Tame Topology ^†

Abstract

Alexander Grothendieck suggested creating a new branch of topology, called “topologie modérée”. In a paper by N. A’Campo, L. Ji, and A. Papadopoulos the authors conclude that no such tame topology has been developed at a purely topological level. We see our theory of sets with distinguished families of subsets, which we call smopologies, as realising Grothendieck’s idea and the demands of the mentioned paper. Dropping the requirement of stability under infinite unions makes it possible to obtain several equivalences of categories of spaces with categories of lattices. We show several variations in Stone duality and Esakia duality for categories of small or locally small spaces and some subclasses of strictly continuous (or bouned continuous) mappings. Such equivalences are better than the spectral reflector functor for usual topological spaces. Some spectralifications of Kolmogorov locally small spaces can be obtained by Stone duality. Small spaces or locally small spaces seem to be generalised topological spaces. However, looking at them as topological spaces with an additional structure is better. The language of smopologies and bounded continuous mappings simplifies the language of certain Grothendieck sites and permits us to glue together infinite families of definable sets in structures with topologies, which was important when developing the o-minimal homotopy theory.

1. Introduction

1.1. Grothendieck’s Programme and Basics of Tame Topology

Alexander Grothendieck suggested, in his famous scientific programme [1], creating a new kind of topology, called “topologie modérée” (“tame topology” in English), which would eliminate pathological phenomena (for example, space-filling curves). His mathematical ideas led to philosophical and physical questions about the nature and structure of space [2]. Grothendieck’s programme was realised in many special situations for many decades, but N. A’Campo, L. Ji and A. Papadopoulos [3] state that no clear definition of tame topology has been given. One could say that, as a part of model theory or real algebraic geometry, we have o-minimality (the main reference is [4]), which is widely recognised as a realisation of Grothendieck’s programme. However, in o-minimality, the definable open sets, not arbitrary open sets, play the main role. This means that, from the tame point of view, the usual notion of a topology is secondary to another concept basic to some algebra-friendly topology. We propose the theory of tame spaces (such as small spaces and locally small spaces) as a realisation of Grothendieck’s postulate on a purely topological level. Seemingly a kind of generalised topology, tame topology is the usual topology with some additional structure.

Definition 1 

([5,6,7]). A locally small space is a pair $\mathcal{X}=(X,{\mathcal{L}}_X)$, 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) 

$\forall x\in X\exists A_x\in {\mathcal{L}}_{X}x\in A_x$ (i.e., $\bigcup {\mathcal{L}}_X=X$).

Elements of ${\mathcal{L}}_X$ are called small open subsets (or smops) of X, while ${\mathcal{L}}_X$ is called a smopology. A small space is such a locally small space $(X,{\mathcal{L}}_X)$ that $X\in {\mathcal{L}}_X$. Then, the smopology is called unitary. The complements of smops are called co-smops, and the Boolean combinations of smops are the constructible sets. The family of all co-smops (constructible sets, resp.) of a small space $(X,{\mathcal{L}}_X)$ is denoted by ${\mathcal{L}}_X^{\prime }$ ($Con\left(\mathcal{X}\right)$, resp.).

The idea is to drop some of the conditions for a topology. The finitary character of small spaces distinguishes them among locally small spaces. A smopology is a basis of a usual topology. Small spaces were used in [5,6,7,8,9,10], while locally small spaces were used in [5,6,9,10], sometimes with another definition.

Notation. For families of subsets $\mathcal{A},\mathcal{B}\subseteq \mathcal{P}\left(X\right)$, we use

$${\mathcal{A}}^o=\{Y\subseteq X:Y\cap A\in \mathcal{A} \mathrm{for} \mathrm{any} A\in \mathcal{A}\},$$

$$\mathcal{A}{\cap }_1\mathcal{B}=\{A\cap B:A\in \mathcal{A},B\in \mathcal{B}\}.$$

1.2. Genealogy and Implicit Use of Tame Spaces

Small spaces are often unnamed in the literature ([11], Definition 7.1.14 or [12], p. 12). Both small and locally small spaces were implicitly used in o-minimal homotopy theory [13,14] under the name of generalised topological spaces (in the sense of Delfs and Knebusch), which, in turn, may be seen as sets with G-topologies (compare [15]) or a particular form of Grothendieck sites (see [8] and ([14], p. 2)). Definable or locally definable spaces, widely used in real algebraic geometry or model theory ([4,8,14,16], and implicitly in [17]), are expansions of small or locally small spaces, respectively.

Definition 1 above provides a simple language for locally small spaces, not using Grothendieck sites, which is analogical to Lugojan’s [18] or Császár’s [19] language of generalised topology, where a family of subsets is required to fulfill some, but not all, of the conditions traditionally required for a topology.

1.3. Categories of Tame Spaces

When using locally small spaces and small spaces, we need to distinguish important mappings between them. This is why we use the following notions.

Definition 2 

([5,6,7]). 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 $f^{-1}\left({\mathcal{L}}_Y\right)$: 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 $f^{-1}\left({\mathcal{L}}_Y\right)\subseteq {\mathcal{L}}_X$,

(d) 

a strict homeomorphism if f is a bijection and $f^{-1}\left({\mathcal{L}}_Y\right)={\mathcal{L}}_X$.

It is suitable to use the category theory language. The spaces satisfying the Kolmogorov separation axiom ($T_0$) are our focus.

Definition 3 

([5,6]). We have the following categories:

(a) 

the category $\mathbf{LSS}$ of locally small spaces and their bounded continuous mappings,

(b) 

the full subcategory ${\mathbf{LSS}}_0$ of $T_0$ locally small spaces,

(c) 

the full subcategory ${\mathbf{SS}}_0$ of $T_0$ small spaces.

(d) 

the subcategory ${\mathbf{LSS}}_0^s$ in ${\mathbf{LSS}}_0$ of (bounded) strongly continuous mappings.

Definition 4 

([7]). The topology $\tau \left({\mathcal{L}}_X\right)$ generated by ${\mathcal{L}}_X$ is called the original topology and the topology $\tau \left(Con\right(\mathcal{X}\left)\right)$ generated by $Con\left(\mathcal{X}\right)$ is called the constructible topology of $(X,{\mathcal{L}}_X)$.

A small space $(X,{\mathcal{L}}_X)$ is called Heyting if it is $T_0$ and the closure in the original topology of any constructible set is a co-smop (i.e., $\overline{A}\in {\mathcal{L}}_X^{\prime }$ for any $A\in Con\left(\mathcal{X}\right)$).

A map between Heyting small spaces f: $\mathcal{X}\to \mathcal{Y}$ is Heyting continuous if it is continuous and satisfies any of the following equivalent conditions:

1. 

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

2. 

$f^{-1}\left(int\left(C\right)\right)=int\left(f^{-1}\left(C\right)\right)$ for $C\in Con\left(\mathcal{Y}\right)$.

We have the category $\mathbf{HSS}$ of Heyting small spaces and Heyting continuous maps.

The name “Heyting small spaces” follows the conventions of [20].

1.4. Stone and Esakia Dualities

Although the language of category theory was not developed in the 1930s, Stone Duality is named after the papers of M. H. Stone ([21,22] for generalised Boolean algebras, Ref. [23] for distributive lattices). There are plenty of available versions, as in [20,24,25], including versions developed by H. Priestley [26,27], named Priestley Duality. Esakia Duality, which emerged from the considerations on modal logics [28,29], can be seen as a restriction of Priestley Duality.

Algebraic and analytic geometry, as well as model theory, use Stone Duality. The spectral topology (also called the Harrison topology) is used in the case of the real spectrum (see [11,12,30]) and the Zariski spectrum (see [31,32], Chapter II), while the constructible topology (also called the patch topology) is used in the case of the space of types [33,34], allowing (in the case of the o-minimal spectrum) for retopologisation to the spectral topology [35].

Many extensions of Stone duality have been published in recent years. For example: the locally compact Hausdorff case [36], removing the zero-dimensionality together with the commutativity assumptions [37], a generalisation of Gelfand–Naimark–Stone Duality to completely regular spaces [38] with an application to the characterisation of normal, Lindelöf and locally compact Hausdorff spaces [39], completely dropping the compactness assumption [40]. From the algebraic perspective, we have extensions to: orthomodular lattices [41], some non-distributive (implicative, residuated, or co-residuated) lattices [42], and left-handed skew Boolean algebras [43]. Esakia Duality can be extended to implicative semilattices [44]. Many applications of Stone duality exist in various contexts [45,46,47].

2. Results

Definition 5 

([6]). A bornology in a bounded lattice $(L,\vee ,\wedge ,0,1)$ is an ideal $B\subseteq L$ such that $\bigvee B=1$. The set of all prime filters in L is denoted by $\mathcal{PF}\left(L\right)$. For each $a\in L$, we have $\tilde{a}=\{F\in \mathcal{PF}\left(L\right)\mid a\in F\}$. We set $\tilde{A}=\{\tilde{a}\mid a\in A\}\subseteq \mathcal{P}\left(\mathcal{PF}\left(L\right)\right)$ for $A\subseteq L$.

2.1. Categories of Distributive Lattices

Definition 6 

([6,7]). An object of $\mathbf{LatBD}$ is a system $(L,L_s,{\mathbf{D}}_L)$ with $L=(L,\vee ,\wedge ,0,1)$ a bounded distributive lattice, $L_s$ a bornology in L and ${\mathbf{D}}_L\subseteq \mathcal{PF}\left(L\right)$ (a decent lump) satisfying the conditions:

(1)

${\mathbf{D}}_L\subseteq \bigcup \widetilde{L_s}$,

(2)

$\forall a,b\in La\ne b\Rightarrow {\tilde{a}}^d\ne {\tilde{b}}^d,$ where ${\tilde{a}}^d=\{F\in {\mathbf{D}}_L\mid a\in F\}$,

(3)

$\tilde{L}{\cap }_1{\mathbf{D}}_L={\left(\widetilde{L_s}{\cap }_1{\mathbf{D}}_L\right)}^o\subseteq \mathcal{P}\left({\mathbf{D}}_L\right)$.

A morphism of $\mathbf{LatBD}$ from $(L,L_s,{\mathbf{D}}_L)$ to $(M,M_s,{\mathbf{D}}_M)$ is such a homomorphism of bounded lattices $h:L\to M$ that:

(a) 

satisfies the condition of domination $\forall a\in M_s\exists b\in L_{s}a\vee h\left(b\right)=h\left(b\right),$

(b) 

respects the decent lump: $\{h^{-1}\left(G\right):G\in {\mathbf{D}}_M\}\subseteq {\mathbf{D}}_L$.

The category $\mathbf{LatD}$ may be identified with the full subcategory in $\mathbf{LatBD}$ generated by objects satisfying $L=L_s$.

Definition 7 

([6,7]). The category $\mathbf{ZLatD}$ has

(1)

pairs $(L,{\mathbf{D}}_L)$ where L is a distributive lattice with zero and ${\mathbf{D}}_L$ is a distinguished decent set of prime filters in $\mathcal{PF}\left(L\right)$ as objects,

(2)

homomorphisms of lattices with zeros respecting the decent sets of prime filters and satisfying the condition of domination as morphisms.

The category $\mathbf{ZLat}$ may be identified with the full subcategory in $\mathbf{ZLatD}$ of objects satisfying ${\mathbf{D}}_L=\mathcal{PF}\left(L\right)$. Moreover, we have the category $\mathbf{HAD}$ of Heyting algebras with decent (i.e., constructibly dense) sets and homomorphisms of Heyting algebras respecting the decent sets.

2.2. Categories of Spectral-Like Spaces

Definition 8 

([6,7]). An object of $\mathbf{SpecBD}$ is a system $((X,{\tau }_X),C{O}_s\left(X\right),X_d)$ where $(X,{\tau }_X)$ is a spectral space, $C{O}_s\left(X\right)$ is a bornology in the bounded lattice $CO\left(X\right)$ and $X_d$ (a decent lump) satisfies the following conditions:

(1)

$X_d\subseteq \bigcup C{O}_s\left(X\right)$,

(2)

$R_d:CO\left(X\right)\ni A\mapsto A\cap X_d\in CO\left(X\right){\cap }_1{X}_d$ is an isomorphism of lattices,

(3)

$CO\left(X\right){\cap }_1{X}_d={\left(C{O}_s\left(X\right){\cap }_1{X}_d\right)}^o\subseteq \mathcal{P}\left(X_d\right)$.

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 $\mathbf{SpecBD}$ is such a spectral mapping between spectral spaces $g:(X,{\tau }_X)\to (Y,{\tau }_Y)$ that:

(a) 

satisfies the condition of boundedness $\forall A\in C{O}_s\left(X\right)\exists B\in C{O}_s\left(Y\right)g\left(A\right)\subseteq B;$

(b) 

respects the decent lump: $g\left(X_d\right)\subseteq Y_d$.

We have the full subcategory $\mathbf{SpecB}$ of objects satisfying $X_d=\bigcup C{O}_s\left(X\right)$. The category $\mathbf{SpecD}$ may be identified with the subcategory in $\mathbf{SpecBD}$ of objects satisfying $C{O}_s\left(X\right)=CO\left(X\right)$. We also have the category $\mathbf{HSpecD}$ of Heyting spectral spaces (see [20]) with decent subsets and spectral mappings respecting the decent subsets.

Definition 9 

([6]). We have the category $\mathbf{uSpec}$ of up-spectral spaces and spectral mappings. The category ${\mathbf{uSpecD}}^s$ has

(1)

pairs $((X,{\tau }_X),X_d)$, where $(X,{\tau }_X)$ is an up-spectral space and $X_d$ is a distinguished decent subset of X as objects;

(2)

bounded strongly continuous mappings respecting the decent subsets as morphisms.

The category ${\mathbf{uSpec}}^s$ may be identified with the full subcategory in ${\mathbf{uSpecD}}^s$ generated by objects satisfying $X_d=X$.

2.3. Main Equivalences

Theorem 1 

([6,7]). We have the following equivalences:

1. 

The categories ${\mathbf{LSS}}_0$, ${\mathbf{LatBD}}^{op}$ and $\mathbf{SpecBD}$ are equivalent.

2. 

The categories ${\mathbf{SS}}_0$, ${\mathbf{LatD}}^{op}$ and $\mathbf{SpecD}$ are equivalent.

3. 

The categories $\mathbf{uSpec}$ and $\mathbf{SpecB}$ are equivalent.

4. 

The categories ${\mathbf{uSpec}}^s$ and $\mathbf{ZLat}$ are dually equivalent.

5. 

The categories ${\mathbf{LSS}}_0^s$, ${\mathbf{ZLatD}}^{op}$ and ${\mathbf{uSpecD}}^s$ are equivalent.

6. 

The categories $\mathbf{HSS}$, $\mathbf{HSpecD}$ and ${\left(\mathbf{HAD}\right)}^{op}$ are equivalent.

A version of Hofmann–Lawson duality for locally small spaces also exists [48].

2.4. The Spectralification Method and Consequences

When analysing small or locally small spaces, the spectral spaces [20,31] are especially helpful. This is achieved by the standard spectralifications, formally introduced in Section 5 of [7]. (In particular, the spectralifications of a Kolmogorov topological space may be constructed by choosing lattice bases of the topology). Theorem 1 can be seen as an extension of the method of taking the real spectrum or the o-minimal spectrum.

Corollaries on spaces: A Kolmogorov small space is essentially a constructibly dense subset of a spectral space, while a Kolmogorov locally small space is essentially a constructibly dense subset of an up-spectral space [6].

Corollaries on mappings: Bounded continuous mappings between $T_0$ locally small spaces are restrictions of spectral mappings between up-spectal (or just spectral) spaces to some constructibly dense subsets [6]. Open continuous definable mappings between definable spaces over o-minimal structures are Heyting continuous as mappings between Heyting small spaces [7].

3. Conclusions

Since families of sets that are closed under finite unions are common in mathematics, a new branch of general topology (in the spirit of Engelking [49]), considering the above or new kinds of tame spaces and relevant mappings between them, is possible. We initiated the development of such a branch by showing the above equivalences, while the usual topology provides only spectral reflections (see [20] (Chapter 11)). The use of smopologies should be helpful in such areas of mathematics as the generalisations of o-minimality and other parts of model theory (especially where definable topologies are used), algebraic geometry, and analytic geometry.