Prototypes, Poles, and Tessellations Towards a Topological Theory of Conceptual Spaces Abstract. The aim of this paper is to present a topological method for constructing discretizations (tessellations) of conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. Alexandroff spaces, as they are called today, have many interesting properties that distinguish them from other topological spaces. In particular, they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, a special type of Alexandroff spaces was used by Ian Rumfitt to elucidate the logic of vague concepts in a new way. According to his approach, conceptual spaces such as the color spectrum give rise to classical systems of concepts that have the structure of atomic Boolean algebras. More precisely, concepts are represented as regular open regions of an underlying conceptual space endowed with a topological structure. Something is subsumed under a concept iff it is represented by an element of the conceptual space that is maximally close to the prototypical element p that defines that concept. This topological representation of concepts comes along with a representation of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical operations that characterize regular open interpretations of classical Boolean propositional logic. In the last 20 years, conceptual spaces have become a popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using prototypes and metrics of similarity spaces, one obtains geometrical discretizations of conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces. Thereby, Rumfitt's and Gärdenfors's constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. This class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher- 2 order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The crucial role of order theory for Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical stimuli in favor of a more fine-grained gradual distinction between more-orless prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called "thickness problem" (Douven et al.) and problems of selecting a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines. Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Digital Topology, Voronoi Tessellations. 1. Introduction. The aim of this paper is to present a topological method for constructing discretizations (tessellations) of conceptual spaces. This method works for a class of spaces that the Russian mathematician Pavel Alexandroff defined some 80 years ago (cf. Alexandroff (1937)). Alexandroff spaces, as they are called today, exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, Ian Rumfitt used a special class of Alexandroff spaces to elucidate the logic of vague concepts in a new way (cf. Rumfitt (2015, chapter 8)). According to his approach, the color spectrum and other similar conceptual spaces give rise to classical systems of concepts that have the structure of complete atomic Boolean algebras1. Rumfitt was not the first to study conceptual systems defined via prototypes of conceptual spaces. For some 20 years or so Peter Gärdenfors and his collaborators have shown that conceptual spaces serve as a useful modeling tool in the fields of cognitive 1 Complete atomic Boolean algebras are isomorphic to 2L, where 2L is the power set of a set L. 3 psychology, artificial intelligence, linguistics, and philosophy. 2 The core idea of the conceptual space approach is that concepts can be represented geometrically as regions of (metrically structured) similarity spaces. Using prototypes and the metrical structure of similarity spaces, Gärdenfors constructed geometrical discretizations of conceptual spaces by so-called Voronoi tessellations. The topological account of conceptual spaces to be presented in this paper has several advantages over Gärdenfors' geometrical account. For instance, the vexing question over choosing the right metric of a conceptual space (from infinitely many candidates) can be avoided. Moreover, the so-called "thickness problem" of Douven et alii vanishes. Gärdenfors's meanwhile "classical" approach of conceptual spaces and the topological approach are not unrelated to each other. As will be shown, Gärdenfors's geometrical construction of conceptual spaces gives rise to the construction of topologically defined Alexandroff spaces. More precisely, Voronoi tessellations are extensionally equivalent to topologically defined discretizations that rely only on the topological features of Alexandroff spaces. Furthermore, Rumfitt's as well as Gärdenfors's constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. For these spaces, the corresponding Boolean algebras of regular open regions yield natural atomic tessellations. This suggests that the class of Alexandroff spaces provides a convenient framework for conceptual spaces in general. Thus, an important task of cognitive science is to understand how they can be endowed with a spatial structure that can serve as a basis for the elaboration of a more or less detailed classification of stimuli or experiences.3 2 For some interesting recent work on the role of prototypes in the theory of conceptual spaces see Douven and Gärdenfors (2019). Recent applications of the theory of conceptual spaces in linguistics, cognitive science and philosophy may be found in Zenker and Gärdenfors (2015). 3 With some good will, Carnap's "attribute spaces" may be considered as a forerunner of conceptual spaces in Gärdenfors's sense (cf. Carnap (1971)). In contrast to attribute spaces, regions of conceptual spaces that correspond to concepts are non-homogeneous in the sense that some (generating) points are more prototypical than others. While Rumfitt and Gärdenfors assume a strict dichotomy between prototypical and non-prototypical elements, this paper shows how to define a 4 For defining a spatial structure on a set of experiences, Gärdenfors proposed to employ so-called Voronoi tessellations based on a Euclidean structure and a finite set of prototypes of the underlying conceptual space: A Voronoi tessellation based on a set of prototypes is a simple way of classifying a continuous space of stimuli. The partitioning results in a discretization of the space. The prime cognitive effect is that the discretization speeds up learning. ... [A] Voronoi tessellation is a cognitively economical way of representing information about concepts. Furthermore, having a space partitioned into a finite number of classes means that it is possible to give names to the classes. (Gärdenfors (2000, 89)) As shown in the following, the topological essence of Gärdenfors's and Rumfitt's discretizations of continuous conceptual spaces is based on a structural correspondence between the order and topological structures discovered by Alexandroff in the 1930s. A geometrically defined Voronoi tessellation uniquely determines a topological tessellation that is extensionally equivalent to a regular open tessellation constructed by Alexandroff's method. The constructions of Rumfitt and Gärdenfors boil down to different, very special cases of Alexandroff's method of constructing topological spaces from partial orders. Thus, Alexandroff spaces may be considered a natural topological habitat of conceptual spaces. They provide a natural framework for conceptual spaces that deal with empirically meaningful concepts. More precisely, this claim can be explicated as follows: (1) Empirically meaningful concepts must be stable in the sense that if such a concept applies to a situation x, it also applies to minor variations x' of x. This stability is accompanied by a certain degree of conceptual vagueness. gradual distinction between "more" or "less prototypical" elements of a conceptual space X. This is done by using the so-called "specialization order" characteristic for Alexandroff spaces. More precisely, in the framework of Alexandroff spaces, prototypical elements are characterized as maximal elements of the specialization (quasi-)order, while all other elements have a more or less high degree of proto–typicality. For a more detailed comparison of the similarities and differences of Carnap's and Gärdenfors's approaches, see Sznajder (2016, section 6). 5 Stable concepts do not single out empirical objects with absolute precision. This should be considered a virtue rather than a vice. Otherwise, concepts would no longer be empirically applicable. (2) Arbitrary conjunctions of stable concepts should be stable. This requirement expresses a reasonable conceptual modesty. Otherwise, we could eliminate the inherent vagueness of empirical concepts by purely logical means by forming (infinite) conjunctions of more and more concepts that eventually result in an absolutely precise conceptualization of reality. Topologically, (1) and (2) can be satisfied by requiring that a conceptual space S has the structure of an Alexandroff space (S, OS) such that concepts are characterized as elements of the Boolean algebra O*S of regular open subsets of S. In this paper, we rely on a topological account of concepts, i.e., concepts are characterized as topologically wellformed regions in contrast to Gärdenfors's geometrical approach to conceptual spaces, which represents natural concepts as convex regions with convexity defined geometrically with the aid of the Euclidean space of an underlying conceptual space. In comparison to the geometrical account, the topological one to be presented in this paper is more austere insofar as different geometrical structures may be considered realizations of one and the same topological structure.4 The topological concepts used in this paper require a conceptual apparatus that likely not all philosophers are acquainted with. Topology, as used in this paper, goes beyond the vague idea that "topology" is just a sort of "generalized geometry" as expressed in the well-known pun: "A topologist is someone for whom the shapes of a coffee mug and a 4 From a more abstract point of view, topological structures and convex structures are not unrelated to each other: Both may be mathematically characterized as closure structures (cf. van de Vel (1994)). 6 donut do not essentially differ." Thus, for the reader's convenience, the necessary rudiments of the mathematical theory of topology are briefly recalled in Appendix A.5 The topology employed in this paper for elucidating the structure of conceptual spaces can be characterized as a "nonclassical". This topology considerably differs from "classical" topology emerging from the study of Euclidean spaces and their relatives. While "classical" topological spaces may be succinctly characterized as spaces that satisfy the Hausdorff axiom6 and often even stronger separation axioms, "nonclassical" topological spaces such as Alexandroff spaces do not satisfy the Hausdorff axiom. Formulated in a positive way, "nonclassical" topology is characterized by the fact that it is strongly related to a certain order structure (called the specialization order7) such that the topological structure is completely characterized completely by the order. For "classical" topological spaces, the specialization order is trivial (cf. Goubault-Larecq (2013), Kuratowski and Mostowski (1976), Steen and Seebach Jr. (1978)). For Alexandroff spaces, it is, however, highly non-trivial and suffices to characterizes the topological structure. Now and then, topology has been mentioned in the literature about conceptual spaces as is, for instanceÎ, exemplified by Gärdenfors's books (2000, 2014). There, topology is 5 All terms defined in the appendix are underlined when they are used for the first time in the main text. 6 The Hausdorff separation axiom for topological spaces requires that two distinct points x and y of the space have disjoint open neighborhoods U(x) and U(y), or, in other words, that x and y can be separated from each other topologically. Many of the familiar topological spaces such as Euclidean spaces, and, more generally, metrical spaces satisfy the Hausdorff axiom. For a precise definition of the axiom and other separation axioms, see A10. 7 Since this structure is very important for the rest of this paper, it may be expedient to give a preliminary informal description just now. Given a topological space (X, OX), the elements a Î OX may be interpreted as properties that the elements x Î X may or may not possess ("x has the property a" iff x Î a). Then an element x may be defined as "more special", "more central" or "more typical" than an element y (denoted by y ≤ x) iff x has at least as many properties as y. In many papers dealing with conceptual spaces such a (quasi-)order of specialization is implicit assumed when geometrical illustrations by Venn-like diagrams are used to distinguish between central and not-socentral cases of concepts. See, for instance, the "bird space" where "penguin" occupies a less central position than "robin" (Osta-Vélez and Gärdenfors (2020, p.6)). As will be explained later, this order structure is based on the topological structure of X, it is called the specialization quasi-order of the topological space (X, OX) and denoted by (X, ≤). A precise topological definition of (X, ≤) is given in A6. . 7 understood in a vague sense as a kind of generalized Euclidean geometry. This attitude is not to be criticized per se. The only point we want to emphasize is that this is not the way in which topological concepts are used in the following. As used in this paper, the concept of topology essentially differs from that of classical Euclidean geometry. Nevertheless, the topological approach put forward in this paper may be considered as a contemporary attempt to respond positively to the admonition that Plato is said to have engraved above the door to his academy: (1.1) AGEWMETRIKOS MHDEIS EISITW Those of us, who consider this maxim still relevant for contemporary philosophy, have no reason to restrict our attention to classical geometry of Euclid. Rather, we should attempt to make sense of it in terms of our modern understanding of geometry. To put it bluntly then, today, geometry understood as a theory of space in general, is topology. Up to now, geometrical methods in this modern sense, i.e., in the sense of topological methods, have been an under-exploited resource in many areas of philosophy and related disciplines such as cognitive psychology, and cognitive science in general. This holds, in particular, for the issue of how (possible) experiences or stimuli are represented in appropriate formal structures aka conceptual spaces. Following a recent proposal of Douven and Gärdenfors, let us conceive a conceptual system as an agreement between the members of a community that a particular meaning domain be partitioned in a particular way. A concept is then a particular element of such a partitioning. (Douven and Gärdenfors (2019, 2)) In a very simplified way, then, a conceptual system as a method of partitioning a domain of events, possible experiences can be described as a partitioning of a set X of experiences, stimuli or something similar. Without giving criteria to distinguish between interesting and uninteresting partitions, however, a theory of conceptual systems as a theory of partitions does not get off the ground. Here, geometry and topology come to 8 the rescue. Instead of being content to characterize a concept system abstractly as a partition, one may conceive it as a sort of map that partitions a space in various regions. Maps are tools for providing orientation for some parts or aspects of the world. Consider, for instance, a Voronoi tessellation of a conceptual space. Quite literally, a Voronoi tessellation indicates where something of interest for us, is located in a certain space and how it is spatially related to other, more or less similar entities. Locating an experience somewhere in a conceptual space helps categorize it: Moreover, provided that two cognitive agents use the same map (in a sense to be specified), they can compare their experiences, assess how similar they are, and deliberate what kinds of (common) actions are advisable to carry out in a situation. Thereby maps also enable us to communicate and, more specifically, even to reason about experiences (cf. Douven and Gärdenfors (2019)). Modern maps do not presuppose a Euclidean structure of space. This is already evident for the many kinds of "topological" mappings and diagrams we are using in our everyday epistemic activities. For instance, the various kinds of subway maps that people use for orientation, the countless types of diagrams and other graphical presentations up to the most indirect, highly sophisticated ways of producing computer-aided images of medical phenomena by methods of positron emission tomography and the like (cf. XXXX, XXXX, XXXX). All these ways of representing and making sense of aspects of the world are based on maps that are not just catalogues of what there is, rather, they are conceptually grounded symbolizations that heavily depend on highly sophisticated geometrical, or better, topological theories. Thus, the issue of geometrical and topological representations of events, experiences and processes for all kinds of sciences is of the outmost importance for modern cognitive sciences and related disciplines. Non-trivial representations require a certain amount of formal, in particular mathematical tools that cannot be justified in 9 advance. Thus, in order to persuade the reader that it is worth the effort to get acquainted with topology at least to a modest degree, it is argued that the concept of Alexandroff spaces is useful in quite a few areas: Alexandroff spaces help elucidate problems related to the logic of vague concepts, in particular, a novel solution of the Sorites paradox (proposed by Rumfitt). In a related way, Alexandroff spaces provide a natural semantic for Bobzien's logic of clearness and help overcome certain problems of the concept of higher-order vagueness. Further, Alexandroff topological spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. Moreover, the crucial role of order theory for Alexandroff spaces enables us to refine the all-or-nothing distinction between the prototypical and the nonprototypical in favor of a more fine-grained gradual distinction between more-or-less proto–typical elements of conceptual spaces. Finally, it is shown that certain shortcomings of the geometrical approach, caused by some inherent inadequacies of the geometrical approach to conceptual spaces, can be avoided by the topological approach, for instance, the so-called "thickness problem" and problems of choosing an appropriate for similarity spaces. Finally, a problem may be mentioned that can be tackled only by a topological account of conceptual spaces, namely, issues of digital conceptual spaces that are necessary to deal with digitally given data that have become ever more important in the areas of artificial intelligence, computer science and related disciplines. This paper is structured as follows. In section 2, as starting point of the project of the topological elucidation of conceptual spaces, we investigate the topological structure of polar spaces in detail. This type of conceptual spaces was recently introduced by Ian Rumfitt in his book The Boundary Stones of Thought (Rumfitt 2015) as a convenient framework for dealing with the logic of vague concepts. Polar spaces may be considered an elementary example of the general topological account elaborated in this paper. As is 10 shown they also provide a natural semantics for Bobzien's logic of clearness that has been designed cope with certain problems of higher-order vagueness. In section 3 the relation between the topologically defined tessellations of polar spaces and the better known geometrically defined Voronoi tessellations of Gärdenfors's conceptual spaces is explicated. Section 4 addresses the topology and order structure of a subclass of Alexandroff spaces that is especially useful for studying conceptual spaces, namely, weakly scattered Alexandroff spaces. It is shown that this class of spaces may be considered the most general class of spaces that gives rise to well-behaved classifications and categorizations of objects. Section 5 argues that the topological approach can contribute to the so-called design theory of (natural) concepts. We conclude with some general remarks on the intricate intertwinement of the aspects of order, algebra, and topology in the framework of Alexandroff spaces in section 6. Moreover, the paper has two appendixes. In Appendix A, for the reader's convenience, basic definitions and facts of topology are collected that are used throughout the paper. Appendix B contains a list of various kinds of Alexandroff spaces that are interesting for the purposes of this paper. 2. The Topological Structure of Polar Spaces. In this section, we begin the explication of the topological structure of conceptual spaces by investigating the topological structure of polar spaces. These spaces were recently introduced by Rumfitt (cf. Rumfitt (2015, chapter (8.4)). In particular, we show that these spaces are very simple Alexandroff spaces. Rumfitt presents his approach by way of example, discussing the polar topology of the well-known color spectrum as a conceptual space of color experiences. He does not, however, calculate its topology in detail. This will be addressed in this section. Let X be a set of colored objects that serves as the underlying set of a conceptual space for color experiences. We are looking for a discretization of X, i.e., a partition of X that 11 allows us to classify color experiences into different categories. This can be done with the aid of certain paradigmatic or prototypical experiences of red, blue, yellow and so on (cf. Gärdenfors (2000)). More precisely, Rumfitt argues that the classification of colors is best conceptualized as a procedure based on a comparison of certain color experiences to be considered as paradigmatic or prototypical: The spectrum enables us to attach senses to colour terms not because it shows boundaries, but because it displays colour paradigms or poles. Sainsbury likens colour paradigms to 'magnetic poles exerting various degrees of influence: some objects cluster firmly to one pole, some to another, and some, though sensitive to the forces, join no cluster'. ... I prefer a simpler analogy, which likens paradigms to gravitational poles, that is, massive bodies. If a small body is sufficiently close to a gravitational pole, it will be drawn towards it, rather as we are drawn to classify as red those objects that are sufficiently close in colour to a paradigm, or pole, of red. Rumfitt (2015, 236) The essential mathematical structure to be extracted from this example is as follows. Assume that there is given a set X of objects to be classified and a subset P of X to be considered as a set of distinguished elements that are "paradigmatic" or "prototypical" objects. In the terminology of Rumfitt (2015), they are called poles. These poles are used to classify the ordinary objects. This procedure is rendered precise by the following definition: (2.1) Definition. Let X be a set and P Í X be a subset of distinguished elements to be interpreted as prototypes or poles. Assume that for every object x Î X, there is a nonempty set m(x) Í P of poles p. The poles p Î m(x) are said to be maximally close to x. For each x, the set m(x) is assumed to satisfy the following two requirements: (i) For all x Î X, Ø ≠ m(x) Í P. (ii) For all x Î X (m(x) = {x} Û x Î P). 12 The map X3⁄43⁄4m3⁄43⁄4>2P defined (i) and (ii) is called a pole distribution and denoted by (X, m, P).  Requirements (2.1) (i) and (ii) entail that poles do some classificatory work by classifying the elements of X according to the poles that are maximally close to them. First, poles are distinguished from non-poles as those elements that are, so to speak, "self-classifying", i.e., m(p) = {p}.8 Second, and this is an observation that is more interesting, a pole distribution (X, m, P) defines a topology on X. More precisely, this is done with the help of the following interior kernel operator 2X3⁄43⁄4int3⁄43⁄4>2X: (2.2) Proposition. Let (X, m, P) be a pole distribution and A Í X. Define the operator int by x Î int(A) =: (x Î A & "pÎP (p Î m(x) Þ p Î A)) Û {x} È m(x) Í A.  The operator int is a topological interior kernel operator that defines an Alexandroff topology. Informally formulated, x Î int(A) iff x Î A and all poles that are maximally close to x, i.e., the set m(x), also belong to A. In other words, the interior of A comprises those elements of A whose maximally close poles also belong to A. Equivalently, the topology corresponding to a pole distribution (X, m, P) is defined by the closure operator operator 2X3⁄43⁄4cl3⁄43⁄4>2X: (2.3) Definition. Let (X, m, P) be a pole distribution and A Í X. Define the operator cl by x Î cl(A) := (x Î A or $pÎP(p Î A and p Î m(x)) Û ({x}È m(x)) Ç A ≠ Ø.  That is, the closure cl(A) of set A comprises the members of A together with all objects for which at least one of their maximally close poles is in A. In other words, cl(A) comprises all elements of A that are in A or that have at least a connection to elements of A. 8 The function m need not be defined with the aid of a full-fledged metric on X as Gärdenfors seems to assume. 13 The topological space (X, OX) defined by the operator int (or cl) is called the polar space of the distribution (X, m, P). The proof that int and cl are topological operators involves a routine check that these operators satisfy the Kuratowski axioms (A.2); see Rumfitt (2015, 243 246). A closer inspection of definitions (2.2)1 or and (2.2)2 reveals that they even satisfy the Alexandroff condition (A.1) (iii), namely, arbitrary intersections (unions) of open (closed) sets are open (closed): (2.4) Proposition. The topology (X, OX) defined by a pole distribution (X, m, P) is an Alexandroff topology.   Rumfitt defines the topology (X, OX) given by a pole distribution (X, m, P), but he does not describe the topology in any detail. In particular, he does not mention that (X, OX) is an Alexandroff space, nor does he explicitly show that O*X is atomic. Endowed with the topology defined by a pole distribution (X, m, P), the color spectrum (X, OX) is a very special Alexandroff space; namely, (X, OX) is a T0-space such that the singletons {p} Í P are open and all singletons {x} for x Î X – P are closed. More precisely, the following proposition holds: (2.5) Proposition. Let (X, OX) be a topological space defined by a pole distribution (X, m, P). Then, X is a weakly scattered T0–Alexandroff space, i.e., cl(P) = X. For all y Î X, let V(y) be the smallest open set that contains y. For elements p Î P and x Î X – P one calculates: int(p) = {p}. int(x) = Ø. V(x) = {x} È m(x). V(p) = {p}. cl(x) = {x}. cl(p) = {x; p Î m(x)}. int(cl(p)) = {x; {p} = m(x)}. The specialization order of (X, OX) is given by x < y := x ≠ y and y Î m(x). Thus, for polar spaces (X, OX) the specialization order (X, ≤) is of depth 1. It just amounts to the distinction between prototypical and non-prototypical elements of X. The set P of poles is 14 a dense set of isolated points of X, i.e., {p} Î OX for p Î P, and cl(P) = Èp ÎP cl(p) = X. The Boolean algebra O*X of regular open sets of (X, OX) is isomorphic to the power set 2P. An isomorphism 2P3⁄43⁄4r3⁄43⁄4>O*X can be inductively defined as follows: r(p) := intcl(p), and if r(A) and r(B) are already defined, r(A È B) := j(i(r(A) È i(r(B))), the maps O*X3⁄4i3⁄4>OX and i and OX3⁄4j3⁄4>O*X as defined in A.3. A canonical minimal basis of OX is given by the set {V(x); x Î X} = {{x} È m(x); x Î X}.  Proof. To prove these claims, one must only check definitions. Be it sufficient to prove that V(x) = {x} È m(x). According to the definition of the interior kernel operator int one has y Î int({x} È m(x)) Û y Î {x} È m(x) & "p (p Î m(y) Þ p Î {x} È m(x)). Clearly, every element in {x} È m(x) satisfies this condition. On the other hand, any smaller set properly contained in {x}È m(x) does not satisfy the condition. For two different elements x and y, V(x) and V(y) are different. Hence, X is a T0-space. Thus, the set {{x} È m(x), x Î X} forms a unique minimal basis for the topology on X.  Proposition (2.5) characterizes Rumfitt's polar spaces as a special class of Alexandroff spaces (X, OX) (cf. Bezhanishvili, Esakia, and Gabelaia (2004) and Bezhanishvili, Mines, and Morandi (2004)). The cardinalities of OX and O*X may be quite different. Already for the color circle S with a finite set P of prototypical colors ("red", "orange", "yellow", ...) OS is of uncountable cardinality while O*S has only finitely many elements 2|P|. This is a compelling argument that O*S rather than OS should be taken as the set that represents the Boolean algebra of concepts of a conceptual space (cf. Rumfitt 2015). Further, one should note that O*X associated with a polar conceptual space (X, OX) come equipped with a binary similarity relation: 15 (2.6) Definition. Let O*X be the Boolean lattice of regular open subsets of the topological space (X, OX). A reflexive and symmetric similarity relation on O*X is defined by A ~ B := cl(A) Ç cl(B) ≠ Ø for A, B Î O*X.  This relation is not necessarily transitive. It can be used to distinguish between different conceptual spaces X and Y that have isomorphic Boolean lattices O*X and O*Y but that differ in their similarity relations ~X and ~Y defined on O*X and O*Y, respectively. An elementary example is given by the linear color spectrum Lin(S) and the circle spectrum Cir(S) (cf. Gärdenfors (2000) and Rumfitt (2015)). Both have the same number of prototypes, say, "red", "yellow", "green", "blue", and "purple", but their similarity relations may differ. For the linear spectrum one obtains (2.7) Lin(S) = < red ~ yellow ~ green ~blue ~ purple > In contrast, for the circular color spectrum one obtains (2.7)' Cir(S) = < red ~ yellow ~ green ~ blue ~ purple ~ red > These two similarity structures are different since in Lin(S) "red" and "purple" are not similar to each other, while in Cir(S) they are similar). Nevertheless, the corresponding Boolean lattices of regular open concepts of the two conceptual spaces are isomorphic (as Boolean algebras) to the power set 2P of the set P = {"red", "yellow", "green", "blue", "purple"}. Rumfitt rightly emphasizes that his topological representation of the color spectrum is just an example analogous results hold for all conceptual spaces that "involve predicates whose meaning is given by reference to paradigms or poles" (Rumfitt 2015, 255). His main concern is not a topological reconstruction of the color spectrum as a (polar) topological space, rather, the topology of the color spectrum is only the basic ingredient 16 for his solution of the Sorites paradox.9 This solution relies on the fact that for regular open interpretations of classical Boolean propositional logic "the truth of a disjunction does not entail the truth of any of its disjuncts"(Rumfitt (215, 254). Before we go on, it may be expedient to explain in more detail the general significance of (2.5) for the theory of conceptual spaces. Succinctly stated, (2.5) ensures that a conceptual space X endowed with a pole distribution (X, m, P) has a Boolean lattice O*X of regular open subsets that is atomic and isomorphic to the power set 2P of the set of its prototypes P. As is pointed out in Rumfitt (2015), the linear and the circular color spectrum Lin(S) and Cir(S) belong to this class of spaces. Proposition (2.5) states that these spaces possess well-behaved conceptual systems O*X. More precisely, O*X is a complete atomic Boolean algebra generated by intcl(p), pÎ P. In other words, the conceptual systems related to this kind of conceptual spaces have a very simple classical structure. Rumfitt's solution of the Sorites paradox convincingly shows that the topological approach may be useful for attacking intricate philosophical problems. For further applications it is expedient, however, to consider a somewhat broader class of topological spaces than polar 9	Cut down to its bones, the topological core of Rumfitt's solution of the Sorites paradox consists in exploiting the peculiar properties of topologically defined regular open interpretations of Boolean logic. The details are as follows. Let O*X be the Boolean algebra of regular open sets of (X, OX). A regular open interpretation of a propositional language L (with propositional variables a, b, ..., and the Boolean connectives Ù, Ú, ...) in O*X is a map L3⁄4r3⁄4>O*X such that (i) r(a Ù b) := r(a) Ç r(b), (ii) r(a Ú b) := int(cl(r(a) È r(b)), and (iii) r(¬a) := intCr(a). The crucial feature of a regular open interpretation is that it yields a semantic of classical Boolean logic that may render a disjunction a Ú b true in X without implying that for all x Î X either a or b is true at x. The underlying topological fact is simply that for a regular open interpretation r a disjunction a Ú b of a and b has the interpretation int(cl(r(a) È r(b)). This set may be strictly larger than the union r(a) È r(b) of the disjuncts r(a) and r(b). This fact may be used to cope with the Sorites paradox, see Rumfitt (2015, p. 253) or Mormann (2020, section 5). As it seems, Rumfitt assumes that the topological product of polar spaces is again a polar space. This is, as can be easily shown, not correct. An elementary example is the Khalimsky plane. This space is the Cartesian product of two polar spaces but not a polar space itself, see (2.10), (2.11), and (A.12)(iv). Rumfitt's arguments are not affected by this slight inaccuracy, however. Be this as it may, the approach of the present paper has no difficulty of dealing with products of polar spaces, see (2.8)ff. 17 spaces, namely, weakly scattered Alexandroff spaces (X, OX), for which polar spaces provide only the simplest example. More generally, one obtains: (2.8) Proposition. (i) The topological product (X   Y, O(X   Y) of the polar spaces (X, OX) and (Y, OY) is a weakly scattered Alexandroff space whose specialization order (X   Y, ≤   ≤) has depth 2. (ii) Finite products of weakly scattered Alexandroff spaces are weakly scattered Alexan– droff spaces. If the specialization orders (X, ≤) and (Y, ≤) are of depth m+1 and n+1, respectively, then the specializiation order (X   Y, ≤) is of depth m + n + 1. (iii) Let ~ be an equivalence relation on weakly scattered Alexandroff space (X, OX) such that the quotient map X3⁄43⁄4q3⁄4>X/~ is open. Then (X/~, OX/~) is a weakly scattered Alexandroff space. Proof. (i) Let (X, m, P) and (Y, n, Q) be the pole distributions of (X, OX) and (Y, OY), respectively. Assume (x, y) Î X   Y, (p, q) Î P   Q, x < p, and y < q. In X   Y the chain (x, y) < (p, y) < (p, q) is of length 2. Hence the specialization order of the topological product (X   X, OX   Y) has depth 2. (ii) If MX and MY are the dense sets of maximal elements of the specialization orders (X, ≤) and (Y, ≤), respectively, then MX   MY is the dense set of isolated elements of the weakly scattered Alexandroff space (X   Y, OX   Y). The depth of the specialization of (X   Y, ≤) is calculated as in (i). (iii) Since the set ISO(X) of isolated points of X is mapped by q onto the open set {[p], p Î P} of isolated points of X/~, the space (X/~, OX/~) is weakly scattered and q(ISO(X)) is dense in X/~. Obviously, the quotient space q(X) is Alexandroff.  Proposition (2.8) ensures that there are plenty of weakly scattered Alexandroff spaces. The following neat result shows that this class of spaces has topological properties that possess interesting modal interpretations: 18 (2.9) Proposition. Let (X, OX) be a weakly scattered Alexandroff space. Then the following equivalent conditions hold: (i) X satisfies the McKinsey axiom, i.e., int(cl(A)) Í cl(int(A)) for all A Í X. (ii) The boundary operator bd of (X, OX) satisfies bd(bd(A)) = bd(A) for all A Í X. Proof. Bezhanishvili, Mines, and Morandi (2003, Propositions 2.1 and 2.4) prove (among other things) that (2.9)(i) and (2.9)(ii) are equivalent with the assumption that (X, OX) is weakly scattered.  In order to see that (2.9) has interesting modal interpretations it is expedient first of all to recall a trail-blazing result of McKinsey and Tarski (1944). According to these authors, the modal system S4 is the logic of topology in the sense that a proposition is valid in S4 if and only it is valid in all topological spaces. McKinsey and Tarski's result has been generalized in many ways, in particular by establishing a correspondence between certain classes of topological spaces on the one hand and certain extensions S4.X of S4 logic on the other. As is well known, the extension of S4.1 of S4 by the McKinsey axiom corresponds to the logic of weakly scattered spaces (cf. Bezhanishvili et al. (2003), (2004), Gabelaia (2001)). In other words, weakly scattered Alexandroff spaces are models for the modal logic S4.1.10 As a second example that weakly scattered Alexandroff spaces have some useful applications in modal logic, Bobzien's logic of clearness may be mentioned. Very succinctly, this can be explicated as follows. For general topological spaces (X, OX) one easily calculates that, instead of (2.9)(ii), only the weaker equation bd(bd(bd(A))) = bd(bd(A)) holds for all A. There may be A Í X with bd(bd(A)) ≠ bd(A).11 Proposition (2.9) (ii) ensures that weakly scattered Alexandroff spaces behave particularly well with respect to 10 Not all weakly scattered spaces are Alexandroff, of course. See appendix B. 11 An elementary example is given by the real line (R, OR): For the set of rational numbers Q one obtains bd(bd(Q) = Ø and bd(Q) = R. 19 boundaries bd, since for them the stronger formula bd(bd(A)) = bd(A) holds. In several papers Susanne Bobzien has argued that the logic of vague concepts should be cast in the framework of a modal logic based on an operator C such that CA is to be read as "It is clear that A" (cf. Bobzien (2012, 2015)). More precisely, she argues that the operator C should satisfy at least the axioms of the modal system S4. With the aid of C an operator U of "unclearness" operator U such that UA is to be read as "It is not clear that A, and it is not clear that not-A". Obviously, the operators C and U are related to each other in the same way as the topological operators int and bd are related. Bobzien compellingly argues that U should satisfy the law U2A = UA, and proves that U2A = UA is equivalent with the assumption that her "clearness" operator C satisfies the McKinsey axiom. In sum, by (2.9) weakly scattered Alexandroff spaces offer a natural topological semantic for Bobzien's modal "logic of clearness". Rumfitt's and Bobzien's solutions of the Sorites paradox and related problems of the theory of vagueness show that weakly scattered Alexandroff spaces and their theory offer interesting insights into the subtleties of the theoretical logic of vagueness. This does not exhaust the usefulness of Alexandroff spaces, however. In the rest of this section I want to hint at the fact that the topology of Alexandroff spaces also has an immense practical importance as foundation of the discipline of digital topology. Digital topology deals with the geometrical and topological investigation of digitized objects or digitized images and providing both theoretical and computational frameworks for image computing. It plays an essential role in various fields related to digital images, such as image analysis, computer graphics, pattern recognition, shape modeling and computer vision. It emerged during the second half of the twentieth century with the birth of computer graphics and digital image processing. As a starting point for digital topology one may consider the "digital line" or "Khalimsky line". The Khalimsky line is a polar space that can be defined as follows: 20 (2.10) Definition. Let Z = {..., -2, -1, 0, 1, 2, ...} be the set of entire numbers. Denote the set of odd numbers by 2Z + 1 = {...-3, -1, 1, 3, ...}. Then a pole distribution (Z, m, 2Z+1) is defined by the map m(2n) = {2n-1, 2n+1} , m(2n+1) := {2n+1}. The corresponding polar topological space (Z, OZ) is called the "digital line" or the "Khalimsky line".  (2.11) Corollary. Let Z2m denote the quotient space Z/2mZ for some integer m. Then the canonical quotient map Z3⁄43⁄4q3⁄43⁄4>Z2m is open (with respect to the Khalimsky topology and the quotient topology) and defines a finite topological space (Z2m, OZ2m). It is called the Khalimsky circle. (Z2m, OZ2m) is a polar space and may be considered as a kind of digital model of the circular color spectrum (X, m, P) with P = { 1, 3, ...2m-1}.  Applying (2.5) to (2.11) the pole distribution (Z, m, 2Z+1) defines a topological space (Z, OZ) such that the singletons {2m} of the "even points" 2m Î Z are closed, and the singletons {2m+1} of the odd points 2m+1Î Z are open subsets. The smallest open set of OZ containing 2m is {2m – 1, 2m, 2m +1}, and O*(Z)) is the power set 2L, L = 2Z+1. (2.12). Definition. The topological product (Z   Z, OZ   Z) of two copies of the digital line (Z, OZ) is called the "digital plane". Higher-dimensional digital spaces (Zn, OZn), n ≥ 3, are defined analogously.  The most important quality of the digital spaces (Zn, OZn) is that they can serve as "discrete" or "digital" models of the Euclidean spaces (Rn, ORn). For the digital line this claim is rendered precise by the following proposition: (2.13) Proposition. Let (R, OR) be the real line with the Euclidean topology. Define the map R3⁄43⁄4q3⁄43⁄4>Z by (i) q(x) = x iff x = 2m, m Î Z and (ii) q(x) = 2m+1 for 2m < x < 2m+2, m Î Z. 21 Then the digital line (Z, OZ) is the quotient space of the real line (R, OR) by the map q. Further, the quotient map (R, OR) 3⁄43⁄4q3⁄43⁄4> (Z, OZ) is an open continuous map. Thereby, the polar topological space (Z, OZ) is shown to be a connected topological space.  (2.14) Corollary. The higher-dimensional digital spaces (Zn, OZn) are weakly scattered connected Alexandroff spaces that are open quotient spaces of the Euclidean spaces (Rn, ORn) by the product maps (Rn, ORn) 3⁄43⁄4qn3⁄43⁄4>(Zn, OZn).  Corollary (2.14) has been said to provide the foundation for the disciplines of digital topology and geometry. In a nutshell, it establishes the fact that digital structures (Zn, OZn) can serve as a (partially faithful) models of the continuous structures (Rn, ORn).12 Or, to emphasize more clearly the relevance of (2.14) for the issue of conceptual spaces Continuous conceptual spaces (i.e., conceptual spaces based on Euclidean spaces Rn and appropriate substructures) can be replaced (at least in principle) by digital conceptual spaces Zn (and appropriate derivatives). If one wants to deal with digital data (which becomes ever more important in many areas in science and elsewhere), then such a replacement is necessary.13 3. Topological and Geometrical Tessellations. The aim of this section is to discuss several types of tessellations (discretizations) of conceptual spaces that define conceptual classifications based on geometrical or topological structures of conceptual spaces. 12 The task of finding out what are the digital counterparts of various kinds of continuous phenomena (if there are any), may be highly non-trivial. An early classical result in this field is the digital Jordan curve theorem according to which a continuous of a digital circle defines a tessellation of the digital plane in two parts. 13 This paper is, of course, not the appropriate place to deal with digital topology and its many applications in any greater depth. The literature on digital topology is immense. Here, it must suffice to mention just some introductory texts, e.g., Rosenfeld (1979), Kovalevsky (2006), Kong, Kopperman, and Meyer (1991), Melin (2008). 22 Because the Boolean algebra O*X is atomic for polar spaces (X, OX), the resulting tessellation is particularly simple and essentially unique. (3.1) Definition. A regular open tessellation of a topological space (X, OX) is a set T of some disjoint regular open subsets Al Î O*X with supremum VAl = X. The Al are called the cells of T. Note that the supremum VAl of the Al is to be taken in O*X (not in OX).14 The set bd(T) = X ÈAl is called the boundary of T. If the Al are atoms of O*X, then T is called an atomic tessellation. Points of X that are not in any cell Al are said to be on the boundary bd(T) of the tessellation T.  (3.2) Examples of Topological Tessellations. (i) Let (X, OX) be a topological space, A Î O*X, A ≠ Ø, X. Denote the Boolean complement of A in O*X by A* (A* = int(CA)). Then, T = {A, A*} is a regular open tessellation of X with two open cells A and A* and boundary bd(T) = bd(A) (= bd(A*)). More generally, let A1, ..., An be n regular open subsets of X with VAi = X. The intersections of the Ai generate a regular open tessellation of X that has m cells, m ≤ 2n – 1. (ii) A particularly important (geometrical and topological) tessellation is the tessellation of the Euclidean line R given by open intervals: T := {(2m, 2m+2); m Î Z}. 15 This tessellation may be called the Khalimsky tessellation. (iii) Tessellations of higher-dimensional Euclidean spaces Rn can be defined analogously.  Regular open tessellations for topological spaces (X, OX) exist in profusion. The point is to find tessellations that are interesting for some reason or other. For instance, one may ask 14 The supremum VAi may be strictly larger than the set-theoretical union UAi of the AI. There should be no real danger to confuse this supremum with the open hull V(A) of a subset A of an Alexandroff space (X, OX). 15 As will be shown in a moment, the tessellation (3.2)(iii) is closely related to the construction of the digital line (Z, 0Z). 23 whether or not a space has a atomic regular open tessellation. As is easily shown, the real line R and, more generally, Euclidean spaces Rn do not possess atomic tessellations. In contrast, polar spaces (X, OX) possess atomic regular tessellations: (3.3) Proposition. Let (X, m, P) be a pole distribution for X. Then, the topological space (X, OX) has a canonical regular open atomic tessellation defined by T := {intcl(p); p Î P} by the atoms of the Boolean algebra O*X and X = VpÎP intcl(p). Proof. Let (X, OX) be defined by (X, m, P). As proved in (2.6), the Boolean algebra O*X is isomorphic to the power set 2P. Thus, the atoms of O*X generate a regular open tessellation T of X. The atoms of T are the regular open sets int(cl(p)) = {x; {p} = m(x)}, p Î P. Elements of X to which more than one pole is maximally close are located at the boundary of T, i.e., bd(T) = {x; {p, p'} Í m(x)} for some p, p' Î P and p ≠ p'.  As already mentioned in the introduction, Rumfitt's topologically defined polar spaces are not the only examples of conceptual spaces. Probably better known are Gärdenfors's geometrically defined conceptual spaces. Hence, it may be expedient now to deal with this class of spaces in more detail and to explain how Rumfitt's and Gärdenfors's conceptual spaces may be conceived as Alexandroff spaces. According to Gärdenfors, conceptual spaces are similarity spaces. A similarity space is a metrical space whose metric is used to define a binary similarity relation on it. Distances in the space are meant to measure similarity: the shorter the distance between objects, the more similar they are. Let us now consider in more detail the most prominent class of tessellations of conceptual spaces, namely, the so-called Voronoi tessellations (cf. Gärdenfors (2000), Decock and Douven (2015), Zenker and Gärdenfors (2015)). Voronoi tessellations are geometrical tessellations defined by the underlying geometrical structure of Euclidean conceptual spaces. More precisely, the general definition is as follows: 24 Given a set P of two or more but a finite number of distinct points in the Euclidean plane, we associate all locations in that space with the closest member(s) of P with respect to the Euclidean distance. The result is a tessellation of the plane into a set of the regions associated with members of P. We call this tessellation the planar ordinary Voronoi diagram generated by P, and the regions constituting the Voronoi diagram ordinary Voronoi polygons. (Okabe et al. (1992, p. 44)) As an illustration of this general definition, consider the simplest Voronoi tessellation of the Euclidean plane E: (3.4) Example. Let E be the Euclidean plane endowed with a Cartesian coordinate system (x, y). For the two points pL = (-1, 0) and pR = (1, 0) the Voronoi tessellation generated by pL and pR is given by the two open half-planes L and R: L: = {(x, y); d((x, y), pL) < d((x, y), pR)}, R : = {(x, y); d((x, y), pR) < d((x, y), pL)} The boundary of this tessellation is just the topological boundary bd(L) = bd(R) of the two half-planes L and R, forming the line: bd(L) = bd(R) = {(x,y); d((x,y), pL) = d((x,y), pR)} = {(x, y); x = 0}.  Let p1, ..., pn be finitely many different points of the Euclidean plane. Then, a general Voronoi tessellation may be conceived as the result of the intersection of n!/(2! (n-2)!) pairs of half-planes each defined by the bisectors of the pairs (pi, pj) of different points pi and pj such that the plane is divided into convex open cells together with their boundaries. Clearly, a geometrically defined Voronoi tessellation of Euclidean space defines a regular open topological tessellation in the sense of (3.1). By construction, all open Voronoi cells are convex and disjoint from each other (cf. Gärdenfors (2000, 88), Okabe et alii (1992)). As is well known, they are not only open but even regular open. From the very definition of Voronoi cells, points not in the interior of any cell are the points positioned at an equal 25 distance to two (or more) paradigmatic points pi. Hence, they are located on the topological boundaries of the cells defined by the pi. This fact can be used to show that a Voronoi tessellation based on the metrical structure of Euclidean space E also yields a pole distribution (E, m, P): (3.5) Proposition. Let T be Voronoi tessellation of the Euclidean plane E defined by a finite set P of prototypes p1, ..., pn. Then a topological pole distribution (E, m, P) is defined as follows: Take the Voronoi generators p1, ..., pn as the set P of poles of a pole distribution X3⁄43⁄4m3⁄43⁄4>2P defined as: m(x) = {pi; x Î cl(<pi>); <pi> the Voronoi cell defined by pi)}. Let (E, OE) be the polar topological space defined by (E, m, P). Since the cells <pi> of the Voronoi tessellation are convex, they are regular open. Hence, (E, m, P) defines a regular open tessellation T = {<pi>; pi Î P}. This tessellation is, of course, not atomic with respect to Euclidean topology. It is, however, by definition, a regular atomic tessellation with respect to the polar topology defined (E, m, P). Its regular open atoms are just the Voronoi cells <pi>. In terms of the pole distribution m, one has x Î E contained in a cell int(cl(p)) iff m(x) = {p}.  In other words, the cells of the topological tessellation of E defined by (E, m, P) coincide with the cells of the Voronoi tessellation of E. Moreover, the geometrically defined boundary of the Voronoi tessellation coincides with the topologically defined boundary. In sum, every geometrical Voronoi tessellation of the Euclidean space E defined by a finite set P of prototypes gives rise to a topological tessellation defined by a pole distribution (E, m, P). The two tessellations are extensionally equivalent in the sense that their cells and boundary areas coincide. Thus, they may be conceived as two different interpretations of the same set-theoretical data. 26 A great advantage of the topological interpretation, i.e., the interpretation of cells <pi> as regular open (atomic) elements of the Boolean lattice O*E of a polar topological space (E, OE), lies in the fact that classical logical operations on concepts (conjunction, disjunction, and negation) are represented in a natural way by topologically defined operators on O*E.16 Moreover, tessellations of a space (X, OX) define an equivalence relation ~ on X in a natural way: (3.6) Definition. Let T be a (topological or geometrical) tessellation of (X, OX). The elements x, y Î X are equivalent with respect to T iff the following holds: x ~ y := x = y or there is a cell A of T and x, y Î A.   (3.7) Example. Let (R, OR) be the Euclidean line and T = {(2m, 2m+2), m Î Z} the tessellation (3.2). Choose the points 2m+1 Î (2m, 2m+2) as representatives of the resulting (non-trivial) equivalence classes of the relation ~ defined by the tessellation on R by x ~ y := x = y or 2m < x, y < 2m+2 Then (R/~, OR/~) is just the Khalimsky digital line (Z, OZ).   (3.8) Corollary. Let (Z, OZ) be the Khalimsky line and m ≥ 2 a natural number. Consider the equivalence relation: x ~ y := x – y Î 2mZ. Then the quotient space (Z/~, OZ/~) is called the Khalimsky digital circle (Z2m, OZ2m) (cf. Melin (2008, 14)). Let [x] Î Z2m denote the class of elements represented by x Î Z2m. Then (Z2m, OZ2m) is a polar space with poles represented by the elements [1], [3], ..., [2m – 1]. An element [2k] with 0 ≤ k ≤ m – 1 represents the class of elements that have the same distance from the poles 2k -1 and 2k + 1. In other words, the Khalimsky circle defined for 2m can be considered as a digital model of the circular color spectrum with m prototypical colors.  16 This holds not only for polar spaces, but for weakly scattered Alexandroff spaces in general, see section 4 for more details. 27 Compared with the geometrical construction of a Voronoi tessellation of the Euclidean plane, a topological tessellation requires much fewer structural presuppositions in the sense that a Euclidean structure of space is much more specific than a topological one. This is a conceptual advantage insofar as certain problems caused by the presence of representational artifacts disappear. For example, for Euclidean spaces, there are many different metrical structures that define the same underlying topological structure.17 With respect to these different metrics, one and the same set P of prototypical points may give rise to different Voronoi tessellations. Which should be chosen as the "right" one? This is a question that may have no unique answer. A topological approach does not have the burden to answer it. Another problem that may be attributed to the peculiarities of the specific mathematical apparatus used for the definition of a Voronoi tessellation T of a conceptual space concerns the boundary area bd(T) of T. This issue has been dubbed the "thickness problem" (cf. Douven et al. (2013) and Douven (2019)). The "thickness problem" can be explicated as follows. Consider a Voronoi tessellation of the Euclidean plane. By its very construction, the boundaries of the Voronoi cells are "thin" compared to their interior since they are composed of lines consisting of points that have equal distances to two (or more) prototypical points. Douven et al. rightly point out that this assumption for most conceptual spaces is not very plausible. For instance, for the conceptual space of the color spectrum, the boundary, say, between "red" and "orange," is defined by points positioned at exactly the same distance from the prototypical points of "red" and "orange." Empirically, this does not make much sense. What does it matter that a certain shade of color is positioned at the same distance from a prototypical "red" and a prototypical "orange"? Moreover, in a general case, there is no reason to assume 17 A prominent case is provided by the family of Minkowski metrics di(x, y) for 1 ≤ i ≤ ¥. This problem is briefly discussed for d1 (Manhattan metric) and d2 (Euclidean metric) in Gärdenfors (2000, chapter 3.9). The Euclidean metric d2 offers a structural advantage in that the cells of its Voronoi tessellations are always convex with respect to the standard convex structure of Euclidean space. This does not hold for d1 (cf. Hernandez Conde 2017). 28 that boundaries are "thin" compared to the regular open cells of Voronoi tessellation. Douven et al. (2013) proposes overcoming this shortcoming by the introduction of "collated Voronoi diagrams" that arise as a result of projecting similar ordinary Voronoi diagrams onto each other such that the set-theoretical union of their boundaries define a blurred and more or less "thick" area to take into account the vagueness of concepts and their boundaries. For topological tessellations, no "thickness" problem arises, since they do not distinguish between "thick" and "thin" as geometrical tessellations do (in an artificial way). The following example shows that the topological approach easily deals with tessellations with cells whose boundaries are "thicker" than the cells themselves: (3.8) Example. Let X be the set {a, w} È N, with N natural numbers and a and w are two objects that are different from all natural numbers and from each other. Take P = {a, w} and define a pole distribution (X, m, P) by m(i) = {a, w}, i Î N, m(a) = {a}, and m(w) = {w}. The corresponding topological structure (X, OX) is given by cl(a) = {a} È N , cl(i) = {i} , cl(w) = {w} È N int(a) = {a} , int(N) = Ø , int(w) = {w} intcl(a) = {a} , intcl(w) = {w} , bd(w) = bd(a) = N. bd(i) = {i} , V(i) = {i, a, w} , cl(N) = N. The specialization order of (X, OX) is given by i < a, w for all i Î N, and there is no other nontrivial relation between these objects. The cardinalities of the boundaries bd(w) and bd(a) of the regular open cells {a} and {w} are much greater than the cardinalities of the regular open cells {a} and {w} themselves. A natural basis for OX is given by {{a}, {w}, {i, a, w}; i Î N}. In contrast, the cardinality of the algebra of regular open sets O*X is much smaller. O*X is isomorphic to the Boolean 29 algebra with 4 elements generated by {a} and {w}). Thus, moving from OX to O*X amounts to a considerable gain of conceptual parsimony (cf. section5).  The example (3.8) shows that the topological approach has no difficulty in dealing with the "thickness" of boundaries. The concept of topological tessellation is flexible enough to allow cells with boundaries that are intuitively much "thicker" than the cells they are boundaries of. 4. Weakly Scattered Alexandroff Spaces as a General Framework for Topological Conceptual Spaces. The aim of this section is to show that weakly scattered Alexandroff spaces may be considered as a convenient topological framework for conceptual spaces in general. Weakly scattered Alexandroff spaces provide a natural generalization of polar spaces. They possess all their nice features but are more flexible and have a larger domain of applications. To put it bluntly, they may be considered as the "right" generalization of polar spaces. To set the stage, let us begin by recalling the essential features of the Alexandroff approach. An Alexandroff space X is defined as a topological space for which arbitrary intersections (unions) of open (closed) sets are open (closed) and not only finite ones) (cf. (2.6)). Clearly, every topological space (X, OX) with only a finite number of elements is an Alexandroff space. Finite spaces do not exhaust the class of Alexandroff spaces, however. Rather, Alexandroff topology becomes a particularly interesting field of topology exactly for spaces of infinite cardinality. Thus, the Alexandroff topology of a color circle and similar conceptual spaces defined by prototype distributions (X, m, P) qualifies as an interesting Alexandroff topology (cf. Rumfitt (2015)) as well as digital topological spaces such as the Khalimsky plane (Z   Z, O Z   Z). All Alexandroff spaces (X, OX) are completely characterized by their specialization orders (X, ≤) defined by x ≤ y := x Î cl(y). A particularly well-behaved subclass of Alexandroff 30 spaces is the class of spaces whose specialization orders (X, ≤) have maximal elements, i.e., for each x Î X there is a y such that x ≤ y and y is maximal in (X, ≤). Clearly, polar spaces and their products are weakly scattered. Given an Alexandroff space (X, OX) (or equivalently, a partial order (X, ≤) one may distinguish two kinds of "extreme" elements: x Î X is an extreme element with respect to ≤ if and only if x is a maximal element of the specialization order, and x Î X is extreme with respect to the topology OX iff {x} Î OX, i.e., iff x Î ISO(X). Fortunately, these two concepts of "extreme elements" coincide: (4.1) Proposition. Let (X, OX) be a weakly scattered Alexandroff space with specialization order (X, ≤). The sets ISO(X) of isolated points and the set MX of maximal points of the specialization order (X, ≤) coincide. Conceiving (X, OX) as a conceptual space, "extreme" elements of X are the prototypical elements p of the regular open concepts int(cl(p)) Î O*X. Proof. Let p be a maximal element of (X, ≤) and assume that {p} is not open. Then int(p) = Ø. By definition of int this is equivalent CclC(p) = Ø, i.e., cl(Cp) = X. Since X is Alexandroff one obtains X = cl(Cp) = Èa≠p cl(a). Hence p Î cl(a) and a ≠ p, i.e., p < a for at least one a. This is a contradiction against the maximality of p. Hence {p} is open. Now assume {p} is open and suppose p is not maximal and p Î cl(a) = CintCa. Clearly, p Î Ca, since a ≠ c. Since {p} is open one obtains int(p) Í intCa. This is a contradiction. Hence p is maximal.  (4.2) Corollary. Weakly scattered Alexandroff space (X, OX) satisfy the McKinsey axiom int(cl(A)) Í cl(int(A)), for all A Í X.  Now we can formulate the main theorem of this paper that characterizes weakly scattered Alexandroff spaces as a convenient class for dealing with order-theoretical, algebraic, and topological aspects of conceptual spaces: 31 (4.3) Theorem. Let (X, OX) be a weakly scattered T0 Alexandroff space with specialization order (X, ≤), and let ISO(X) be the set of maximal elements. Then (X, OX) satisfies the McKinsey axiom, and the Boolean lattice O*X of regular open elements of OX is an atomic Boolean algebra with atoms intcl(p), p Î ISO(X) as generators. One obtains a regular open atomic tessellation of X by X = VpÎISO(X) intcl(p), i.e., O*X = 2L, with L being the set of atoms of O*X, i.e., L = {int(cl(p)), p Î ISO(X)}. Proof. Let ISO(X) be the set of maximal elements of the specialization order (X, ≤) of (X, OX). By definition, for each x Î X there is at least one p Î ISO(X) such that x ≤ p. Since the Alexandroff topology is the upper topology of the specialization order (X, ≤), singletons {p} are open, and closures cl(p) of {p} are the down sets  p := {x; x ≤ p}. The sets int(cl(p)) := {x; -x Í  p} for p Î ISO(X) are atoms of O*X. For different p, p*, sets intcl(p) and intcl(p*) are disjoint and regular open, as int(cl(p)) Ç int(cl(p*)) = intcl({p} Ç {p*}) = Ø because the operator intcl is a nucleus, i.e., it distributes over finite intersections of open sets (cf. Johnstone (1982 (ch. II, 48)) To prove that intcl(p) is an atom in O*X is seen as follows: assume that x Î A = intcl(A) Í intcl(p). Since intcl(p) is open, one has -x Í cl(p)). This entails that x ≤ p and therefore that p Î -x Í A. Hence, intcl(p) Í intcl(A) = A and A = intcl(p). We now prove that any regular open A Î O*X has the form A = VnÎM' intcl(n) for some M' Í M. Let MA := {p; p Î A Ç M}. Clearly, VnÎÎAM intcl(p) Í A. If we can show that A Í VnÎÎAM intcl(p), we are done. Assume that x Î A and define Mx := {p; x ≤ p and n Î ISO(X)}. Clearly, Mx Í AM. Hence, x Î cl(AM). Assume that y Î -x. This is the case iff x ≤ y and this entails that My Í Mx. Thus, we obtain -x Í cl(AM). This means that x Î intcl(AM) = intcl(ÈnÎAM{p}) = VnÎAM intcl(p). Rather, A Í VpÎÎAM intcl(p). 18 18 According to theorem (4.3), for O*X to be an atomic Boolean lattice, it suffices that (X, OX) is Alexandroff and weakly scattered. These two requirements are, however, not necessary to ensure that O*X is atomic. There are non-weakly scattered Alexandroff spaces and non-Alexandroff weakly 32 Theorem (4.3) shows that the topological account of conceptual spaces has another important advantage over the geometrical account of conceptual spaces based on the concept of Euclidean convexity. To put it bluntly, in Gärdenfors's account of conceptual spaces the logical, i.e., the syllogistic aspects of concept systems are virtually absent. These syllogistic aspects of concept systems are encapsulated in the traditional logical calculus dealing with connectives such as "AND", "OR", "NOT", etc. The geometrical approach, interested mainly in constructing discretizations of a conceptual space (by Voronoi tessellations or otherwise) has no means to adequately represent most of the classical logical operations on concepts defined for them from the time of Aristotelian syllogistics. This is due to the fact that these operations do not go well with convexity. Among the classical logical operations of concepts such as disjunction, conjunction, negation and others, only the conjunction of concepts has a well-behaved geometrical representation in conceptual spaces structured by convexity. That is, if A and B are concepts represented by convex regions of a conceptual space (X, co) the conjunction A & B of A and B is represented by the set-theoretical intersection of these regions. Other logical operations such as disjunction and negation of concepts cannot be represented in a plausible way by convex regions. The most obvious case is negation. Assume that a concept A is represented by a convex region of a conceptual space (X, co). For the sake of definiteness, take X to be the Euclidean plane endowed with standard Euclidean vector convexity (cf. Gärdenfors (2000, (2009), Douven (2019)). If only convex subsets of X are recognized as (natural) concepts the set-theoretical complement CA of A is not a concept since CA is usually not convex. The convex hull co(CA) does not work much better, since it is usually much larger than CA and has a non-trivial intersection with A. Disjunctions do not score better. The set-theoretical union A È B of convex sets A and scattered spaces with regular open atomic tessellations O*X, see Appendix B. 33 B is usually not convex. Hence, it cannot represent a concept. On the other hand, the convex hull of co(A È B) is usually too great to serve as a supremum of A and B, since it does not satisfy the law of distributivity: (4.4) A Ú (B Ù C) = (A Ú B) Ù (A Ú C) does not hold in the lattice Co(X) of convex subsets of X. In sum, the familiar classical logical connectives of concepts have no place in the framework of conceptual spaces defined with the aid of convexity operators. In contrast, the topological account of conceptual spaces has no problems with the Boolean logic of concepts. Every conceptual space X endowed with a topology OX comes along with a naturally defined regular open interpretation of the Boolean lattice O*X. Moreover, if (X, OX) is weakly scattered Alexandroff, then O*X is even atomic with distinguished isolated elements a Î ISO(X) as generators of its atoms.19 Let us conclude this section with some brief remarks on the relation between geometrically and topologically defined conceptual spaces. Very succinctly then, the relation between geometrically and topologically defined conceptual spaces may be expressed as follows. Starting from a "classical" geometrically defined conceptual space (endowed with Voronoi tessellation defined via prototypes and Euclidean convexity), one obtains a corresponding topologically defined polar space (X, OX) with the same set of prototypes and the same set of open cells. This set of cells topologically defines a regular atomic tessellation of (X, OX) whose elements are the atoms of the Boolean algebra O*X of the underlying 19 The fact that negations, disjunctions and other familiar combinations of concepts can be represented naturally in a topological framework should perhaps not be taken as the ultimate and definite argument that "not red", "red or blue or green" are natural concepts in exactly the same sense as "red", "blue", and "green" are natural concepts. But for every even minimally useful calculus of concepts negations and disjunctions of concepts are indispensable. An example treated in this paper in some detail is Rumfitt's solution of the Sorites paradox dealing with "non-red" etc. Thus, a theory of concepts that does not deal with the issue of logical connectives has to be assessed as seriously incomplete. In sum, I'd tend to answer the question (asked by a reviewer of an earlier version of this paper) whether the fact that Gärdenfors' account of conceptual spaces does not deal with negative, disjunctive and other combinations of concepts is to be judged as "a bug or a feature" in favor of the first option. 34 topological space (X, OX). Thus, in contrast to a conceptual space defined by a geometrical convexity, a conceptual space endowed with a topological structure comes with a readymade and well-behaved classical system of concepts, namely, the complete atomic Boolean algebra O*X. Moreover, a closer look at the construction of the conceptual space reveals that the full-fledged apparatus of Euclidean geometry is not necessary to construct a topological discretization of X. Rather, a more austere structure suffices (i.e., a pole distribution (X, m, P) in the sense of (2.1)). Like polar spaces weakly scattered Alexandroff spaces possess regular atomic tessellations that can be used to construct discretizations of conceptual spaces. As distinguished from polar spaces, for weakly scattered Alexandroff spaces, the dichotomy between prototypical and non-prototypical elements is replaced by a gradual distinction between "more" and "less prototypical" elements defined by the specialization order.20 Already Rumfitt's elucidation of the logic of vague concepts has shown that the very simple class of polar spaces offers a fruitful explication of many aspects of conceptual spaces. The more comprehensive class of weakly scattered Alexandroff spaces offers a sufficiently flexible framework for dealing with various aspects of conceptual discretization and categorization arising in cognitive science and related disciplines. 5. Towards a topological design theory of conceptual spaces. The basic assumption of the conceptual spaces approach is that concepts can be usefully represented as well-formed subsets of a conceptual space. In order to speak meaningfully about well-formedness requires that the space in question is structured in one way or another. Then, the basic task of this approach is to find appropriate structures that allow us to characterize empirically useful concepts as structurally well-formed subsets. Topological structures 20 As stated already in the introduction, such a gradual distinction has been assumed often more or less implicitly by many authors, nice examples can be found in the recent paper Osta-Vélez and Gärdenfors (2020). 35 have shown to be basic for all kinds of "spaces" that are used in numerous and variegated realms of knowledge (physics (of course), game theory, biology, neural sciences, economics, logic) (see for example Adams and Franzosa (2008), Curto (2017), Rabadán and Blumberg (2020)). Thus, it appears reasonable to expect that topological structures may also play a role in the theory of conceptual spaces. Topological concepts are flexible enough to be adapted to various empirical necessities. It is a matter of "theory-guided" empirical research to find out which topological structures for which types of conceptual spaces are the most useful ones. Topological structures are flexible enough to take into account a variety of criteria (which sometimes pull in opposite directions) that a system of "good," i.e., "natural concepts," should satisfy. This means that topology may be a helpful device for setting up a general "design theory" that aims to determine how "good" discretizations of conceptual spaces by "natural concepts" should look like. An account of such a theory has recently been put forward by Douven (2019) and Douven and Gärdenfors (2019). They present a list of desiderata for the design of conceptual spaces that "good" (or even "optimal") systems of concepts should satisfy. More precisely, according to these authors, a good conceptual system should satisfy the requirements of parsimony, informativeness, contrast, and learnability. These design principles may pull in different directions. Thus, the overall task is to find a kind of equilibrium or balance between the various principles. The following shows how Douven and Gärdenfors's list of design principles are accounted for by the topological approach (based on weakly scattered Alexandroff spaces): (5.1) Parsimony. The conceptual structure should not overload the system's memory. Topological response: Already elementary examples like the color spectrum show that the cardinality of the Boolean lattice O*X of regular open sets is much lower than the cardinality of the Heyting lattice OX of open sets. This is evidence that the choice of O*X (instead of OX) as the set of representatives of concepts is in line with the principle 36 of parsimony.  (5.2) Informativeness. Concepts should be informative, meaning that they should jointly offer good and roughly equal coverage of the domain of classification cases. Topological Response: According to the topological approach, the extensions intcl(p) of concepts jointly offer complete coverage of the domain of classification cases since O*X provides an atomic topological tessellation VpÎpintcl(p) of X. In a somewhat different vein, informativeness is expressed by the fact that the set P of prototypes is dense in X, i.e., cl(P) = X.  (5.3) Presentation. The conceptual structure should be such that it allows one to choose for each concept a prototype that is a good representative of all items falling under the concept. Topological response: According to the topological approach, the prototypes for all concepts are defined as isolated points of (X, OX) or equivalently as maximal elements of the specialization order (X, ≤).  (5.4) Contrast: The conceptual structure should be such that prototypes of different concepts can be so chosen such that they are easy to tell apart. Topological response: Different prototypes p and p' are topologically separated in the sense that their extensions intcl(p) and intcl(p') are separated, i.e., intcl(p) Ç intcl(p') = Ø.   (5.5) Learnability: The conceptual structure should be learnable, ideally from a small number of instances. Topological response: This can topologically be taken into account by requiring that for "good" conceptual spaces X, for each concept x Î X there is a path x < x1 < ... < xm from x to xm, and xm maximal with respect to the specialization order (X, ≤) of (X, OX). 37 For polar spaces this requirement is satisfied by very short paths of length 2 that have prototypes as their maximal elements. For more general spaces, for instance, topological products of polar spaces, the paths are longer and pass through intermediate elements that may be characterized as "more or less prototypical".21 This amounts to a more complex categorization than that which is determined by just one prototypical pole.  The task of designing a good (or even optimal) conceptual structure for a conceptual space may be understood as problem of conceptual engineering. Every engineer who wants to be taken seriously, has to take into consideration certain constraints that restrict the feasibility of his projects. Not everything that appears to be desirable or possible in principle from an ideal point of view, is practically feasible. This holds, in particular, for the cognitive scientist who aspires to design conceptual spaces that are to be used for building programs, robots, or other digital devices that are designed for accomplishing various cognitive tasks. To be specific, consider the growing importance of computer simulations in many areas of science. The conceptual spaces involved in this endeavor use to be digital spaces of one kind or another. Expressed in an extremely simplified way, this amounts to the replacement of continuous conceptual spaces (usually derived from Euclidean spaces Rn) by digital substitutes such as Zn or something similar. Thus, a comprehensive design theory of conceptual systems has to take into account the feasibility and even the necessity of discretizations on various levels. This is a desideratum that goes far beyond the level of concretization that Douven and Gärdenfors's design theory has achieved up to now. For instance, in (Douven (2019)) the author presents several examples of "conceptual structures for disc-shape similarity spaces" (Douven (2019, 125 127)). These examples, intuitively appealing 21 Detailed discussions about the essential role of "intermediate elements" in digital spaces may be found in Adams and Franzosa (2008, chapter 11.3), Melin (2008) and elsewhere. 38 and plausible as they may be, are not much more than intuitive illustrations or geometrical metaphors for design problems. In order to render them more precise and less metaphorical, the conceptual engineers have to take seriously the constraints that determine the architecture of conceptual spaces. One important constraint that gains more and more relevance in contemporary cognitive science and artificial intelligence is the recognition of the digital ("discrete") character of many conceptual spaces. The other side of this coin is the necessity to admit the limited relevance of considerations based on plausible intuitive Euclidean spaces. Perhaps the task of finding appropriate design criteria for conceptual systems can be compared with the task of finding in physics fruitful "relative priori principles" that characterize general features of theories concerning aspects such as continuity, probability, and the causality of laws (cf. Cassirer (1937), Friedman (2001)). For instance, in physics states spaces of systems have often been assumed a priori to satisfy certain geometrical constraints, e.g., the constraint of being Riemannian manifolds. This Riemannian constraint is mathematically highly non-trivial and not very intuitive, at least not for someone who is accustomed only to elementary Euclidean geometry. Analogously as in physics, appropriate design criteria for conceptual spaces, allow us to steer a middle course between arbitrary relativism that does not distinguish between better and worse systems and an overly rigid universalism that recognizes only one conceptual system as the one and only correct one: Contra relativism, this renders the notion of a natural concept non-arbitrary, while also allowing for the occurrence of (sometimes substantial) differences among conceptual systems used in different cultures. But, contra universalism, the best designed conceptual systems, and hence natural concepts, are not held to reflect some fundamental blueprint of the physical or mental world. (Douven and Gärdenfors (2019, 2-3)). 39 6. Concluding Remarks. In this paper, arguments have been put forward in favor of the thesis that conceptual spaces can be usefully endowed with the topological structures of weakly scattered Alexandroff spaces. Alexandroff topologies are an expedient conceptual device for addressing the important role that prototypes and paradigmatic elements play in human and non-human categorization. In the simplest case, this is evidenced already by polar spaces defined by pole distributions. Weakly scattered Alexandroff spaces are more general than polar spaces but inherit most of the useful properties exhibited by the former. In weakly scattered Alexandroff spaces the mathematical structures of order, algebra, and topology are interwoven in an intricate manner that can be used to tackle a variety of problems that arise for issues of conceptualization and categorization. Topological structures are fundamental spatial structures and arguably even the most fundamental ones. Thus, if one subscribes to Gärdenfors's thesis that "to understand the structure of our thoughts ... we should aim at unveiling our conceptual spaces" (Gärdenfors (2000, 262)), one should invest some effort in understanding the topological structures of conceptual spaces. APPENDIX A: ELEMENTS OF TOPOLOGY. For the reader's convenience, this appendix lists some basic definitions and facts of topology that are used in this paper. (A.1). Definition. Let X be a set with power set 2X. A topological space (X, OX) is a relational structure with OX Í 2X satisfying the following: (i) Ø, X Î OX; (ii) Finite intersections and arbitrary unions of elements of OX are elements of OX. The elements of OX are called open sets of (X, OX). The set-theoretical complements of 40 open sets are called closed sets.22 As usual, when there is no danger of confusion, a topological space (X, OX) is simply denoted by X. (iii) A topological space (X, OX) is an Alexandroff space iff arbitrary intersections (unions) of open (closed) sets are open (closed).  If X has more than one element, then different topological structures exist on X. In particular, there are two extreme topological structures (X, O0X) and (X, O1X) defined by O0X := {Ø, X} and O1X := 2X. The topology (X, O0X) is called the indiscrete topology on X, and the topology (X, O1X) is called the discrete topology. With respect to set-theoretical inclusion all topological structures (X, OX) on X lie between these two (rather uninter– esting) extremal topologies: O0X Í OX Í O1X. A topology OaX is coarser than a topology ObX iff OXa Í ObX. Equivalently, ObX is said to be finer than OaX. Thus, O0X is the coarsest topology on X, and O1X is the finest topology on X.  The following definition collects some standard methods to construct new topologies from old ones: (A.2). Definition. Let (X, OX) and (Y, OY) be two topological spaces. Recall that a (settheoretical) map X3⁄43⁄4f3⁄43⁄4>Y is continuous iff f-1(OY) Í OX. (i) The product topology O(X   Y) on X   Y is the finest topology such that the projections X   Y 3⁄43⁄4pX3⁄43⁄4>X and X   Y 3⁄43⁄4pY3⁄43⁄4>Y are continuous with respect to O(X   Y) and O(X) and O(X   Y) and O(Y), respectively. (ii) Let Z3⁄43⁄4i3⁄4>X be an inclusion map of a subset Z Í X. If (X, OX) is a topological space, the induced topological structure (Z, OZ) is the coarsest topology on Z such that the map i is continuous, i.e., OZ = Z Ç OX. (iii) Let ~ be an equivalence relation on X and X3⁄43⁄4q3⁄43⁄4>X/~ the canonical quotient map. The quotient topology OX/~ on X/~ is the finest topology such that q is continuous.  Topological structures (X, OX) can be defined in many equivalent ways. For our purposes, particularly useful is a definition in terms of closure operators cl or interior kernel operators int. These operators must satisfy the so-called Kuratowski axioms: 22 A set may be open and closed. For instance, the sets Ø and X are open and closed for all topological structures (X, OX). Sets that are open and closed, are sometimes called clopen. A topological space is called connected iff Ø and X are the only clopen subsets of X. 41 (A.3). Definition. A topological closure operator is an operator 2X3⁄43⁄4cl3⁄43⁄4>2X satisfying the four requirements (i) – (iv) below. Dually, a topological interior kernel operator is a map 2X3⁄43⁄4int3⁄43⁄4>2X satisfying requirements (i)* - (iv)*: (i) cl(A È B) = cl(A) È cl(B) (i)* int(A Ç B) = int(A) Ç int(B). (ii) cl(cl(A)) = cl(A). (ii)* int(int(A)) = int(A). (iii) A Í cl(A). (iii)* int(A) Í A. (iv) cl(Ø) = Ø. (iv)* int(X) = X. The requirements (i), (ii), (iii) and (iv) are sometimes called (i) distributivity, (ii) idempotence, (iii) extension, and (iv)(normality). Closure operators and interior kernel operators are interdefinable: Denoting the set-theoretical complement of A by CA, one obtains cl(A) = Cint(CA)) and int(A) = Ccl(CA)). Every topological closure operator cl uniquely defines a topological structure (X, OX) and vice versa. Given cl, the class of open sets OX is defined by OX := {B; B = Ccl(A); A Í X}. Dually, given a topological interior kernel operator int, the corresponding topological structure OX is defined by OX := {A; A = int(A), A Í X}. For A Í X, the boundary bd(A) of A is defined as bd(A) := cl(A) Ç cl(CA) = C(int(A) È int(CA)). Moreover, it is possible to define cl (and int) in terms of bd.  Topological closure operators are only one of many different types of closure operators used in mathematics. As is well known, also the concept of convexity may be defined in terms of closure operators: (A.4). Definition. A convex closure operator (or convexity) on a set X is defined as an operator 2X 3⁄43⁄4co3⁄43⁄4>2X that satisfies the following requirements: (i) A Í B Þ co(A) Í co(B). (ii) co(co(A)) = co(A). (iii) A Í co(A). (iv) co(Ø) = Ø. (v) For all y Î co(A) there is a finite set F Í A such that y Î co(F). (Algebraicity). A set A is called convex with respect to the operator co iff co(A) = A. The convex operator co is of arity ≤ n provided its convex sets are precisely the sets A with the property that 42 co(F) Í A for each F Í A with cardinality #F ≤ n.  The familiar Euclidean convexity is a convex operator of arity 2.23 More interesting is the observation that the topological closure operator cl of an Alexandroff space (X, OX) is also a convex closure operator in the sense of (A.4): By definition a set A is closed in the Alexandroff topology iff A =  A. Then x Î  A iff there is an a Î A with x ≤ a. In other words, x Î  a = co(a). Thus, one may choose F(x) = {a} as the finite set F(x) with F(x) Í A and x Î co(F(x)). Hence, the "lower convexity", defined by the Alexandroff topological operator, is of arity 1. In other words, a conceptual space (X, OX) endowed with an Alexandroff topological structure OX (defined by the operator cl) may be conceived as a conceptual space endowed with a convex structure (defined as well by cl).24 Admittedly, this "Alexandroff convexity" is rather different from the Euclidean convexity that most conceptual spaces are assumed to be endowed with. Nevertheless, this fact suggests that Alexandroff's and the Gärdenfors's approaches to conceptual spaces are not totally alien to each other. Quite often, a topological structure (X, OX) and a convex structure (X, co) co-exist on the same set X. Prominent example are the Euclidean spaces Rn. In this situation it is expedient to require appropriate compatibility conditions of the two structures (for details see van 23 Indeed, all convexity operators that Gärdenfors considers in his many contributions to the approach of conceptual spaces are of arity 2. More precisely, his basic primitive concept is a ternary relation B(x, y, z) of elements x, y, and z of a conceptual space C. The relation B(x, y, z) is to be read as "y is between x and z". Of course, B has to satisfy certain intuitively plausible conditions in order to be accepted as a "good" definition of betweenness. For any two points x and z the relation B defines a subset [x, z] of elements between x and z, i.e., [x, z] := {y; B(x, y, z)}. Then a set A Í C is defined as convex with respect to B iff x, z Î A entails [x, z] Í A. This defines a convex closure operator co for which A.3(5) is just the requirement that there is F(y) = B(x, y, z) for some x, z Î A. 24 The standard convex operators of Euclidean spaces and the topological operators of Alexandroff spaces are both convex closure operators. Thus, they can be treated as two cases of a general theory of convex structures (cf. van de Vel (1993)). This suggests that topology and convexity, used as devices for structuring conceptual spaces should be considered as special cases of the more general approach of an approach that conceives conceptual spaces as closure structures. 43 de Vel (1993, ch. III). (A.5). Proposition. Let (X, OX) be a topological space. An open subset A Î OX is regular open iff A = int(cl(A)). The set of all regular open subsets of X is denoted by O*X. O*X is well known to be a complete Boolean algebra. There is a map OX3⁄43⁄4j3⁄43⁄4>O*X defined by j(A) := int(cl(A)) and an inclusion map O*X3⁄4i3⁄4>OX such that j • i = idO*X and idOX Í i • j.  (A.6). Definition. Let (X, OX) be a topological space. (i) An element x Î X is isolated iff {x} Î OX. The set of isolated points of X is denoted by ISO(X). (ii) A subset A Í X is dense in X iff cl(A) = X. (iii) The space (X, OX) is weakly scattered iff ISO(X) is dense in X, i.e., cl(ISO(X)) = X. (iv) The space (X, OX) satisfies the McKinsey axiom iff int(cl(A)) Í cl(int(A)) for all A Í X.  (A.7). Proposition. Let (X, OX) be a topological space, A, B Í X. The following assertions are equivalent: (i) X is weakly scattered. (ii) X satisfies the McKinsey axiom int(cl(A)) Í cl(int(A)). (iii) For all A Í X the boundary bd(A) of satisfies bd(bd(A)) = bd(A). (iv) If A and B are dense in X, then A Ç B is dense in X. This proposition is a part of the stronger propositions (2.1) and (2.4) of (Bezhanishvili, Mines, Morandi (2003)).  (A.8). Definition (Specialization quasi-order of a topology). Let X be a set. A quasi-order on X is a binary relation ≤ such that for all x, y, z Î X, the following conditions (i) and (ii) are satisfied: (i) x ≤ x. (Reflexivity) (ii) x ≤ y and y ≤ z implies x ≤ z. (Transitivity) 44 (iii) If also x ≤ y and y ≤ x implies x = y is satisfied the quasi-order ≤ is said to be a partial order, and the structure (X, ≤) is called a poset. (iv) A subset C of a partial order (X, ≤) is a chain iff all elements x, y Î C are comparable, i.e., x ≤ y or y ≤ x. If C is a finite chain in X with #C = n +1, the length of C is n. The partial order (X, ≤) is said to have finite depth, if the length of the longest chain in X is n. A topological space (X, OX) defines a quasi-order (X, ≤) by x ≤ y := x Î cl(y). This quasiorder is called the specialization quasi-order of (X, OX). The set of maximal elements of (X, ≤) with respect to this order is denoted by MX.  For many traditional topological spaces such as Euclidean spaces (E, OE), the specialization order (E, ≤) is trivial, i.e., x ≤ y iff x = y, or, equivalently, iff MX = X. In contrast, for nontrivial Alexandroff spaces (X, OX) the specialization quasi-order is non-trivial, i.e., MX ≠ X. (A.9). Proposition (Upper topology defined by a quasi-order (X, ≤)). Let (X, ≤) be quasiorder. For A Í X, define the upper set of A by -A := {x; a ≤ x for some a Î A}. The upper topology (X, OX) corresponding to (X, ≤) is defined by OX := {-A; A Í X}. The set X endowed with the upper topology of the quasi-order (X, ≤) is an Alexandroff topological space (X, OX).25  (A.10). Proposition. Let (X, OX) be an Alexandroff space with the specialization quasiorder (X, ≤). Then, the upper topology of X defined by (X, ≤) is isomorphic to (X, OX). In other words, an Alexandroff space (X, OX) is completely determined by its specialization quasi-order (X, ≤). An element a Î X is maximal with respect to the specialization order if and only if a Î ISO(X), i.e., {a} Î OX.  (A.11). Separation axioms. Let (X, OX) be topological space. (i) X is a T0-space iff for every x Î X and every y ≠ x there exists an open set A Î OX such that either x Î A and y Ï A or x Ï A and y Î A. 25 Analogously, the lower set  A of A Í X is defined by  A := {y; y ≤ a for some a Î A}. Thereby, the so-called lower topology of (X, ≤) is defined as the set of all lower sets { A; A Í X}. In this paper, however, there is no need to consider this topology of (X, ≤). 45 (ii) X is a T1/2-space iff every point x Î X is either open or closed. (iii) X is a T1-space iff every point x Î X is closed. (iv) (X, OX) is a T2-space (or Hausdorff space) iff for distinct points x and y there are open sets A Î OX and B Î OX containing x and y such that x Ï A and y Ï B. (v) The separation axioms T0 – T2 satisfy a chain of proper implications: T2 Þ T1 Þ T1/2 Þ T0.  The following examples show that Euclidean and Alexandroff spaces behave quite differently with respect to separation axioms: (A.12). Examples. (i) The standard Euclidean topology OR of the real line R is generated by open intervals (a, b) = {x; a < x < b}. Two distinct points x and y can be separated by open intervals U(x) and U(y), which are disjoint from each other. Hence, (R, OR) is a T2–space. A fortiori, all points are closed, and no point is open. (ii) Let (N, ≤) be the set of natural numbers endowed with their natural order ≤. A topological space (N, ON) is defined by stipulating that Ø and the sets -n := {m; n ≤ m} are open for each n Î N. Then (N, ON) is an Alexandroff space that satisfies T0 but not T1/2. No point of (N, ON) is open, and the only closed point of (N, ON) is 0. (iii) The Khalimsky line (Z, OZ) (as a polar space) satisfies the axiom T1/2 but not T1. (iv) The Khalimsky plane (Z   Z, OZ   Z) satisfies T0 but not T1/2. The even points (2m, 2n) Î Z   Z are closed, the odd points (2m+1, 2n+1) Î Z   Z are open, and the "mixed points" (2m, 2n+1), (2m +1, 2n) Î Z   Z are neither open nor closed.  (A.13). Proposition. (i) An Alexandroff space (X, OX) satisfies T1 iff it is discrete, i.e., OX = 2X. (ii) A topological space (X, OX) is a T0-Alexandroff space iff its specialization quasi-order (X, ≤) is a partial order.26 26 In classical topology, the separation axiom T1 is considered a minimal requirement that must be satisfied for a topological space to be considered reasonable. (A.13)(i) shows that Alexandroff spaces fall outside the realm of classical theory: Only trivial (discrete) Alexandroff spaces are T1, and some important Alexandroff spaces such as the digital plane (Z   Z, O(Z   Z)) fail to be T1/2. 46 (iii) Let (X, OX) be an Alexandroff space with specialization quasi-order (X, ≤). Define an equivalence relation on X by x ~ y := x ≤ y and y ≤ x. Then (X/~, OX/~) is a T0-Alexandroff space.  Appendix B. (EXAMPLES OF ALEXANDROFF SPACES). (B.1). All finite topological spaces (X, OX) are Alexandroff space with O*X atomic. The Sierpinski space (X, OX) with X = {a, b}, OX = {Ø, {a}, {a, b}} is weakly scattered. In general, finite topological are not weakly scattered. The smallest example is the space X = {a, b, c} with topology OX = {Ø, {a, b}, {c}, {a, b, c}}. The only isolated point of (X, OX) is {c}, but {c} is not dense in (X, OX), since clearly cl(c) = {c}. Hence, (X, OX) is not weakly scattered. Nevertheless, O*X is atomic, namely, the Boolean algebra with four elements {Ø, {a, b}, {c}, {a, b, c}}, generated by the atoms {a, b} and {c}. (B.2). Polar spaces X provide the simplest class of Alexandroff spaces that may have infinitely many elements. Examples treated in detail include the linear color spectrum and the circular color spectrum (color circle) with finitely many poles but infinitely many shades of colors. Perhaps the most important example of a polar space with "real-world"-applications is the Khalimsky line (or digital line) (Z, OZ) with pole distribution (Z, m, 2Z+1) (cf. 2.X). Indeed, the Khalimsky line may be considered as the "foundation of digital topology" (cf. Kopperman (1994)). (B.3). Finite products of polar spaces are weakly scattered Alexandroff spaces. More generally, finite products of weakly scattered Alexandroff spaces are weakly scattered Alexandroff spaces. (B.4). Not all Alexandroff spaces with infinitely many elements are weakly scattered. An example is the standard linear order (N, ≤) as a specialization order of the corresponding 47 Alexandroff topology (N, ON). As is easily observed, the set of isolated points of this space is empty. Hence, (N, ON) is not weakly scattered. Nevertheless, the Boolean algebra O*N is atomic, namely, the unique Boolean algebra of 2 elements generated by the atom N. (B.5). More generally, there are infinite trees (X, ≤) whose Alexandroff topologies (X, OX) have regular open lattices O*X that the atomic Boolean algebras 2n, n = 1, 2, ... Just take X to be the disjoint union of n copies of N endowed with the natural order. Identify the minimal elements ("0") of all copies of N with each other. The result is a tree with n infinite linear branches. Clearly, (X, OX) is not weakly scattered, because X has no isolated points at all. However, the Boolean algebra O*X of regular open sets of the Alexandroff space (X, OX) is the atomic Boolean algebra 2n generated by the regular open upper sets -x1, ..., -xn, where each xi generates a branch of the tree as its open hull-xi. (B.6). Not all Alexandroff spaces are regular atomic. An example is given by the specialization order (X, ≤) of the infinite binary tree.27 Let X be the set of finite 0-1sequences (e1, ..., en), ei = 0, 1, endowed with the following partial order: For x, y Î	X one has x ≤ y := x is an initial subsequence of y, i.e. y = (x, z). Then (X, ≤) is an infinite binary tree with root the empty sequence Ø. The two children of Ø are the sequences (0) and (1), the children of (0) and (1) are (0, 0) and (0,1), and (1, 0) and (1, 1), respectively, and so on. For all x Î X, the subtrees -x are regular open subsets of (X, OX). They are not atomic since for any x, one may find x < x'< x'' < ... such that -x É -x' É -x'' É .... One then obtains infinite strictly decreasing sequences of regular open elements of O*X.  27 I owe this example to Imanol Mozo. 48 (B.7). There are weakly scattered spaces (X, OX) with atomic O*X that are not Alexandroff: Let (R, OR) be the set of real numbers R endowed with the topology engendered by the standard Euclidean topology and the elements of the set Q of rational numbers. Then, the rationals are isolated points of (R, OR) such that cl(Q) = R since every open neighborhood U(s) of an irrational number s contains a rational number q Î Q. The singletons {s} of irrational numbers s Î R – Q are closed but not open: if {s} were open, s Î intcl(s) Í intcl(R – Q). However, clearly intcl(R – Q) = int(R – Q) = Ø, since Q is open in this topology. Hence, {s} is closed but not open. This shows that (R, OR) is not Alexandroff, since the intersection {s} of the open neighborhoods of s is not open. The Boolean lattice O*R is atomic, since clearly O*R = PQ due to the fact that the only atomic elements of O*R are singletons {q} with q Î Q.  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