There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou’s probabilistic bisimilarity, whereas the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator that there is such a process with an uncountable Zhou ordinal.

We investigate separation properties for neighbourhood spaces in some details within a framework of constructive mathematics, and define corresponding separation properties for quasi-apartness spaces. We also deal with separation properties for spaces with inequality.

In his book, John McDowell on Worldly Subjectivity, Tony Cheng argues that recent changes to McDowell’s theory of perceptual justification should lead him to accept that experiences possess non-conceptual content. In this paper, I take issue with Cheng’s conclusion. Instead, I argue that McDowell should adopt Travis’s position, where experiences aren’t taken to possess content at all. I argue that we can distinguish two separate Myths of the Given in McDowell’s writings. While McDowell often seamlessly moves from one to the (...) other, I argue that it is difficult to see how he can justify this due to his recent alterations to his position. I argue that if we reject one Myth and retain the other, then McDowell can both hold on to a version of his view that the space of reasons is the space of the conceptual, all the while he accepts Travis’s arguments to the effect that experiences are devoid of content. Finally, I consider some arguments McDowell might present against accepting Travis’s position. While these objections do not convince me, what is notable about them is that, if they are sound, then they will equally count against Cheng’s intermediary position where non-conceptual contents are ascribed to experiences. Hence, I conclude that there are no grounds on which McDowell can reasonably endorse that experiences possess non-conceptual contents. Either he should stick to his conceptualism, or he should follow Travis and reject that experiences possess contents at all. (shrink)

Given a cofinal cardinal function $h\in {}^{\kappa }\kappa $ for $\kappa $ inaccessible, we consider the dominating h-localisation number, that is, the least cardinality of a dominating set of h-slaloms such that every $\kappa $ -real is localised by a slalom in the dominating set. It was proved in [3] that the dominating localisation numbers can be consistently different for two functions h (the identity function and the power function). We will construct a $\kappa ^+$ -sized family of functions h (...) and their corresponding localisation numbers, and use a ${\leq }\kappa $ -supported product of a cofinality-preserving forcing to prove that any simultaneous assignment of these localisation numbers to cardinals above $\kappa $ is consistent. This answers an open question from [3]. (shrink)

Although environmental philosophy and the human exploration of space share common beginnings, scholars from either field have not given adequate attention to the possible connections between them. In this essay, we seek to spur the rapprochement and cross-fertilization of philosophy and space policy by highlighting the philosophic dimensions of space exploration, pulling together issues and authors that have had insufficient contact with one another. We do so by offering an account of three topics: planetary exploration, planetary protection (...) and the search for extraterrestrial life, and terraforming. The resulting synthesis seeks to change our thinking about earthbound environmental ethics as it considers the philosophical dimensions of space exploration, and introduces the possible benefits of a humanities-oriented approach to space policy. (shrink)

In this paper we will consider two possible definitions of projective subsets of a separable metric space X. A set A subset of or equal to X is Σ11 iff there exists a complete separable metric space Y and Borel set B subset of or equal to X × Y such that A = {x ε X : there existsy ε Y ε B}. Except for the fact that X may not be completely metrizable, this is (...) the classical definition of analytic set and hence has many equivalent definitions, for example, A is Σ11 iff A is relatively analytic in X, i.e., A is the restriction to X of an analytic set in the completion of X. Another definition of projective we denote by ΣX1 or abstract projective subset of X. A set of A subset of or equal to X is ΣX1 iff there exists an n ε ω and a Borel set B subset of or equal to X × Xn such that A = {x ε X:there existsy ε Xn ε B}. These sets ca n be far more pathological. While the family of sets Σ11 is closed under countable intersections and countable unions, there is a consistent example of a separable metric space X where ΣX1 is not closed under countable intersections or countable unions. This takes place in the Cohen real model. Assuming CH, there exists a separable metric space X such that every Σ11 set is Borel in X but there exists a Σ11 set which is not Borel in X2. The space X2 has Borel subsets of arbitrarily large rank while X has bounded Borel rank. This space is a Luzin set and the technique used here is Steel forcing with tagged trees. We give examples of spaces X illustrating the relationship between Σ11 and ΣX1 and give some consistent examples partially answering an abstract projective hierarchy problem of Ulam. (shrink)

We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [2], Mayer-Wolf [5], and Weaver [8]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer–Specker group Zω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}^{\omega}}}$$\end{document}. The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and (...) Steprāns :52–59, 2006). (shrink)

Editor’s NoteThanks to the initiative of Alan Norrie, we are pleased to present here a symposium on Nick Wilson’s book The Space that Separates: A Realist Theory of Art. Several authors have contri...

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture (...) can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means (...) the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. (shrink)

The Cartesian concept of nature, which has determined modern thinking until the present time, has become obsolete. It shall be shown that Hegel's objective-idealistic conception of nature discloses, in comparison to that of Descartes, new perspectives for the comprehension of nature and that this, in turn, results in possibilities of actualizing Hegel's philosophy of nature. If the argumentation concerning philosophy of nature is intended to catch up with the concrete Being-of-nature and to meet it in its concretion, then this is (...) impossible for the finite spirit in a strictly a priori sense – this is the thesis supported here which is not at all close to Hegel. As the argumentation rather has to consider the conditions of realization concerning the Being-of-nature, too, it is compelled to take up empirical elements – concerning the organism, for instance, system-theoretical aspects, physical and chemical features of the nervous system, etc. With that, on the one hand, empirical-scientific premises are assumed (e.g. the lawlikeness of nature), which on the other hand become (now close to Hegel) possibly able to be founded in the frame of a Hegelian-idealistic conception. In this sense, a double strategy of empirical-scientific concretization and objective-idealistic foundation is followed up, which represents the methodical basic principle of the developed considerations. In the course of the undertaking, the main aspects of the whole Hegelian design concerning the philosophy of nature are considered – space and time, mass and motion, force and law of nature, the organism, the pro-blem of evolution, psychic being – as well as Hegel's basic thesis concerning the philosophy of nature, that therein a tendency toward coherence and idealization manifests itself in the sense of a (categorically) gradually rising succession of nature: from the separateness of space to the ideality of sensation. In the sense of the double strategy of concretization and foundation it is shown that on the one hand possibilities of philosophical penetration concerning actual empirical-scientific results are opened, and on the other hand – in turn – a re-interpretation of Hegel's theorem on the basis of physical, evolution-theoretical and system-theoretical argumentation also becomes possible. In this mutual crossing-over and elucidation of empirical and Hegelian argumentation not only do perspectives of a new comprehension of nature become visible, but also, at the same time – as an essential consequence of this methodical principle – thoughts on the possibilities of actualizing Hegel's philosophy of nature. (shrink)

We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal $$\delta $$ δ -separated sets in metric and pseudometric spaces from the (...) point of view the Axiom of Choice and its weaker forms. (shrink)

Let ≤r and ≤sbe two binary relations on 2ℕ which are meant as reducibilities. Let both relations be closed under finite variation and consider the uniform distribution on 2ℕ, which is obtained by choosing elements of 2ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤rX, differs from the lower ≤s-cone of X. By c osure (...) under finite variation, the Kolmogorov 0-1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2ℕ which can be defined by a countable set of total continuous functionals on 2ℕ in the same way as the usual resource-bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations. (shrink)

Information space and time in their socio-cultural dimension act as knowledge components of being, and, consequently, the most important regulator of social relations. At the same time, the social and cultural dominants of the information space and time are determined by the creative activity of a person. In principle, a person is always focused on knowledge, where its elements interact in a comprehensive way, and not on information, in which there is a separation of its elements from the (...) spiritually observed cross-section of reality, and thus from reality, which does not lend itself to the construction of connections, and the construction somewhat reduces the elasticity of the structure of information space and time. (shrink)

The growing body of research on temporal and spatial experience lacks a comprehensive theoretical approach. Drawing on Giddens’ framework, we present time-space distanciation as a construct for theorizing the relations between culture, time, and space. TSD in a culture may be understood as the extent to which time and space are abstracted as separate dimensions and activities are extended and organized across time and space. After providing a historical account of its development, we outline a multi-level (...) conceptualization of TSD supported by research on cultural differences in the experience of time and space. We impact this conceptualization by examining two ethnographic case studies. We conclude by highlighting future research directions. TSD is an integrative, interdisciplinary, multi level construct with the potential to guide the burgeoning social science of time and space. (shrink)

This paper pushes back against the Democritean-Newtonian tradition of assuming a strict conceptual dichotomy between spacetime and matter. Our approach proceeds via the more narrow distinction between modified gravity/spacetime and dark matter. A prequel paper argued that the novel field Φ postulated by Berezhiani and Khoury's 'superfluid dark matter theory' is as much matter as anything could possibly be, but also below the critical temperature for superfluidity as much spacetime as anything could possibly be. Here we introduce and critically evaluate (...) three groups of interpretations that one should consider for such Janus-faced theories. The consubstantiality interpretation holds that Φ is both matter and a modification of spacetime, analogously to the sense in which Jesus is both human and god. The fundamendalist interpretations consider for each of these roles whether they are instantiated fundamentally or emergently. The breakdown interpretations focus on the question of whether Φ signals the breakdown, in some sense to be specified, of the MG-DM dichotomy and perhaps even the broader spacetime–matter distinction. More generally, it is argued that hybrid theories urge a move towards a single space of theories, rather than two separate spaces of spacetime theories and matter theories, respectively. (shrink)

According to High-Dimensional Wavefunction Fundamentalism (HDWF) the wavefunction field evolving in configuration space is all that exists fundamentally. The main argument in favor of HDWF is an argument from separability and locality: separability is a desirable feature of a fundamental metaphysics and HDWF is indeed such a separable metaphysics. Separability in turn is desirable because it is simple and intuitive. Tim Maudlin has recently argued that intuitiveness and simplicity cannot motivate separability. In particular, our intuitions stem from our (...) interactions with the three-dimensional world which is non-separable. Therefore, he concludes, there is _nothing else_ HDWF theorists can appeal to motivate separability. I call this Maudlin’s challenge. The present paper addresses Maudlin’s challenge by showing how the facts that some plurality of entities are separable entail that its constituents are fundamental, for well-motivated notions of fundamentality. (shrink)

Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian group (...) R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document} and a ‘utility function’ u:X⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$\end{document} such that, for any x,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document} which differ in only finitely many coordinates, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∑i∈Iu-u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i\in \mathcal{I }} \left[u-u\right]\ge 0$$\end{document}. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} also satisfies a weak continuity condition, then the paper shows that, for anyx,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document}, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∗∑i∈Iu≥∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u\ge {}^*\!\sum _{i\in \mathcal{I }}u$$\end{document}. Here, ∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u$$\end{document} represents a nonstandard sum, taking values in a linearly ordered abelian group ∗R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\mathcal{R }$$\end{document}, which is an ultrapower extension of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document}. The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. (shrink)

Much has been written about the history of the work of men and women in the premodern past. It is now generally acknowledged that early modern ideological assumptions about a strict division of work and space between men and productive work outside the house on the one hand, and women and reproduction and consumption inside the house, on the other, bore little relation to reality. Household work strategies, out of necessity, were diverse. Yet what this spatial complexity meant in (...) particular households on a day-to-day basis and its consequences for gender relationships is less clear and has received relatively little historical attention. The aim of this paper is to add to our knowledge through a case study of the way that men and women used and organized space for work in the county of Essex during the “long seventeenth century”. Drawing on critiques of the concept of “separate spheres” and the models of economic change to which it relates, together with local/micro historical methods, it places evidence within an appropriate regional context to argue that spatial patterns were enormously varied in early modern England and a number of factors—time, place, occupation, and status, as well as gender—determined them. Understanding of the dynamic, complex, uneven purchase of patriarchy upon the organization, imagination, and experience of space has important implications for approaches to gender relations in early modern England. It raises additional doubts about the utility of the separate spheres analogy, and particularly the use of binary oppositions of male/female and public/private, to describe gender relations and their changes in this period and shows that a deeper understanding demands more research into the local contexts in which the gendered division and meaning of work was negotiated. (shrink)

We often think of normal childhood as a progressive development towards a fixed—and often tacitly individualistic and masculine—model of what it is to be an adult. By contrast, phenomenologists, psychoanalysts, sociology of childhood, and feminist thinkers have set out to offer richer accounts both of childhood development and of mature existence. This paper draws on accounts of childhood development from phenomenologist Maurice Merleau-Ponty and object relations theorist D. W. Winnicott in order to argue that childhood development takes place in “transitional (...) spaces”; explores typical gendered patterns in the formation of selfhood that “split” relationality and separateness into the “feminine” and the “masculine”; and offers a phenomenology of perception, love, and objectivity in order to show the manner in which, contra individualistic and masculine visions of adulthood, maturity requires an embrace rather than eschewal of ambiguity, and the capacity to continue to dwell in the transitional space between relatedness and separateness. (shrink)

Summary In the visual arts, the constructions of the spatial and chromatic structures of pictures can hardly be separated from each other. The phenomenological approach from the perspective of the arts provides an independent and worthwhile approach to the topic of colour and space. We address some matters of composition in design and, to some extent, in naturalistic painting. The phenomenological approach from the perspective of the arts reveals various topics that invite further investigation by the means of generic (...) vision science. However, although such studies might lead to increased academic insight, a thorough inquiry by the means of experimental phenomenology might well lead to formal descriptions that come somewhat closer to possible applications in the visual arts. (shrink)

A model of reality is called separable if the state of a composite system is equal to the union of the states of its parts, located in different regions of space. Spekkens has argued that it is trivial to reproduce the predictions of quantum mechanics using a separable ontological model, provided one allows for arbitrary violations of ‘dynamical locality’. However, since dynamical locality is strictly weaker than local causality, this leaves open the question of whether an ontological (...) model for quantum mechanics can be both separable and dynamically local. We answer this question in the affirmative, using an ontological model based on previous work by Deutsch and Hayden. Although the original formulation of the model avoids Bell’s theorem by denying that measurements result in single, definite outcomes, we show that the model can alternatively be cast in the framework of ontological models, where Bell’s theorem does apply. We find that the resulting model violates local causality, but satisfies both separability and dynamical locality, making it a candidate for the ‘most local’ ontological model of quantum mechanics. (shrink)

This paper takes up the themes in the articles and examines not only the environmental changes that are taking place in relation to universities, but also the dynamics of their organizational implications. It argues that there are parallels between managerially and academic professionalism in that both deny context. Arguing for a context-sensitivity that is not dependant, issues of space and governance become important in order to understand forms of knowledge and the relationship between the contexts of production and the (...) contents of what is produced. Universities have different capacities to play at the game of scales and they are judged according to abstract indicators that provide little or no opportunity for learning. Instead of examining these relations, expertise is assumed to be spatial, whilst universities transform themselves in the slipstream of imagined futures as if they were separate from the present and past. Understanding is lost in the process and so too are the opportunities to adequately examine the differences in types of knowledge's that are produced for sustainable futures in contemporary societies. (shrink)

We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces.

This research examines the space phenomenon in the novel "Dreams in the Sea", which is accepted as the first novel of Aslam Reşit Jankır. The study has focused on the phenomenon of "space" and has chosen this topic for many technical reasons. Because there are many and far places in the novel, and the phenomenon of "space" dominates the events in the work. The author's emphasis on space is clearly evident from the first stage, with the (...) text of the work represented by the title "Dreams in the Sea". The semiotics of the title arouses the curiosity of entering the place (sea) where the drama of a Syrian family takes place. She is the family of the boy who became the subject of the agenda by being found drowned in his elegant dresses on the Istanbul coast. The events in the novel are productively linked to many places as they are fused with the structure of the text and are up-to-date: Village-homeland-the borders separating two neighboring countries-Mardin-Istanbul-sea-overseas countries."Real" and "imagination" are intertwined within this structure. Because the author has changed the main element in his text and has given it to the imagination with the desire to work on a new subject that excites and trembles the reader and drags them into the event. The author thus created a flavor and beauty for his text. The analyzes of this research are compatible with the mechanics of the analytical descriptive method and intersect with semiotics to the extent required by the situation. I have divided the chapters of this review into sections according to the main locations in the novel, and I have concluded it by examining the title and citing the most obvious conclusions I have reached. (shrink)

Conceptual integration theory (Fauconnier & Turner, 1998, 2002) is re-examined in the light of recent criticisms (e.g., Gibbs, 2000, 2001). It is argued that the assumption of four independent "cognitive spaces" enters the model primarily as an entailment of the "space," "blending," and "construction" metaphors, leads to unnecessary ambiguity, and works against statement of the theory in a form that supports derivation of testable hypotheses. Several examples from Fauconnier and Turner (2002) are analyzed to show that they can be (...) interpreted more simply, without need of four separate "spaces." Suggestions are made for reformulating conceptual integration theory without the problematic metaphors. (shrink)

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding (...) in [4], where the authors prove these theorems under stronger assumptions. Our proof is also somewhat simpler. (shrink)

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, (...) and that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

This article examines MichËle Roberts' novel In the Red Kitchen in the light of contemporary spatial theory: in particular, the geographer Doreen Massey's critique of space and time. It argues that Roberts' text is particularly susceptible to such readings because of its multilayered portrayal of space, examining women's enclosure within the home across three different time periods, and through five different voices, which are overlaid one upon another in order to create a sense of simultaneity. In so doing, (...) the text shows that while the nature of women's oppression may change over time, their fundamental association with the domestic space does not. The article explores three different recurring spaces in the text: the kitchen, the bedroom and the crypt, the last of which acts as the receptacle for the trauma experienced by the female subject within the other two. The Gothic trope of the crypt is associated with incestuous abuse at the hands of a father-figure whom the woman both fears and desires; and thus it also signifies the precarious security to be found within the home. ‘Homelessness’, however, becomes another traumatic concept within a text which, while problematizing the relationship between home and identity, the subject and the space that contains and confines her, can envisage no other way of sustaining a secure sense of subjectivity. In the final analysis, while the boundaries that separate space from time are rendered porous and permeable in this novel, the complete collapse of all spatial boundaries is not conceptualized as desirable. (shrink)

Moore attempts to show that privacy, conceived as "control over access to oneself and to information about oneself" is "necessary" for human well-being. Moore grounds his argument in an analysis of the need for physical separation, which Moore suggests is universal among animal species. Moore notes, "One basic finding of animal studies is that virtually all animals seek periods of individual seclusion or small-group intimacy." Citing several studies involving rats and other animals, Moore points out that a lack of such (...) separate space frequently results in threats to survival. Moore goes on to suggest, quite plausibly, that since we evolved from such animals, we share some need for separation. I argue such reasoning involves a conceptual mistake, as a need for physical space and separation is not obviously tantamount to a need for privacy of any kind - much less a need for information privacy. (shrink)

Personal space is the distance that people tend to maintain from others during daily life in a largely unconscious manner. For humans, personal space-related behaviors represent one form of non-verbal social communication, similar to facial expressions and eye contact. Given that the changes in social behavior and experiences that occurred during the COVID-19 pandemic, including “social distancing” and widespread social isolation, may have altered personal space preferences, we investigated this possibility in two independent samples. First, we compared (...) the size of personal space measured before the onset of the pandemic to its size during the pandemic in separate groups of subjects. Personal space size was significantly larger in those assessed during the onset of the pandemic. Lastly, we found that the practice of social distancing and perceived risk of being infected with COVID-19 were linked to this personal space enlargement during the pandemic. Taken together, these findings suggest that personal space boundaries expanded during the COVID-19 pandemic independent of actual infection risk level. As the day-to-day effects of the pandemic subside, personal space preferences may provide one index of recovery from the psychological effects of this crisis. (shrink)

Utility functions often lack additive separability, presenting an obstacle for decision theoretic axiomatizations. We address this challenge by providing a representation theorem for utility functions of quasi-separable preferences of the form $$u(x,y,z)=f(x,z) + g(y,z)$$ on subsets of topological product spaces. These functions are additively separable only when holding z fixed but are cardinally comparable for different values of z. We then generalize the result to spaces with more than three dimensions and provide applications to belief elicitation, inequity aversion, (...) intertemporal choice, and rank-dependent utility. (shrink)

We define qualia space Q to be the space of all possible conscious experience. For simplicity we restrict ourselves to perceptual experience only, though other kinds of experience could also be considered. Qualia space is a highly idealized concept that unifies the perceptual experience of all possible brains. We argue that Q is a closed pointed cone in an infinite-dimensional separable real topological vector space. This quite technical structure can be explained for the most part (...) in a simple, intuitive way. The structure of qualia space allows us to consider and even answer in a precise way such questions as: Is there a continuous path from the sensation of blue to the sensation of pain? Once we fix a desired accuracy of approximation, do there exist finitely many perceptual experiences such that any possible perceptual experience is approximately equal to one of them? What should be meant by ‘fundamentally different’ perceptual experiences? There is the possibility of additional structure, such as a Hilbert space structure on the vector space in which Q is embedded. (shrink)

We show that metric spaces and continuous functions between them are domain representable using the category of Scott-Ershov domains. A notion of effectivity for metric spaces is thereby inherited from effective domain theory. It is shown that a separable metric space with an effective metric can be represented by an effective domain. For a class of spaces, including the Euclidean spaces, the usual notions of effectivity are obtained. The Banach fixed point theorem is a consequence of the least (...) fixed point theorem for domains. A notion of semieffective domains is introduced and used to give a new proof of Ceitin's theorem. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:SPACE-TIME AND THE COMMUNITY OF BEINGS: SOME COSMOLOGICAL SPECULATIONS INTRODUCTION XERT EINSTEIN, in his essay "Relativity and the Problem of Space," makes several interesting comments on the implications of relativity theory for the Newtonian concepts of absolute space and time. Among these are the following: Since the special theory of relativity revealed the physical equivalence of all inertial systems, it proved the untenability of the hypothesis (...) of an aether at rest. It was therefore necessary to renounce the idea that the electromagnetic field is to be regarded as a state of a material carrier. The field then becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter in the theory of Newton.1 On the basis of the general theory of relativity..., space, as opposed to" what fills space," which is dependent on the coorrdinates, has no separate existence.2 There is no such thing as empty space, i.e., a space without field. Space-time does not claim existence on its own but only as a structural quality of the field. Thus Descartes was not so far from the truth when he believed he must exclude the existence of an empty space. The notion indeed appears absurd, as long as physical reality is seen exclusively in ponderable bodies. It requires the idea of the field as the representative of reality, in combination with the general principle of relativity, to show the true kernel of Descartes' idea; there exists no space " empty of field." 3 1 Albert Einstein, Relativity: The Special and the General Theory (New York, 1961), pp. 149-150. 2 Ibid., p. 155. s Ibid., pp. 155-156. 480 SPACE-TIME AND THE COMMUNITY OF BEING 481 Space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way, the concept "empty space " loses its meaning.4 These statements raise some interesting questions for the philosopher working within the Judaeo-Christian metaphysical and cosmological tradition. Einstein's concept of the spacetime continuum constituted by the field (or by the plurality of fields seen as interrelated) amounts in effect to a reduction of space-time to relation, to that which is constituted by a universal relational network. One may ask whether this concept is, as many seem to believe, a demolition of the classical and Christian view of the world, or whether, on the contrary, it amounts to a rediscovery of that view from the perspective of the vocabulary of contemporary physics. To ask the question in a slightly different way: Does relativity have anything to do with relativism and hence with the anti-ontological thrust of the latter? This question inevitably involves us in the more general question how the physical structure of the cosmos does or does not reflect the ontological structure of being-i.e., how is the cosmos as known by physics related to the world-order as the intelligible object of a philosophical cosmology? How are the various levels of cosmic order, from inorganic matter to the human level, related to one another? It will be the contention of the present essay that the relativistic concept of the cosmos not only is not destructive of the classical and Christian one, but converges with it, because the latter is grounded in the concept of being as being, and being in turn must be understood as the universal concept and reality which has reality by virtue of the community of beings, i.e., by virtue of relation. These concepts will be further clarified in the course of the discussion. Let it be clearly understood, however, that what follows does not pretend to constitute a rigorous logical derivation of a cosmological system but is 4 Ibid.," Note to the Fifteenth Edition," p. vi. 482 GEORGE A. KENDALL rather a series of speculations, of brief glimpses of areas of convergence which have drawn attention to themselves in the course of the struggle for ontological and cosmological truth. As such, they are presented as a step along the way in the search for understanding, not as... (shrink)

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of (...) van Douwen, and gives partial answers to questions of Terada and Medvedev. (shrink)

Defining privacy is problematic because the condition of privacy appears simultaneously to require separation from others, and the possibility of choosing not to be separate. This latter feature expresses the inherent normative dimension of privacy: the capacity to control the perceptual and informational spaces surrounding one’s person. Clearly the features of separation and control as just described are in tension because one may easily enough choose to give up all barriers between oneself and the public space. How could the (...) capacity for privacy give rise to its absence? Yet both the separation and control features of privacy do seem indispensable to any sensible understanding of it. In this paper I set out an approach to defining privacy that keeps these features and avoids the tension between them. (shrink)

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding (...) in [4], where the authors prove these theorems under stronger assumptions. Our proof is also somewhat simpler. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Space, Place and Capitalism: The Literary Geographies of “The Unknown Industrial Prisoner” by Brett HeinoDavid McLaughlinSpace, Place and Capitalism: The Literary Geographies of “The Unknown Industrial Prisoner” BY BRETT HEINO Singapore: Palgrave Macmillan, 2021I would not be the first to describe Brett Heino’s new book as timely. Its publication in 2021 coincided with the fiftieth anniversary of the first publication of David Ireland’s The Unknown Industrial Prisoner (...) (1971). This experimental novel, centered on a “kaleidoscopic vision” of its characters and their lives, tells the story of a group of workers at the Puroil refinery in a fictionalized part of Sydney. In Heino’s own words, this project emerged from a desire to write about industrialization and representation in Australian fiction. Working primarily as a legal scholar and historian, this work represents his first foray into literary geography.At the heart of Heino’s argument is a question: What does literature know about space? In his pursuit of a spatially-directed reading of The Unknown Industrial Prisoner, Heino eschews work in literary geography that cultural geographers might recognize— particularly the writings of Sheila Hones, the doyenne of an emerging interdiscipline of literary geography, whose practitioners work at the interstices between literary studies and geography. Heino criticizes this interdisciplinary literary geography as being too focused on establishing the “sovereignty” or “imperialism” of fictional space at the expense of recognizing the [End Page 132] role of literature in the world. Arguably, such a stance is a misreading of the thinking within literary geography that “reading literature is essentially an experimental process in which the categorical distinction between the frames of ‘fact’ and ‘fiction’ become blurred,”1 and where the “ontological distinction between literary and non- literary spaces” is undermined.2 Neither of these statements supposes that reading literature is a practice by which the world falls away and nothing remains but the book. Instead, following Doreen Massey’s idea that space is a complicated process of events and happenings, literary geographers in this mold argue that fiction is a part of life as it happens, and reading is a constitutive practice in the co- production of space.In its place, Heino emphasizes a form of literary geographical thinking that understands literature’s role as being separate from, but intimately connected to, the actual world. His theoretical framework is multi-layered, borrowing from Marxist literary scholars, including Terry Eagleton and Fredric Jameson, and Marxist geographers, including Henri Lefebvre. In essence, however, it can be expressed neatly in the following quotation from geocritic Bertrand Westphal:Fiction does not reproduce the real, but actualises new virtualities that had remained unformulated, and then goes on to interact with the real according to the hypertextual logic of interfaces... fiction detects possibilities buried in the folds of the real, knowing that these folds have not been temporised.3I think that Heino’s rejection of what he characterizes as poststructural literary geography, of the kind represented in his analysis of Sheila Hones’s work, misses some opportunities. First, there are similarities between his conception of literary geography and those he leaves aside— not least, their shared acceptance of the productive tension that sits between literature as text and world as practical, non- representational, and exceeding narrative. Secondly, the shared aim of both approaches is to understand how “representation ‘takes place’ and precisely how texts are part of what happens.”4 This is particularly powerful in the light of Heino’s aim to understand the development and contents of literary texts in materialist terms. Despite these areas [End Page 133] of convergence, Heino’s argument that literary geography’s potential to analyze class and political structures remains untapped is well founded. This is where Heino’s contribution is at its strongest.Heino’s work is at its most convincing when he discusses the realities of “abstract space” that both permeate the spaces of The Unknown Industrial Prisoner and structure the very fabric of latter- twentieth century Australia. Running through both Heino’s critique and Ireland’s novel is an understanding “that the inherent drive of capital is towards the creation of a fragmented, homogenised and hierarchised space within which it can produce and... (shrink)

Suppose \ is a computable real. We extend previous work of Clanin, Stull, and McNicholl by determining the degrees of categoricity of the separable \ spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we ascertain the complexity of associated projection maps.