Working in constructive set theory we formulate notions of constructive topological space and set-generated locale so as to get a good constructive general version of the classical Galois adjunction between topological spaces and locales. Our notion of constructive topological space allows for the space to have a class of points that need not be a set. Also our notion of locale allows the locale to have a class of elements that need not be a set. Class sized mathematical structures (...) need to be allowed for in constructive set theory because the powerset axiom and the full separation scheme are necessarily missing from constructive set theory. We also consider the notion of a formal topology, usually treated in Intuitionistic type theory, and show that the category of set-generated locales is equivalent to the category of formal topologies. We exploit ideas of Palmgren and Curi to obtain versions of their results about when the class of formal points of a set-presentable formal topology form a set. (shrink)

Homologous operational localization processes are effectuated in terms of generalized topological covering systems on structures of physical events. We study localization systems of quantum events' structures by means of Gtothendieck topologies on the base category of Boolean events' algebras. We show that a quantum events algebra is represented by means of a Grothendieck sheaf-theoretic fibred structure, with respect to the global partial order of quantum events' fibres over the base category of local Boolean frames.

In 1906, Frigyes Riesz introduced a preliminary version of the notion of a topological space. He called it a mathematical continuum. This development can be traced back to the end of 1904 when, genuinely interested in taking up Hilbert’s foundations of geometry from 1902, Riesz aimed to extend Hilbert’s notion of a two-dimensional manifold to the three-dimensional case. Starting with the plane as an abstract point-set, Hilbert had postulated the existence of a system of neighbourhoods, thereby introducing the notion of (...) an accumulation point for the point-sets of the plane. Inspired by Hilbert’s technical approach, as well as by recent developments in analysis and point-set topology in France, Riesz defined the concept of a mathematical continuum as an abstract set provided with a notion of an accumulation point. In addition, he developed further elementary concepts in abstract point-set topology. Taking an abstract topological approach, he formulated a concept of three-dimensional continuous space that resembles the modern concept of a three-dimensional topological manifold. In 1908, Riesz presented his concept of mathematical continuum at the International Congress of Mathematicians in Rome. His lecture immediately won the attention of people interested in carrying on his research. They promoted his ideas, thus assuring their gradual reception by several future founders of general topology. In this way, Riesz’s work contributed significantly to the emergence of this discipline. (shrink)

Elementary submodels of some initial segment of the set-theoretic universe are useful in order to prove certain theorems in general topology as well as in algebra. As an illustration we give proofs of two theorems due to Arkhangelskii concerning cardinal invariants of compact spaces.

In this paper, I present a general theory of topological explanations, and illustrate its fruitfulness by showing how it accounts for explanatory asymmetry. My argument is developed in three steps. In the first step, I show what it is for some topological property A to explain some physical or dynamical property B. Based on that, I derive three key criteria of successful topological explanations: a criterion concerning the facticity of topological explanations, i.e. what makes it true of a particular (...) system; a criterion for describing counterfactual dependencies in two explanatory modes, i.e. the vertical and the horizontal; and, finally, a third perspectival one that tells us when to use the vertical and when to use the horizontal mode. In the second step, I show how this general theory of topological explanations accounts for explanatory asymmetry in both the vertical and horizontal explanatory modes. Finally, in the third step, I argue that this theory is universally applicable across biological sciences, which helps to unify essential concepts of biological networks. (shrink)

The logical theory of branching space-times, which provides a relativistic framework for studying objective indeterminism, remains mostly disconnected from discussions of space-time theories in philosophy of physics. Earman has criticized the branching approach and suggested “pruning some branches from branching space-time.” This article identifies the different—order-theoretic versus topological—perspective of both discussions as a reason for certain misunderstandings and tries to remove them. Most important, we give a novel, topological criterion of modal consistency that usefully generalizes an earlier criterion, and we (...) introduce a differential-geometrical version of branching space-times as a non-Hausdorff (generalized) manifold. (shrink)

Sierpinski networks are networks of fractal nature having several applications in computer science, music, chemistry, and mathematics. These networks are commonly used in chaos, fractals, recursive sequences, and complex systems. In this article, we compute various connectivity polynomials such as M -polynomial, Zagreb polynomials, and forgotten polynomial of generalized Sierpinski networks S k n and recover some well-known degree-based topological indices from these. We also compute the most general Zagreb index known as α, β -Zagreb index and several other (...) general indices of similar nature for this network. Our results are the natural generalizations of already available results for particular classes of such type of networks. (shrink)

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent (...) topological reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness. (shrink)

In the natural sciences higher order structures often occur. There seems to be a need for good methods of describing what we mean by higher order structures in various contexts. This is what hyperstructures are intended to do. We motivate and introduce this new concept. Next we illustrate how it can be applied in various types of genomic analysis—particular the correlations between single nucleotide polymorphisms and diseases. The suggested structure is quite general and may be applied to a variety (...) of situations. Finally we discuss how data sets (f. ex. genomic) may lead to topological spaces, giving new invariants and lead to the prediction of hyperstructures. (shrink)

The topology optimization of computer communication network is studied based on improved genetic algorithm, a network optimization design model based on the establishment of network reliability maximization under given cost constraints, and the corresponding improved GA is proposed. In this method, the corresponding computer communication network cost model and computer communication network reliability model are established through a specific project, and the genetic intelligence algorithm is used to solve the cost model and computer communication network reliability model, respectively. It (...) has been proved that GA can solve the complex problems of computer working environment better, which is 80% higher than the general algorithm, and can select the optimal scheme pertinently. (shrink)

Hobbes emphasized that the state of nature is a state of war because it is characterized by fundamental and generalized distrust. Exiting the state of nature and the conflicts it inevitably fosters is therefore a matter of establishing trust. Extant discussions of trust in the philosophical literature, however, focus either on isolated dyads of trusting individuals or trust in large, faceless institutions. In this paper, I begin to fill the gap between these extremes by analyzing what I call the (...) class='Hi'>topology of communities of trust. Such communities are best understood in terms of interlocking dyadic relationships that approximate the ideal of being symmetric, Euclidean, reflexive, and transitive. Few communities of trust live up to this demanding ideal, and those that do tend to be small (between three and fifteen individuals). Nevertheless, such communities of trust serve as the conditions for the possibility of various important prudential epistemic, cultural, and mental health goods. However, communities of trust also make possible various problematic phenomena. They can become insular and walled-off from the surrounding community, leading to distrust of out-groups. And they can lead their members to abandon public goods for tribal or parochial goods. These drawbacks of communities of trust arise from some of the same mecha-nisms that give them positive prudential, epistemic, cultural, and mental health value – and so can at most be mitigated, not eliminated. (shrink)

We show that a standard axiomatization of mereology is equivalent to the condition that a topological space is discrete, and consequently, any model of general extensional mereology is indistinguishable from a model of set theory. We generalize these results to the Cartesian closed category of convergence spaces.

How do concepts of topology such as that of a boundary apply to the empirical world? Take the example of a chess board, represented here with black squares in black and red squares in white. We see by looking at the board that the squares of any one color have common boundaries only with squares of the opposite color, but each square has corners in common with other squares of the same color, which are points at which their common (...) boundaries intersect. For example, the white square labelled ‘A’ has common boundaries with the black squares that surround it, and common corners with the white squares like square B that are diagonally adjacent to it. (shrink)

Dynamic Topological Logic ( ) is a modal logic which combines spatial and temporal modalities for reasoning about dynamic topological systems , which are pairs consisting of a topological space X and a continuous function f : X → X . The function f is seen as a change in one unit of time; within one can model the long-term behavior of such systems as f is iterated. One class of dynamic topological systems where the long-term behavior of f is (...) particularly interesting is that of minimal systems ; these are dynamic topological systems which admit no proper, closed, f -invariant subsystems. In such systems the orbit of every point is dense, which within translates into a non-trivial interaction between spatial and temporal modalities. This interaction, however, turns out to make the logic simpler, and while s in general tend to be undecidable, interpreted over minimal systems we obtain decidability, although not in primitive recursive time; this is the main result that we prove in this paper. We also show that interpreted over minimal systems is incomplete for interpretations on relational Kripke frames and hence does not have the finite model property; however it does have a finite non-deterministic quasimodel property. Finally, we give a set of formulas of which characterizes the class of minimal systems within the class of dynamic topological systems, although we do not offer a full axiomatization for the logic. (shrink)

We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of “admissible topologies” with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an (...) interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Qp is definably isomorphic to a definable group. (shrink)

Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, from the central (...) position it used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)

Different characterizations of classes of shift dynamical systems via labeled digraphs, languages, and sets of forbidden words are investigated. The corresponding naming systems are analyzed according to reducibility and particularly with regard to the computability of the topological entropy relative to the presented naming systems. It turns out that all examined natural representations separate into two equivalence classes and that the topological entropy is not computable in general with respect to the defined natural representations. However, if a specific labeled (...) digraph representation - namely primitive, right-resolving labeled digraphs - of some class of shifts is considered, namely the shifts having the specification property, then the topological entropy gets computable. (shrink)

Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In (...) fact, many formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined. (shrink)

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing (...) the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor generating p-points which are k-arrow but not \-arrow, and in a partial order of Blass producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. (shrink)

It is a celebrated result of McKinsey and Tarski [28] thatS4is the logic of the closure algebraΧ+over any dense-in-itself separable metrizable space. In particular,S4is the logic of the closure algebra over the realsR, the rationalsQ, or the Cantor spaceC. By [5], each logic aboveS4that has the finite model property is the logic of a subalgebra ofQ+, as well as the logic of a subalgebra ofC+. This is no longer true forR, and the main result of [5] states that each connected (...) logic aboveS4with the finite model property is the logic of a subalgebra of the closure algebraR+.In this paper we extend these results to all logics aboveS4. Namely, for a normal modal logicL, we prove that the following conditions are equivalent: (i)Lis aboveS4, (ii)Lis the logic of a subalgebra ofQ+, (iii)Lis the logic of a subalgebra ofC+. We introduce the concept of a well-connected logic aboveS4and prove that the following conditions are equivalent: (i)Lis a well-connected logic, (ii)Lis the logic of a subalgebra of the closure algebra$\xi _2^ + $over the infinite binary tree, (iii)Lis the logic of a subalgebra of the closure algebra${\bf{L}}_2^ + $over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logicLaboveS4is connected iffLis the logic of a subalgebra ofR+, and transfer our results to the setting of intermediate logics.Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics forS4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames forS4and descriptive spaces are isomorphic. It follows that every logic aboveS4is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of bothCandR. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients ofL2is an interior image ofR. (shrink)

We modify the semantics of topological modal logic, a language due to Moss and Parikh. This enables us to study the corresponding theory of further classes of subset spaces. In the paper we deal with spaces where every chain of opens fulfils a certain finiteness condition. We consider both a local finiteness condition relevant to points and a global one concerning the whole frame. Completeness of the appearing logical systems, which turn out to be generalizations of the well-known modal system (...) G, can be obtained in the same manner as in the case of the general subset space logic. It is our main purpose to show that the systems differ with regard to the finite model property. (shrink)

The standard presentation of topological spaces relies heavily on (naïve) set theory: a topology consists of a set of subsets of a set (of points). And many of the high-level tools of set theory are required to achieve just the basic results about topological spaces. Concentrating on the mathematical structures, category theory offers the possibility to look synthetically at the structure of continuous transformations between topological spaces addressing specifically how the fundamental notions of point and open come about. As (...) a byproduct of this, one may look at the different approaches to topology from an external perspective and compare them in a unified way. Technically, the category of sober topological spaces can be seen as consisting of (co)algebraic structures in the exact completion of the elementary category of sets and relations. Moreover, the same abstract construction of taking the exact completion, when applied to the category of topological spaces and continuous functions produces an extension of it which is cartesian closed. In other words, there is one general mathematical construction that, when applied to a very elementary category, generates the category of topological spaces and continuous functions, and when applied to that category produces a very suitable category where to deal with all sorts functions spaces. Yet, via such free constructions it is possible to give a new meaning to Marshall Stone’s dictum: “always topologize” as the category of sets and relations is the most natural way to give structure to logic and the category of topological spaces and continuous functions is obtained from it by a good mix of free – i.e. syntactic – constructions. (shrink)

In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure (...) and the latter the archetypal stable structure. In this sense we try here to situate our work ono-minimal structures [PS] in a general topological context. Note, however, that thep-adic numbers, and structures definable therein, will also fit into our analysis.In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions. (shrink)

The topological semantics for modal logic interprets a standard modal propositional language in topological spaces rather than Kripke frames: the most general logic of topological spaces becomes S4. But other modal logics can be given a topological semantics by restricting attention to subclasses of topological spaces: in particular, S5 is logic of the class of almost discrete topological spaces, and also of trivial topological spaces. Dynamic Topological Logic interprets a modal language enriched with two unary temporal connectives, next and (...) henceforth. DTL interprets the extended language in dynamic topological systems: a DTS is a topological space together with a continuous function used to interpret the temporal connectives. In this paper, we axiomatize four conservative extensions of S5, and show them to be the logic of continuous functions on almost discrete spaces, of homeomorphisms on almost discrete spaces, of continuous functions on trivial spaces and of homeomorphisms on trivial spaces. (shrink)

The article is devoted to the nature of the topological intuition and disclosure of the specifics of topological heuristics in the framework of philosophical theory of knowledge. As we know, intuition is a one of the support categories of the theory of knowledge, the driving force of scientific research. Great importance is mathematical intuition for the solution of non-standard problems, for which there is no algorithm for such a solution. In such cases, the mathematician addresses the so-called heuristics, built on (...) the basis of guesswork, obtained by intuition. The author substantiates the conclusion that topological intuition significantly specific compared to a traditional mathematical intuitions of Euclidean geometry. Today topology is a rapidly developing field of modern mathematics, integrates nicely with other sections of mathematical science. In its most general form of the topology can be defined as the branch of mathematics that studies the properties of spatial figures, does not change under deformations. The topological intuition is an instrument for development of topology on the basis of typological heuristics, which is the result of applying topological intuition to the objects topology. The author demonstrates in detail providing with the examples the specificity of topological heuristics and establishes its interconnection with Euclidean geometry. The author draws the conclusion about the fundamentality of topological intuition, and that it, perhaps, is primary in relation to traditionally understood mathematical intuition. (shrink)

In this paper I will try to argue for a new version in philosophy entitled as Philosophical Topology. It is inspired by the thought of Peter Strawson as well as ones of some of so-called Continental philosophers like Heidgger. Unlike any of metaphilosophy in general, the philosophical topology focuses rather on analyses of processes of make-up in philosophers’ thinking, especially by revealing the internal logic of philosophical ideas in making and processing in order to explain the intrinsic (...) continuation of philosophical ideas in particular philosophers’ thinking. In this sense the philosophical topology is not one of philosophical methods but a new branch in philosophy that is characterized as being concerned with continuation of ideas as its basic task. So, in this way, it is available to analyses of various doctrines in philosophy. Main issues in the philosophical topology are as follows: 1) to analyze the internal law-likes in the development of philosophy with the perspective of topology; 2) to interpret the general routes of Western philosophy in terms of the philosophical topology; 3) to view the philosophical topology as a way to inquire into metaphysics; 4) to deal properly with the relation of the philosophical topology to other branches in contemporary philosophy such as the philosophy of language, of logic, of science and ethics. (shrink)

We investigate several ideal versions of the pseudointersection number \(\mathfrak {p}\), ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant \(\mathtt {cov}^*({\mathcal I})\) has a crucial influence on the studied notions. For an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal J})\) introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant \(\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J})\) introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have $$\begin{aligned} \min \{\mathfrak (...) {p}_\mathrm {K}({\mathcal I}),\mathtt {cov}^*({\mathcal I})\}=\mathfrak {p},\qquad \min \{\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}\le \mathtt {cov}^*({\mathcal I}), \end{aligned}$$ respectively. In addition to the first inequality, for a slalom invariant \(\mathfrak {sl_e}({\mathcal I},{\mathcal J})\) introduced in Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that $$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathfrak {sl_e}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}=\mathfrak {p}. \end{aligned}$$ Finally, we obtain a consistency that ideal versions of the Fréchet–Urysohn property and the strictly Fréchet–Urysohn property are distinguished. (shrink)

To generalize as in [7] the constructions of [2] and [5], let L be a nondegenerate join-semilattice with unit. With A · B = {x ∨ y ∈ L : x ∈ A, y ∈ B} and A⊥ = {y ∈ L : x ∨ y = 1 for all x ∈ A}, the structure , ·,∪, ⊥ , L, ∅) is the algebra of subsets for L. Let R be the maximal ideals of L. Interpreting L as the totality (...) of elementary situations, and 1 ∈ L as the impossible one , the members of R will be called realizations, and R itself – the logical space associated with L. (shrink)

Dynamic Topological Logic ( $\mathcal{DTL}$ ) is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function. In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to (...) be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on $\mathbb{Q}$ . We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space. (shrink)

The general aim of this paper is to introduce some ideas of the theory of infinite topological games into the philosophical debate on supertasks. First, we discuss the elementary aspects of some infinite topological games, among them the Banach-Mazur game.Then it is shown that the Banach-Mazur game may be conceived as a Newtonian supertask.In section 4 we propose to conceive physical experiments as infinite games. This leads to the distinction between determined and undetermined experiments and the problem of how (...) it is related to that between determinism and indeter-minism. Finally the role of the Axiom of Choice as a source of indetermi-nacy of supertasks is discussed. (shrink)

Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.

This paper sets out some new skeleton mathematically-couched models for dealing with supervenience in some, if not all, its many guises. Our models are based around a naïve invocation of a ‘topology’ induced on object sets by property sets. We have two aims: one is to provide an overview of supervenience with enough rigour and detail to act as a self-contained introduction to the subject; and the other is to set out our new approach—but without getting too bogged down (...) in excessive mathematical, or indeed philosophical, niceties—so that it can be used as a foundation for further work. Throughout we have scattered examples and comments to help clarify the points we make. The early sections use our topological approach to recast what appears elsewhere in the literature. And in the latter sections of the paper, we introduce some new ideas and techniques which help in exploring the general structures underpinning supervenience. (shrink)

We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.