A topological space

(X, τ_{X})

isup-spectralif it is (

T_{0}

) sober and coherent but not necessarily compact.

Fact 8

(compare [22], Corollary 1.5.5). In an up-spectral topological space, the following holds for any

A \subseteq X

$\bar{A} = S p e z ({\bar{A}}^{c o n})$ where $τ_{c o n} = ℧ (C O (X) ∖_{1} C O (X))$ ,
${\bar{A}}^{i n v} = G e n ({\bar{A}}^{c o n})$ where $τ_{i n v} = ℧ c o - I C O (X)$ .

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

Theorem 1

([27,28,30], characterisation of up-spectral spaces). For any topological space

(X, τ_{X})

, the following conditions are equivalent:

X is almost-spectral,
X is open dense in a spectral space,
X is open in a spectral space,
X is a sober BD-space,
X is homeomorphic to the prime spectrum of a distributive lattice with minimum,
X is up-spectral,
X admits a trivial one-point spectralification,
X is semi-spectral and locally spectral,
X is the underlying topological space of a scheme,
X is the underlying topological space of an open subscheme in an affine scheme.

Proof.

This is Theorems 7 and 8 in [30], Theorem 2.1 in [28] and Proposition 16 in [27]. □

Definition 9

([23]). A mapping

g : (X, τ_{X}) \to (Y, τ_{Y})

between up-spectral spaces is calledspectralif the following conditions are satisfied:

g isbounded: $g (C O (X))$ refines $C O (Y)$ ,
g iss-continuous: $g^{- 1} (I C O (Y)) \subseteq I C O (X)$ .

Following [23], we shall denote by

uSpec

the category of up-spectral topological spaces and spectral mappings between them.

Definition 10.

Anup-spectral locally small spaceis a locally small space

(X, L_{X})

where

(X, ℧ L_{X})

is an up-spectral topological space and

L_{X} = C O (X)

. We get the category

uSpLSS

of up-spectral locally small spaces and bounded continuous mappings.

Remark 4.

For objects of

uSpLSS

we have (see [29] and [22]):

A d O p (X) = I C O (X), A d C l (X) = c o - I C O (X), L C o n (X) = C l O p (X, τ_{c o n}) .

Definition 11.

Anup-Priestley spaceis a system

(X, σ_{X}, \leq_{X})

where

(X, σ_{X})

is a Boolean (i.e. zero-dimensional Hausdorff locally compact, [45]) topological space and

\leq_{X}

is a partial order on X satisfying thePriestley separation axiom

i f x \neg \leq_{X} y, t h e n \exists V \in C O (X) s u c h t h a t V = ↑ V, x \in V, y \notin V .

APriestley mappingbetween up-Priestley spaces is a continuous non-decreasing mapping. We have the category

uPri

of up-Priestley spaces and Priestley mappings.

Definition 12.

APriestley locally small spaceis a system

(X, L_{X}^{σ}, \leq_{X})

where

(X, L_{X}^{σ})

is a Boolean locally small space (Definition 16) and

\leq_{X}

is a partial order on X satisfying thePriestley separation axiom

i f x \neg \leq_{X} y, t h e n \exists V \in L_{X}^{σ} s u c h t h a t V = ↑ V, x \in V, y \notin V .

APriestley morphismbetween Priestley locally small spaces is a strictly (equivalently: weakly) continuous non-decreasing mapping. We have the category

PLSS

of Priestley locally small spaces and Priestley morphisms.

Theorem 2.

All the following categories are concretely isomorphic:

uSpec, uPri, PLSS, uSpLSS .

Proof.

We have four concrete functors:

the functor $k_{1} : uSpec \to uPri, k_{1} (X, τ_{X}) = (X, τ_{c o n}, ⇝_{i n v})$

For an up-spectral space $(X, τ_{X})$ , the constructible topology $τ_{c o n}$ is Boolean and the generalisation relation $⇝_{i n v}$ (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 generization relation.
the functor $k_{2} : uPri \to PLSS, k_{2} (X, σ_{X}, \leq) = (X, C O (X, σ_{X}), \leq)$

For an object $(X, σ_{X}, \leq)$ of $uPri$ , the space $(X, C O (X, σ_{X}))$ is clearly locally small ( $C O (X, σ_{X})$ is a smopology) and Boolean since $A d O p (X) = C l O p (X, σ_{X}) = A d C l (X)$ . Obviously, ≤ satisfies the Priestley separation axiom.

Assume $f : X \to Y$ is non-decreasing and continuous. Then

$f^{- 1} (I C O (Y)) = f^{- 1} (C l O p (Y)) \subseteq C l O p (X) = I C O (X) .$

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 $f (C O (X))$ is a refinement of $C O (Y)$ . Hence f is bounded between the locally small spaces.
the functor $k_{3} : PLSS \to uSpLSS, k_{3} (X, L_{X}^{σ}, \leq) = (X, U p_{\leq} \cap L_{X}^{σ})$

The family $U p_{\leq} \cap L_{X}^{σ}$ is the compact-open basis of an up-spectral topology $τ_{X}$ (compare the characterisation of spectral topologies in Theorem 1.5.11 of [22]). This compact-open basis in the up-spectral topology is, in particular, a smopology. Hence $(X, U p_{\leq} \cap L_{X}^{σ})$ is an up-spectral locally small space.

Assume $f : X \to Y$ is non-decreasing and bounded continuous in the constructible smopologies $L_{X}^{σ}, L_{Y}^{σ}$ . Each compact-open in $τ_{X}$ is compact-open in $σ_{X}$ so its image under f is contained in a small constructible in $τ_{Y}$ , a subset of a compact-open in $τ_{Y}$ . We also have $f^{- 1} (I C O (Y, τ_{Y})) \subseteq I C O (X, τ_{X})$ 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 $k_{4} : uSpLSS \to uSpec, k_{4} (X, L_{X}) = (X, ℧ L_{X})$

For an up-spectral locally small space $(X, L_{X})$ , the topological space $(X, ℧ L_{X})$ 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

τ_{X} = ℧ (U p_{⇝_{i n v}} \cap C O (X, σ_{X})), σ_{X} = τ {(U p_{\leq} \cap C O (X, σ_{X}))}_{c o n},

L_{X}^{σ} = C O (X, τ {(U p_{\leq} \cap L_{X}^{σ})}_{c o n}), L_{X} = U p_{⇝_{i n v}} \cap C O (X, τ {(L_{X})}_{c o n})

and

\leq = ⇝_{i n v}

is the generalisation relation of the up-spectral topology

τ_{X}

. □

We give an analogue of (1.3) in [22] and Example 3 in [21].

Proposition 3.

For an up-spectral topological space

(X, τ_{X})

, the corresponding up-spectral locally small space

(X, C O (X))

has the property, that the small constructible weakly open sets are exactly the compact open sets, so smops, i.e.,

L_{X} = C O (X)

and

s C o n (X) \cap ℧ L_{X} = L_{X}

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. □

Remark 5.

From [21], Examples 4–7, we know that:

A weakly open constructible set in a locally small space may not be a smop.
A $T_{0}$ locally small space may have all smops (topologically) compact but not be sober.
A $T_{0}$ locally small space may be (topologically) sober but not compact.
A $T_{0}$ locally small space may be sober and compact with not all smops 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

L_{X} \subseteq C O (X)

, we have

C O (X) = 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 $L_{X}$ are in a bijective correspondence with the weakly open sets, so also with the weakly closed sets,
the prime ideals of $L_{X}$ are in a bijective correspondence with the non-empty, irreducible, weakly closed sets.

Proof.

The proof is similar to that of Proposition 3 of [21]. □

Recall that if not all smops are compact, then by Example 8 from [21] it may happen that

I \subset i (s (I))

, using notation from the above mentioned proof.

5. Stone Duality for Up-spectral Spaces

Corollary 1

(Stone Duality for spectral spaces). The categories

Spec

SpSS

Pri

PSS

are dually equivalent to

Lat

Proof.

Follows from Theorem 2 and the classical Stone Duality. □

Definition 13.

Abornologyin a bounded lattice

(L, \lor, \land, 0, 1)

is an ideal

B \subseteq L

such that

⋁ B = 1 .

Recall that each distributive bounded lattice L corresponds to the spectral space

(PF (L), τ (\tilde{L}))

where

\tilde{L} = {\tilde{a} : a \in L}

. Similarly, we define

\tilde{B} = {\tilde{a} : a \in B}

. A bornology B will be calledspecialif

{(\tilde{B} \cap_{1} ⋃ \tilde{B})}^{o} = \tilde{L} \cap_{1} ⋃ \tilde{B}

as subsets of

P (⋃ \tilde{B})

and

⋃ \tilde{B}

is constructibly dense in

PF (L)

Example 4.

Let

\tilde{R}

be the real spectrum

{Spec}_{r} R [Y]

of the ring

R [Y]

. Consider the following families of compact open sets in this spectral space:

C O (\tilde{R}) = f i n i t e u n i o n s o f i n t e r v a l s o f t y p e [r^{+}, s^{-}] w h e r e r < s \in R

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

B = f i n i t e u n i o n s o f i n t e r v a l s [r^{+}, s^{-}] w h e r e r < s \in R .

Then

B

is a bornology in the distributive bounded lattice

C O (\tilde{R})

. Notice that

C O (\tilde{R}) \subset B^{o} =

= l o c a l l y f i n i t e u n i o n s o f i n t e r v a l s [r^{+}, s^{-}], r < s \in R a n d s i n g l e t o n s {- \infty}, {+ \infty} .

Define

X_{s} = ⋃ \tilde{B} = \tilde{R} ∖ {- \infty, + \infty} .

Still

C O (\tilde{R}) \cap_{1} X_{s} \subset (B^{o}) \cap_{1} X_{s} = {(B \cap_{1} X_{s})}^{o} =

= t h e f a m i l y o f a l l l o c a l l y f i n i t e u n i o n s o f i n t e r v a l s [r^{+}, s^{-}], r < s \in R .

Hence the bornology

B

is not special.

Definition 14

([23]). The category

SpecB

has

as objects: systems $((X, τ_{X}), C O_{s} (X), X_{s})$ where $(X, τ_{X})$ is a spectral topological space, $C O_{s} (X)$ is a special bornology in $C O (X)$ and $X_{s} = ⋃ C O_{s} (X)$ ,
as morphisms: spectral mappings $g : X \to Y$ such that $g (X_{s}) \subseteq Y_{s}$ and satisfying thecondition of boundedness: $C O_{s} (X)$ is a refinement of $g^{- 1} (C O_{s} (Y))$ .

Definition 15.

Anobject of $LatB$ is a system $(L, L_{s})$ with $L = (L, \lor, \land, 0, 1)$ a bounded distributive lattice and $L_{s}$ a special bornology in L.
Amorphism of $LatB$ from $(L, L_{s})$ to $(M, M_{s})$ is such a homomorphism of bounded lattices $h : L \to M$ that isdominating: $\forall a \in M_{s} \exists b \in L_{s} a \leq h (b)$ .

Corollary 2

(Stone Duality for up-spectral spaces). The concretely isomorphic categories

uSpec

uSpLSS

uPri

PLSS

are equivalent to

SpecB

and dually equivalent to

LatB

Proof.

Follows from Theorem 2 and a special case (

X_{d} = X_{s}

) of the Stone Duality for locally small spaces ([23], Theorem 1). □

6. Boolean Locally Small Spaces

Proposition 6.

For a locally small space, the following are equivalent:

$A d O p (X) = A d C l (X)$ ,
$A d O p (X) = L C o n (X)$ ,
$A d C l (X) = L C o n (X)$
$A d O p (X)$ is a Boolean subalgebra of $P (X)$ .

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.

If a locally small space satisfies the above conditions, then we will call it apre-Boolean locally small space.
A Hausdorff pre-Boolean locally small space with all smops compact will be called aBoolean locally small space.
The category of Boolean locally small spaces and bounded (weakly) continuous maps will be denoted by $BLSS$ .

Definition 17.

We have the following categories:

1): $BoolSp$ is the category of zero-dimensional Hausdorff locally compact spaces and continuous mappings (see Dimov [45]),
2): $BAB$ is the category of Boolean algebras with special bornologies and dominating Boolean homomorphisms.

Proposition 7.

BLSS

and

BoolSp

are concretely isomorphic categories. In particular, Boolean locally small spaces are up-spectral.

Proof.

For a Boolean topological space

(X, σ)

the family

L^{σ} = C O (X, σ)

is a smopology for which

L^{σ} \subseteq C l O p (X, σ) \subseteq A d O p (X, L^{σ})

. On the other hand,

A d C l (X) \subseteq A d O p (X) \subseteq C l O p (X)

, hence

A d O p (X) = A d C l (X) = C l O p (X)

and

(X, L^{σ})

is a Boolean locally small space.

For a Boolean locally small space

(X, L^{σ})

, each smop is compact, so

L^{σ} \subseteq C O (X, ℧ L^{σ})

and

A d O p (X) = A d C l (X) \subseteq C l O p (X)

. The topology

℧ L^{σ}

is Hausdorff, locally compact and has a clopen basis

A d O p (X) = A d C l (X)

, 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 ([23]). □

Theorem 3.

BAB

is dually equivalent to

BoolSp

, so also to

BLSS

Proof.

The Dimov’s version of Stone Duality [45] says that

BoolSp

is dually equivalent to his category

Z L B A

. One needs to notice that

Z L B A

is our category

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 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 $a \in A ∖ {0}$ . Then $a^{'} \neq 1 = ⋁_{b \in B} b$ . There exists $b \in B$ such that $b \neg \leq a^{'}$ , so $a \neg \leq b^{'}$ . We have $a = a \land (b \lor b^{'}) = (a \land b) \lor (a \land b^{'})$ . Since $a \land b^{'} < a$ , we have $a \land b \neq 0$ and B is dense.
Each distinguished dense ideal is a bornology.

If $⋁_{A} I = a \neq 1$ then $a^{'} \neq 0$ . Take $i \in I$ such that $0 \neq i \leq a^{'}$ . Then $i = (i \lor a) \land (i \lor a^{'}) = a \land a^{'} = 0$ . Contradiction proves that $⋁_{A} I = 1$ .
Special bornologies are distinguished.

If B is a special bornology, then each clopen set in the constructibly dense open $⋃ \tilde{B}$ is of the form ${\tilde{a}}^{s} = \tilde{a} \cap ⋃ \tilde{B}$ where $a \in A$ . It follows from Proposition 2.6 of [45] 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 $⋃ \tilde{I}$ is constructibly dense. The condition ${(\tilde{I} \cap_{1} ⋃ \tilde{I})}^{o} = A d O p (⋃ \tilde{I}) = L C o n (⋃ \tilde{I}) = C l O p (⋃ \tilde{I}) = \tilde{L} \cap_{1} ⋃ \tilde{I}$ is satisfied by Proposition 2.6 of [45].

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

7. The Standard Up-spectralification

We extend the facts from Section 5 of [21] about the standard spectralification of

T_{0}

small spaces to the locally small case.

Definition 18.

Anembeddingof locally small spaces is an injective map

e : X \to Y

such that

e (L_{X}) = L_{Y} \cap_{1} e (X)

Definition 19

([23]). Anup-spectralificationof 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 (X)

dense in the constructible topology of Y.

Definition 20.

Thestandard up-spectralificationof a

T_{0}

locally small space

(X, L_{X})

is the locally small space (using notations from Theorem 1 of [23])

X^{u s p} = (⋃ \tilde{L_{X}}, \tilde{L_{X}})

with the embedding

X ∋ x \mapsto \hat{x} \in \hat{X} \subseteq ⋃ \tilde{L_{X}}

Remark 6.

Identifying X with

\hat{X}

, we have:

$C O (X^{u s p}) \cap_{1} X = L_{X^{u s p}} \cap_{1} X = L_{X}$ ,
$I C O (X^{u s p}) \cap_{1} X = A d O p (X^{u s p}) \cap_{1} X = A d O p (X)$ ,
$C l O p (X_{c o n}^{u s p}) \cap_{1} X = L C o n (X^{u s p}) \cap_{1} X = L C o n (X) = C l O p (X_{c o n})$ .

Since X is constructibly dense in

X^{u s p}

, we have the following bijection:

L C o n (X) ∋ A \mapsto A^{u s p} \in L C o n (X^{u s p}),

where

A^{u s p}

is the only member of

L C o n (X^{u s p})

such that

A^{u s p} \cap X = A

. One can see that

A^{u s p} = {\bar{A}}^{c o n}

taken in

X^{u s p}

and

\bar{A^{u s p}} \cap X = {\bar{A}}^{X} .

By bijectivity, we have:

i): $V \in A d O p (X)$ iff $V^{u s p} \in A d O p (X^{u s p})$ ,
ii): $F \in A d C l (X)$ iff $F^{u s p} \in A d C l (X^{u s p})$ .

Fact 9.

If X is small, then

X^{u s p} = X^{s p}

is an object of

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:

C is irreducible in X,
the closure $\bar{C}$ in the standard up-spectralification $X^{u s p}$ is irreducible.

Remark 7.

We have the functor

u s p : {LSS}_{0} \to uSpLSS

given by formulas

u s p (X, L_{X}) = (⋃ \tilde{L_{X}}, \tilde{L_{X}}), u s p (f) = {({(L^{o} f)}^{•})}_{s}

where

{({(L^{o} f)}^{•})}_{s}

is the restriction of

{(L^{o} f)}^{•}

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 calledpre-semi-Heytingif any of the following equivalent conditions are satisfied:

the interior in the original topology of any admissible closed set is an admissible open set, i.e., $i n t_{1} A d C l (X) \subseteq A d O p (X)$ ,
the closure in the original topology of any admissible open set is an admissible closed set, i.e., $c l_{1} A d O p (X) \subseteq A d C l (X)$ .

(2) A locally small space will be calledpre-Heytingif any of the following equivalent conditions are satisfied:

the interior in the original topology of any locally constructible set is an admissible open set, i.e., we have $i n t_{1} L C o n (X) \subseteq A d O p (X)$ .
the closure in the original topology of any locally constructible set is an admissible closed set, i.e., we have $c l_{1} L C o n (X) \subseteq A d C l (X)$ .

(3) A locally small space will be calledHeytingif it is

T_{0}

and pre-Heyting.

Inspired by [22] and [21], we have the following propositions.

Proposition 8.

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

the following maps are well defined:
- the open regularisation map $N : A d O p (X) \to A d O p (X)$ given by $N (A) = i n t \bar{A}$ ,
- the closed regularisation map $\bar{N} : A d C l (X) \to A d C l (X)$ given by $\bar{N} (F) = \bar{i n t F}$ ,
for each $V \in A d O p (X)$ , the subspace $(V, L_{X} \cap_{1} V)$ is pre-semi-Heyting.

Proof. (1) Obvious by definition.

(2) For

A, V \in A d O p (X)

, we have that the set

{\bar{A \cap V}}^{V} = V \cap (\bar{A \cap V}) \in V \cap_{1} A d C l (X) = A d C l (V)

is a local co-smop in V. Notice that

(V, τ (L_{X} \cap_{1} V))

is a topological subspace of

(X, τ (L_{X}))

□

Proposition 9

(characterisation of pre-Heyting spaces). For a locally small space

X = (X, L_{X})

, the following conditions are equivalent:

X is pre-Heyting,
for each $B \in L_{X} ∖_{1} L_{X}$ , $\bar{B} \in A d C l (X)$ ,
for each $A \in L C o n (X)$ , the subspace $(A, L_{X} \cap_{1} A)$ is pre-Heyting,
for each $A \in L C o n (X)$ , the subspace $(A, L_{X} \cap_{1} A)$ is pre-semi-Heyting,
for each $F \in A d C l (X)$ , the subspace $(F, L_{X} \cap_{1} F)$ is pre-semi-Heyting.

Proof.

(1) \Leftrightarrow (2)

Of course, (2) follows from (1). Assume

C \in L C o n (X)

. Then, for each

A \in L_{X}

, we have

\bar{C} \cap A = \bar{C \cap A} \cap A

. Since

C \cap A \in C o n (A)

, this set is a finite union of sets of the form

B_{1} ∖ B_{2}

with

B_{1}, B_{2} \in L_{A}

. Thus

\bar{C \cap A} \cap A \in A d C l (A)

and

\bar{C} \in A d C l (X)

(1) \Rightarrow (3)

For

A \in L C o n (X)

and

D \in L C o n (A) = L C o n (X) \cap_{1} A \subseteq L C o n (X)

, we have

\bar{D} \in A d C l (X)

. Now

{\bar{D}}^{A} = \bar{D} \cap A \in A d C l (A)

(3) \Rightarrow (4)

Trivial.

(4) \Rightarrow (5)

Trivial.

(5) \Rightarrow (2)

Each element of

L_{X} ∖_{1} L_{X}

is of the form

A \cap F

with

A \in L_{X}, F \in A d C l (X)

. Since

\bar{A \cap F} = \bar{A \cap F} \cap F \in A d C l (X) \cap_{1} F

by (5), we have

\bar{A \cap F} \in A d C l (X) \cap_{1} A d C l (X) \subseteq A d C l (X)

. □

Remark 8.

There exist pre-semi-Heyting spectral small spaces that are not pre-Heyting. See Example 8.3.11(iii) in [22].

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

Fact 11.

If the locally small space

(X, L_{X})

is (pre-)Heyting, then its smallification

(X, 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 aHeyting (bounded continuous) mapif 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

$f^{- 1} (i n t (C)) = i n t (f^{- 1} (C))$ for $C \in L C o n (Y)$ ,
$f^{- 1} (\bar{C}) = \bar{f^{- 1} (C)}$ for $C \in L C o n (Y)$ .

Definition 23.

HLSS

we denote the category of Heyting locally small spaces and Heyting bounded continuous maps.

Since

HLSS

is a full subcategory of

LSS

, we have

Fact 12.

Strict homeomorphisms between Heyting locally small spaces are isomorphisms of

HLSS

9. Heyting Up-spectral Spaces

Definition 24

([22], Section 8.3). A topological up-spectral space X will be calledHeytingif 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 aHeyting spectralmap if g is spectral and any of the equivalent the conditions holds:

$f^{- 1} (\bar{C}) = \bar{f^{- 1} (C)}$ for $C \in L C o n (Y) = C l O p_{c o n} (Y)$ ,
$f^{- 1} (i n t (C)) = i n t (f^{- 1} (C))$ for $C \in L C o n (Y) = C l O p_{c o n} (Y)$ .

We have the category

HuSpec

of Heyting up-spectral (topological) spaces and Heyting spectral mappings.

Corollary 3.

Each homeomorphism between Heyting up-spectral spaces is an isomorphism in

HuSpec

Definition 25.

We have the category

HuSpLSS

of Heyting up-spectral locally small spaces and Heyting strictly continuous maps.

Proposition 10.

If the standard up-spectralification

X^{u s p}

of a locally small space X is Heyting, then X is Heyting.

Proof.

Assume

\forall A \in L C o n (X^{u s p}) \bar{A} \in A d C l (X^{u s p})

. Then by taking traces with X, we have

\forall B \in L C o n (X) {\bar{B}}^{X} \in A d C l (X)

(see Remark 6). □

Proposition 11.

If X is a Heyting locally small space (object of

HLSS

), then:

$A d O p (X)$ is a Heyting algebra and $(\tilde{L_{X}^{o}}, \tilde{L_{X}}, \hat{X})$ is an object of $LatBD$ ,
$X^{u s p}$ is a Heyting up-spectral locally small space (object of $HuSpLSS$ ).

Proof. (1) For

U, V \in A d O p (X)

define

U \to V = i n t (V \cup (X ∖ U))

. Then, by the Heyting assumption on the locally small space X, we have

U \to V \in A d O p (X)

and this set is the largest element

Z \in A d O p (X)

with the property

V \cap Z \subseteq W

. That

(\tilde{L_{X}^{o}}, \tilde{L_{X}}, \hat{X})

is an object of

LatBD

was proved in Step 3 of the proof of Theorem 1 of [23].

(2) Since X is Heyting and the lattices

A d O p (X)

and

I C O (X^{u s p})

are isomorphic Heyting algebras,

X^{u s p}

is Heyting by Remark 6. □

Remark 9.

The functor

u s p

from Remark 7 has a restriction that will be denoted by

u s p : HLSS \to HuSpLSS

Definition 26.

Anup-Esakia space

(X, σ_{X}, \leq_{X})

is an up-Priestley space such that

f o r a l l C \in C l O p (X, σ_{X}) w e h a v e ↓ C \in C l O p (X, σ_{X}) .

AnEsakia mappingbetween up-Esakia spaces is a Priestley mapping

f : X \to Y

such that

f (↑ x) = ↑ f (x)

. We have the category

uEsa

of up-Esakia spaces and Esakia mappings.

Definition 27.

AnEsakia locally small spaceis such a Priestley locally small space

(X, L_{X}^{σ}, \leq_{X})

that satisfies the condition

f o r a l l A \in L C o n (X, L_{X}^{σ}) w e h a v e ↓ A \in L C o n (X, L_{X}^{σ}) .

AnEsakia morphismbetween Esakia locally small spaces is a Priestley morphism

f : X \to Y

that is ap-morphism, i.e., satisfies the condition

f (↑ x) = ↑ f (x)

. We have the category

ELSS

of Esakia (locally small) spaces and Esakia morphisms.

Corollary 4.

The categories

HuSpec, HuSpLSS, uEsa, ELSS

are concretely isomorphic.

Proof.

Follows from Theorem 2. □

Definition 28.

The category

HSpecB

has:

systems $((X, τ_{X}), C O_{s} (X), X_{s})$ where $(X, τ_{X})$ is a Heyting spectral topological space, $C O_{s} (X)$ is a special bornology in $C O (X)$ and $X_{s} = ⋃ C O_{s} (X)$ (called thedecent lump) as objects,
Heyting spectral mappings between spectral spaces satisfying the condition of boundedness and respecting the decent lump as morphisms.

Definition 29.

The category

HAB

has

systems $(L, L_{s})$ with $L = (L, \lor, \land, \to, 0, 1)$ a Heyting algebra, $L_{s}$ a special bornology in L as objects,
homomorphisms of Heyting algebras $h : L \to M$ that aredominating(i.e., $\forall a \in M_{s} \exists b \in L_{s} a \leq h (b)$ ) and respect the decent lump (i.e., $h^{•} (⋃ \tilde{M_{s}}) \subseteq ⋃ \tilde{L_{s}}$ ) as morphisms from $(L, L_{s})$ to $(M, M_{s})$ .

10. Categories of Spaces with Decent Subsets

While the category SpecD was defined in [23] and the category ESSD was defined in [21], we introduce similarly:

Definition 30.

We have the following categories:

the category $PriD$ whose objects are Priestley spaces with distinguished dense sets (calleddecent sets) and whose morphisms are Priestley mappings respecting the decent sets,
the category $EsaD$ whose objects are Esakia spaces with distinguished dense sets (calleddecent sets) and whose morphisms are Esakia mappings respecting the decent sets,
the category $PSSD$ 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.

SpecD, PriD, PSSD

are concretely isomorphic.

Corollary 6.

HSpecD, EsaD, ESSD

are concretely isomorphic.

Two versions of dualities follow.

Corollary 7

(Stone Duality for small spaces). The categories

SpecD

{SS}_{0}

PriD

PSSD

are dually equivalent to

LatD

Corollary 8

(Esakia Duality for small spaces). The categories

HSpecD

HSS

EsaD

ESSD

, are dually equivalent to

HAD

Definition 31.

We have the following categories:

the category $uSpecD$ 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,
the category $uPriD$ whose objects are up-Priestley spaces with distinguished dense sets (calleddecent sets) and whose morphisms are Priestley mappings respecting the decent sets,
the category $uEsaD$ whose objects are up-Esakia spaces with distinguished dense sets (calleddecent sets) and whose morphisms are Esakia mappings respecting the decent sets,
the category $PLSSD$ whose objects are Priestley locally small spaces with decent sets and whose morphisms are Priestley morphisms respecting the decent sets,
the category $ELSSD$ 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

{LSS}_{0}

uSpecD

uPriD

PLSSD

SpecBD

are dually equivalent to

LatBD

Proof.

Follows from Theorem 1 in [23] and Theorem 2. □

Definition 32.

Anobject of $HuSpecD$ is a system $((X, τ_{X}), X_{d})$ where $(X, L_{X})$ is a Heyting up-spectral space and $X_{d} \subseteq X$ is a constructibly dense subset.
Amorphismfrom $((X, τ_{X}), X_{d})$ to $((Y, τ_{Y}), Y_{d})$ in $HuSpecD$ is such a Heyting spectral mapping between Heyting up-spectral spaces $g : (X, τ_{X}) \to (Y, τ_{Y})$ that $g (X_{d}) \subseteq Y_{d}$ .

Definition 33.

Anobject of $HSpecBD$ is a system $((X, τ_{X}), C O_{s} (X), X_{d})$ where $(X, τ_{X})$ is a Heyting spectral space, $C O_{s} (X)$ is a bornology in the Heyting algebra $C O (X)$ and $X_{d} \subseteq ⋃ C O_{s} (X)$ (adecent lumpof X) is constructibly dense and such that $C O {(X)}_{d} = {(C O_{s} {(X)}_{d})}^{o} \subseteq P (X_{d})$ .
Amorphismfrom $((X, τ_{X}), C O_{s} (X), X_{d})$ to $((Y, τ_{Y}), C O_{s} (Y), Y_{d})$ in

$HSpecBD$ is such a Heyting spectral mapping $g : (X, τ_{X}) \to (Y, τ_{Y})$ between Heyting spectral spaces that satisfies thecondition of boundedness $\forall A \in C O_{s} (X) \exists B \in C O_{s} (Y) g (A) \subseteq B,$ and respects the decent lump: $g (X_{d}) \subseteq Y_{d}$ .

Similarly to Lemma 3, we have

Lemma 2.

The categories

HuSpecD

and

HSpecBD

are equivalent.

Proof.

Theorem 5 of [23] gives us equivalence between

uSpec

and

uSpLSS

. Similarly,

uSpecD

is equivalent to

uSpLSSD

, which is clearly equivalent to

{LSS}_{0}

. Now Theorem 1 of [23] gives us equivalence between

{LSS}_{0}

and

SpecBD

. Restricting to Heyting objects and Heyting morphisms, we get the equivalence between

HuSpecD

and

HSpecBD

. □

Definition 34.

Anobject of $HABD$ is a system $(L, L_{s}, D_{L})$ with

$L = (L, \lor, \land, \to, 0, 1)$ a Heyting algebra, $L_{s}$ a bornology in L and $D_{L} \subseteq ⋃ \tilde{L_{s}} \subseteq PF (L)$ a constructibly dense set satisfying $\tilde{L} \cap_{1} D_{L} = {(\tilde{L_{s}} \cap_{1} D_{L})}^{o} \subseteq P (D_{L})$ .
Amorphism of $HABD$ from $(L, L_{s}, D_{L})$ to $(M, M_{s}, D_{M})$ is such a homomorphism of Heyting algebras $h : L \to M$ that isdominating(i.e., $\forall a \in M_{s} \exists b \in L_{s} a \leq h (b)$ ) and $h^{•} (D_{M}) \subseteq D_{L}$ .

Fact 13

([21], Theorem 2). If

h : A \to B

is a Heyting algebra homomorphism, then

h^{•} : PF (B) \to PF (A)

is a Heyting spectral map between Heyting spectral spaces.

Proposition 12.

f : X \to Y

is a Heyting bounded continuous map between Heyting locally small spaces, then

L^{o} f : L_{Y}^{o} \to L_{X}

is a homomorphism of Heyting algebras.

Proof.

We only check up the condition with the intuitionistic implication. For

V, W \in L_{Y}^{o}

, we have

(L^{o} f) (W \to V) = f^{- 1} (e x t (W ∖ V)) = e x t (f^{- 1} (W) ∖ f^{- 1} (V)) = (L^{o} f) (W) \to (L^{o} f) (V)

. □

11. Esakia Duality for Locally Small Spaces

Corollary 10.

Concretely isomorphic categories

HuSpecD, uEsaD, ELSSD

are equivalent to

HLSS

Proof.

Follows from Theorem 2. □

Theorem 4

(Esakia Duality for locally small spaces). The categories

HLSS

HuSpecD

uEsaD

ELSSD

and

HSpecBD

are dually equivalent to

HABD

Proof.

By the corollary above, we need only to prove that

HLSS

and

HSpecBD

are dually equivalent to

HABD

. Since our proof is about the restrictions of functors from the proof of Theorem 1 in [23], we concentrate on objects and morphisms being Heyting. We shall use the notations from that proof.

Step 1: The restriced functor

\bar{R} : HSpecBD \to HLSS

Assume

((X, τ_{X}), C O_{s} (X), X_{d})

is an object of

HSpecBD

. Then

C O_{s} (X)

is a basis of the induced topology on

X_{s}

and the original topology of

(X_{d}, C O_{s} {(X)}_{d})

is the topology induced from

(X, τ_{X})

. We are to check the Heyting closure condition for the locally small space

(X_{d}, C O_{s} {(X)}_{d})

. Assume

B \in L C o n (X_{d}, C O_{s} {(X)}_{d})

. For

V \in C O_{s} (X)

, we have

B \cap V \in s C o n (X_{d}, C O_{s} {(X)}_{d}) \subseteq C o n (X_{d}, C O {(X)}_{d})

, so

i n t (B \cap V) \in C O {(X)}_{d} \cap_{1} C O_{s} {(X)}_{d} = C O_{s} {(X)}_{d}

. Since

C O_{s} {(X)}_{d}

is a basis of the induced topology of

X_{d}

stable under finite intersections,

i n t (B) = ⋃_{V \in C O_{s} (X)} i n t (B \cap V) \in A d O p (X_{d}, C O_{s} {(X)}_{d})

Assume

f : X \to Y

is a morphism of

HSpecBD

. Since

\forall C \in C o n (Y, C O (Y)) f^{- 1} (\bar{C}) = \bar{f^{- 1} (C)},

we also have

\forall D \in C o n (Y_{d}, C O {(Y)}_{d}) f_{d}^{- 1} ({\bar{D}}^{d}) = {\bar{f_{d}^{- 1} (D)}}^{d} .

Indeed, for any

D = C \cap Y_{d} \in C o n (Y_{d}, C O {(Y)}_{d})

, we have

{\bar{D}}^{d} = \bar{C \cap Y_{d}} \cap Y_{d} = \bar{C} \cap Y_{d}

. Applying this to the Heyting mapping condition, we have

f_{d}^{- 1} ({\bar{D}}^{d}) = f_{d}^{- 1} (\bar{C} \cap Y_{d}) = f^{- 1} (\bar{C}) \cap X_{d} = \bar{f^{- 1} (C)} \cap X_{d} = \bar{f^{- 1} (C) \cap X_{d}} \cap X_{d} = \bar{f^{- 1} (D) \cap X_{d}} \cap X_{d} = {\bar{f_{d}^{- 1} (D)}}^{d}

Take now

K \in L C o n (Y_{d}, C O_{s} {(Y)}_{d})

. We can express its topological closure in

Y_{d}

(omitting

^{d}

) as

\bar{K} = ⋃ {\bar{K \cap V} : V \in C O_{s} {(Y)}_{d}}

. That is why

f_{d}^{- 1} (\bar{K}) = ⋃ {f_{d}^{- 1} (\bar{K \cap V}) : V \in C O_{s} {(Y)}_{d}}

. Each

K \cap V

belongs to

C o n (Y_{d}, C O_{s} {(Y)}_{d}) \subseteq C o n (Y_{d}, C O {(Y)}_{d})

. Hence

f_{d}^{- 1} (\bar{K \cap V}) = \bar{f_{d}^{- 1} (K \cap V)}

. We get

⋃ {f_{d}^{- 1} (\bar{K \cap V}) : V \in C O_{s} {(Y)}_{d}} = ⋃ {\bar{f_{d}^{- 1} (K \cap V)} : V \in C O_{s} {(Y)}_{d}} = ⋃ {\bar{f_{d}^{- 1} (K) \cap W} : W \in C O_{s} {(X)}_{d}} = \bar{f_{d}^{- 1} (K)}

and

f_{d}

is a Heyting bounded continuous mapping.

Step 2: The restricted functor

\bar{S} : {HABD}^{o p} \to HSpecBD

By the classical Esakia Duality (see Remark 10 in [21]), the topological space

PF (A)

is Heyting spectral for a Heyting algebra A. Moreover,

h^{•} : PF (M) \to PF (L)

a Heyting spectral map for a morphism

h : L \to M

HABD

Step 3: The restricted functor

\bar{A} : HLSS \to {HABD}^{o p}

For a Heyting locally small space

(X, L_{X})

, the intuitionistic implication in the bounded lattice

L_{X}^{o}

can be introduced by the formula

U \to V = e x t (U ∖ V)

, making it a Heyting algebra by Proposition 11. For a morphism f in

HLSS

, the mapping

L^{o} f

is a homomorphism of Heyting algebras by Proposition 12.

Step 4: The functor

\bar{R} \bar{S} \bar{A}

is naturally isomorphic to

I d_{HLSS}

The mapping

η_{X} : (\hat{X}, {\tilde{L_{X}}}^{d}) \to (X, L_{X})

is a strict homeomorphism by Proposition 2 of [23] and an isomorphism of

HLSS

by Fact 12.

Step 5: The functor

\bar{S} \bar{A} \bar{R}

is naturally isomorphic to

I d_{HSpecBD}

Each

θ_{X}

is an isomorphism in

SpecBD

as well as an isomorphism of

HSpec

by Fact 11 in [21], so an isomorphism of

HSpecBD

Step 6: The functor

\bar{A} \bar{R} \bar{S}

is naturally isomorphic to

I d_{{HABD}^{o p}}

Each

κ_{L} : L \to {\tilde{L}}^{d}

is an order isomorphism hence, by ([32], 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. □

Lemma 3.

The categories

HuSpec

and

HSpecB

are equivalent and dually equivalent to

HAB

Proof.

That

HuSpec

and

HSpecB

are equivalent follows from Theorem 5 in [23] by restricting to Heyting objects and Heyting morphisms.

That

HSpecB

is dually equivalent to

HAB

follows from Theorem 4 by restricting to the case

X_{d} = X_{s}

. □

Corollary 12

(Esakia Duality for up-spectral spaces). The categories

HuSpec

HuSpLSS

uEsa

ELSS

HSpecB

are dually equivalent to

HAB

Proof.

Follows from Lemma 3 and Corollary 4. □

Corollary 13

(Esakia Duality for spectral spaces). The categories

HSpec

HSpSS

Esa

ESS

are dually equivalent to

HA

12. 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.

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

In each row of the second table, the categories in 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.

${SS}_{0}$	$SpecD$	$PriD$	$PSSD$		$LatD$
$HSS$	$HSpecD$	$EsaD$	$ESSD$		$HAD$
${LSS}_{0}$	$uSpecD$	$uPriD$	$PLSSD$	$SpecBD$	$LatBD$
$HLSS$	$HuSpecD$	$uEsaD$	$ELSSD$	$HSpecBD$	$HABD$

This makes another step, after [23] and [21], 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 2 using the theory of closure algebras. Moreover, we have shown (Theorem 3) how Dimov’s version (from [45]) 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 [21] to the locally small context.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The author declares no conflict of interest.

References

Esakia, L. Topological Kripke models. Dokl. Akad. Nauk SSSR 1974, 214, 298–301. [Google Scholar]
Esakia, L. Heyting Algebras. Duality Theory; Springer Nature: Cham, Switzerland, 2019. [Google Scholar]
Bezhanishvili, G. Leo Esakia on Duality in Modal and Intuitionistic Logics; Springer: Dordrecht, The Netherland, 2014. [Google Scholar]
Bezhanishvili, N. Lattices of intermediate and cylindric modal logics. PhD thesis, Universiteit van Amsterdam, Amsterdam, 2006.
Bezhanishvili, G.; Jansana, R. Esakia Style Duality for Implicative Semilattices. Appl. Categor. Struct. 2013, 21, 181–208. [Google Scholar] [CrossRef]
Bezhanishvili, G.; Bezhanishvili, N.; Moraschini, T.; Stronkowski, M. Profiniteness and representability of spectra of Heyting algebras. Adv. Math. 2021, 391, 107959. [Google Scholar] [CrossRef]
Celani, S.A.; Jansana, R. Esakia Duality and Its Extensions. In Leo Esakia on Duality in Modal and Intuitionistic Logics; Springer: Dordrecht, The Netherland, 2014; pp. 63–98. [Google Scholar]
Stone, M.H. Topological representations of distributive lattices and Brouwerian logics. Časopis Pěst. Mat. Fiz. 1938, 67, 1–25. [Google Scholar] [CrossRef]
Priestley, H. Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 1970, 2, 186–190. [Google Scholar] [CrossRef]
Bice, T.; Starling, C. Locally Compact Stone Duality. J. Log. Anal. 2018, 10, 1–36. [Google Scholar] [CrossRef]
Bice, T.; Starling, C. General non-commutative locally compact locally Hausdorff Stone duality. Adv. Math. 2019, 341, 40–91. [Google Scholar] [CrossRef]
Bezhanishvili, G.; Morandi, P.J.; Olberding, B. A generalization of Gelfand-Naimark-Stone duality to completely regular spaces. Topol. Appl. 2020, 274, 107123. [Google Scholar] [CrossRef]
Bezhanishvili, G.; Morandi, P.J.; Olberding, B. Gelfand-Naimark-Stone duality for normal spaces and insertion theorems. Topol. Appl. 2020, 280, 107256. [Google Scholar] [CrossRef]
Dimov, G.; Ivanova-Dimova, E. Two Extensions of the Stone Duality to the Category of Zero-Dimensional Hausdorff Spaces. Filomat 2021, 35, 1851–1878. [Google Scholar] [CrossRef]
Cannon, S.; Döring, A. A. A Generalisation of Stone Duality to Orthomodular Lattices. In Reality and Measurement in Algebraic Quantum Theory; Springer Nature: Singapore, 2018; pp. 3–65. [Google Scholar]
Hartonas, C. Duality Results for (Co)Residuated Lattices. Log. Universalis 2019, 13, 77–99. [Google Scholar] [CrossRef]
Kudryavtseva, G. A refinement of Stone duality to skew Boolean algebras. Algebra Universalis 2012, 67, 397–416. [Google Scholar] [CrossRef]
Goldblatt, R. Canonical extensions and ultraproducts of polarities. Algebra Universalis 2018, 79, 80. [Google Scholar] [CrossRef]
Hartonas, C. Canonical Extensions and Kripke-Galois Semantics for Non-distributive Logics. Log. Universalis 2018, 12, 397–422. [Google Scholar] [CrossRef]
Kwaśniewski, B.K.; Meyer, R. Stone duality and quasi-orbit spaces for generalised C^*-inclusions. Proc. Lond. Math. Soc. 2020, 121, 788–827. [Google Scholar] [CrossRef]
Piękosz, A. Esakia duality for Heyting small spaces, Symmetry, 14(12) (2022), 2567.
Dickmann, M.; Schwartz, N.; Tressl, M. Spectral Spaces; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
Piękosz, A. Stone duality for Kolmogorov locally small spaces, Symmetry, 13(10) (2021), 1791.
Grothendieck, A. Esquisse d’un Programme [Sketch of a Program]. In Geometric Galois Actions 1; Cambridge University Press: Cambridge, UK, 1997; pp. 5–48, (In French, with an English translation on pp. 243–283.). [Google Scholar]
A’Campo, N.; Ji, L.; Papadopoulos, A. On Grothendieck’s tame topology. In Handbook of Teichmüller Theory, Volume VI; EMS Press: Berlin, Germany, 2016; Volume 27, pp. 521–533. [Google Scholar]
Cruz Morales, J.A. The Notion of Space in Grothendieck. In Handbook of the History and Philosophy of Mathematical Practice; Springer Nature: Cham, Switzerland, 2020; pp. 1–21. [Google Scholar]
Hochster, M. Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 1969, 142, 43–60. [Google Scholar] [CrossRef]
Belaid, K.; Echi, O. On a conjecture about spectral sets, Top. Appl. 2004, 139, 1–15. [Google Scholar]
Echi, O.; Abdallahi, M.O. On the spectralification of a hemispectral space. J. Algebra Appl. 2011, 10, 687–699. [Google Scholar] [CrossRef]
Acosta, L.; Rubio Perilla, I.M. Nearly spectral spaces. Bol. Soc. Mat. Mex. 2019, 25, 687–700. [Google Scholar] [CrossRef]
Acosta, L. Temas de teoría de retículos; Universidad Nacional de Colombia: Bogota, Colombia, 2016. [Google Scholar]
Balbes, R.; Dwinger, P. Distributive Lattices; University of Missouri Press: Columbia, MO, USA, 1974. [Google Scholar]
Bochnak, J.; Coste, M.; Roy, M.-F. Real Algebraic Geometry; Springer: Berlin/Heidelberg, Germany, 1998. [Google Scholar]
Andradas, C.; Bröcker, L.; Ruiz, J. Constructible Sets in Real Geometry; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
Delfs, H.; Knebusch, M. Locally Semialgebraic Spaces; Springer: Berlin/Heidelberg, Germany, 1985. [Google Scholar]
van den Dries, L. Tame Topology and O-minimal Structures; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
Piękosz, A. O-minimal homotopy and generalized (co)homology. Rocky Mt. J. Math. 2013, 43, 573–617. [Google Scholar] [CrossRef]
Piękosz, A. On generalized topological spaces I. Ann. Polon. Math. 2013, 107, 217–241. [Google Scholar] [CrossRef]
Piękosz, A. On generalized topological spaces II. Ann. Polon. Math. 2013, 108, 185–214. [Google Scholar] [CrossRef]
Pillay, A. On groups and fields definable in o-minimal structures. J. Pure Appl. Algebra 1988, 53, 239–255. [Google Scholar] [CrossRef]
Mac Lane, S. Categories for the Working Mathematician; Springer: New York, NY, USA, 1998. [Google Scholar]
Özel, C.; Piękosz, A.; Al Shumrani, M.; Wajch, E. Partially paratopological groups. Topol. Appl. 2017, 228, 68–78. [Google Scholar] [CrossRef]
Piękosz, A. Locally small spaces with an application. Acta Math. Hung. 2020, 160, 197–216. [Google Scholar] [CrossRef]
Adámek, J.; Herrlich, H.; Strecker, G.E. Abstract and Concrete Categories. The Joy of Cats; Wiley: New York, NY, USA, 1990. [Google Scholar]
Dimov, G. Some generalizations of the Stone Duality Theorem. Publ. Math. Debrecen 2012, 80, 255–293. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

Heyting Locally Small Spaces and Esakia Duality

Abstract

1. Introduction

2. Generalities about Locally Small Spaces

3. Specialisation

4. Up-spectral Locally Small Spaces

5. Stone Duality for Up-spectral Spaces

6. Boolean Locally Small Spaces

7. The Standard Up-spectralification

8. Heyting Locally Small Spaces

9. Heyting Up-spectral Spaces

10. Categories of Spaces with Decent Subsets

11. Esakia Duality for Locally Small Spaces

12. Conclusions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

Important Links

Subscribe