We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals.Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are : Every product of a concentrated space with a Hurewicz S1S1 space satisfies S1S1. On the other hand, assuming (...) the Continuum Hypothesis, for each Sierpiński set S there is a Luzin set L such that L×SL×S can be mapped onto the real line by a Borel function. Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits–Nagy. Assuming d=ℵ1d=ℵ1, every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers.A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than addadd, the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy >ℵ1.Our results improve upon and unify a number of results, established earlier by many authors. (shrink)

Results of Sierpiski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is ‘narrow’ in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set ω × ω is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite (...) set. Such partitions or coverings can exist only when the sets forming the product are of limited size.This paper considers such coverings for products of infinitely many sets . In this case, a covering of the product by narrow sets, one for each coordinate direction, will exist no matter how large the factor sets are. But if one restricts the sets used in the covering , then the existence of narrow coverings is related to a number of large cardinal properties: partition cardinals, the free subset problem, nonregular ultrafilters, and so on.One result given here is a relative consistency proof for a hypothesis used by S. Mrówka to produce a counterexample in the dimension theory of metric spaces. (shrink)

Shehtman introduced bimodal logics of the products of Kripke frames, thereby introducing frame products of unimodal logics. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalize this idea to the bimodal logics of the products of topological spaces, thereby introducing topological products of unimodal logics. In particular, they show that the topological product of S4 and S4 is S4 \ S4, i.e., the fusion of S4 and S4: this logic is strictly weaker than the frame product S4 × S4. (...) Indeed, van Benthem et al. show that S4 \ S4 is the bimodal logic of the particular product space , leaving open the question of whether S4 \ S4 is also complete for the product space . We answer this question in the negative. (shrink)

Shehtman introduced bimodal logics of the products of Kripke frames, thereby introducing frame products of unimodal logics. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalize this idea to the bimodal logics of the products of topological spaces, thereby introducing topological products of unimodal logics. In particular, they show that the topological product of S4 and S4 is S4 ⊕ S4, i.e., the fusion of S4 and S4: this logic is strictly weaker than the frame product S4 × S4. (...) Indeed, van Benthem et al show that S4 ⊕ S4 is the bimodal logic of the particular product space Q × Q, leaving open the question of whether S4 ⊕ S4 is also complete for the product space R × R. We answer this question in the negative. (shrink)

For any unit vector in an inner product space S, we define a mapping on the system of all ⊥-closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems (...) of subspaces. Finally, the result will be generalized to general inner product spaces. (shrink)

With a few notable exceptions disability studies has not taken account of intersexuality, and it is principally through the lenses of feminist and queer-theory oriented ethical discussions but not through ‘straight’ bioethics that modes valuing intersex difference have been proposed. Meanwhile, the medical presupposition that intersex characteristics are inherently disabling to social viability remains the taken-for-granted truth from which clinical practice proceeds. In this paper I argue against bioethical perspectives that justify extensive and invasive pre- and post-natal medical interference to (...) eradicate intersex. I argue instead that to constitute the necessary conditions for the recognition of the intersexed child as a person, a life valid in its own right, clinicians must refrain from aggressive interference. Clinical specialists presuppose that intersexed children will be socially disabled and unrecognizable as persons; frustrated by the general failure of traditional interventions to assign a sex, clinicians are now pursuing prenatal technologies, including selective termination, to erase intersex. (shrink)

Henri Lefebvre has considerable claims to be the greatest living philosopher. His work spans some sixty years and includes original work on a diverse range of subjects, from dialectical materialism to architecture, urbanism and the experience of everyday life. The Production of Space is his major philosophical work and its translation has been long awaited by scholars in many different fields. The book is a search for a reconciliation between mental space and real space. In the course (...) of his exploration, Henri Lefebvre moves from metaphysical and ideological considerations of the meaning of space to its experience in the everyday life of home and city. He seeks, in other words, to bridge the gap between the realms of theory and practice, between the mental and the social, and between philosophy and reality. In doing so, he ranges through art, literature, architecture and economics, and further provides a powerful antidote to the sterile and obfuscatory methods and theories characteristic of much recent continental philosophy. This is a work of great vision and incisiveness. It is also characterized by its author's wit and by anecdote, as well as by a deftness of style which Donald Nicholson-Smith's sensitive translation precisely captures. (shrink)

In Geography and the Production of Space in Nineteenth-Century American Literature, Hsuan L. Hsu examines how literature represents different kinds of spaces ranging from the single-family home to the globe. He focuses on authors such as Charles Brockden Brown, Edgar Allan Poe, Herman Melville and Sarah Orne Jewett, who drew on literary tools such as rhetoric, setting, and point of view to mediate between individuals and different kinds of spaces. These authors used forms such as the regional sketch, the (...) domestic novel, and the detective story to re-examine how local spaces and communities would change when incorporated into global economic and political networks. Geography and the Production of Space in Nineteenth-Century American Literature is valuable reading for American literature scholars, and for all concerned with intersections between literature and geography"--Provided by publisher. (shrink)

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel’s conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

We study the Tychonoff Compactness Theorem for several different definitions of a compact space.

We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product (...) [0, 1] I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

This article is based on a perspective on circulation of knowledge that allows the consideration of science as the result of the encounter between diverse communities. We tell a story that constantly changes places, scales, and cultures in order to stress the importance of networks as an alternative to the centre/periphery trope, which entangles world histories of science. The result is a picture much more complex and intertwined than the one suggested by these simplifying dichotomies. We focus on a case (...) study that illuminates the process of knowledge production in non-European spaces of modernity. The return of the Society of Jesus to the newly independent nation-states of Latin America is the point of departure to analyse the circulation of a specific scientific idea in Ecuador: Darwin’s theory of biological evolution through natural selection. The article follows the paths of three different knowledge makers whose encounters are seen as sites of knowledge production: a religious order, a Latin American... (shrink)

It is shown that the product of finitely many sequential, CLP-compact spaces is CLP-compact.

This essay applies Henri Lefebvre’s notion of the production of space, particularly his conceptualization of the tension formed by the perceived-conceived-lived triad to analyze how space is produced in wartime hospitals as demonstrated in Richard Selzer’s “The Whistlers’ Room.” Wounded soldiers participate in producing the triad of the social space of military hospitals through their multilayered performances as fighting soldiers serving the nation and as living human beings longing for human connections. Contradictory performances demonstrate the strategic positioning (...) of wounded soldiers as active producers of different social spaces. By representing the lived spaces produced by wounded soldiers, Selzer challenges the hegemonic representations of the conceived spaces of military hospitals. Selzer’s exploration of soldiers’ contradictory performances opens up opportunities for discussion of writing physicians who assume a similar role by positioning themselves strategically within the medical institution. (shrink)

Let {: i ∈I } be a family of compact spaces and let X be their Tychonoff product. [MATHEMATICAL SCRIPT CAPITAL C] denotes the family of all basic non-trivial closed subsets of X and [MATHEMATICAL SCRIPT CAPITAL C]R denotes the family of all closed subsets H = V × Πmath imageXi of X, where V is a non-trivial closed subset of Πmath imageXi and QH is a finite non-empty subset of I. We show: Every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT (...) CAPITAL C]R extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ if and only if every family H ⊂ [MATHEMATICAL SCRIPT CAPITAL C] with the finite intersection property extends to a maximal [MATHEMATICAL SCRIPT CAPITAL C] family F with the fip. The proposition “if every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ, then X is compact” is not provable in ZF. The statement “for every family {: i ∈ I } of compact spaces, every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R, Y = Πi ∈IYi, extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ” is equivalent to Tychonoff's compactness theorem. The statement “for every family {: i ∈ ω } of compact spaces, every countable filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R, X = Πi ∈ωXi, extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ” is equivalent to Tychonoff's compactness theorem restricted to countable families. The countable Axiom of Choice is equivalent to the proposition “for every family {: i ∈ ω } of compact topological spaces, every countable family ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C] with the fip extends to a maximal [MATHEMATICAL SCRIPT CAPITAL C] family ℱ with the fip”. (shrink)

In this book we approach the development of the main doctrine of Trinitarianism. It is important to remind as we do so that that doctrine must be studied by both the criteria of Christian theology and history, and that as we study the development of doctrine, we need to establish a connection between what is confessed, or dogmas, and what is believed and taught, and go back diachronically from what is confessed to what was taught and to what was believed.

The 2023 edition of the Venice Architecture Biennial Time Space Existence will draw attention to the emerging expressions of sustainability in their numerous forms, ranging from a focus on the environment and urban landscape to the unfolding conversations on innovation, reuse, community, and inclusion. In response to climate change, exhibited projects will investigate new technologies and construction methods that reduce energy consumption through circular design and develop innovative, organic, and recycled building materials. Participants will also address social justice by (...) presenting living solutions envisioned for displaced communities and minorities, while others will examine the tensions between the built urban environment and the nature surrounding it, identifying opportunities for coexistence. Time Space Existence not only aims to be a display but also a living workshop where established and emerging practitioners can work together and present their own visions of the status of architecture and the built environment nowadays. By bringing together ideas from across the world, the exhibition presents itself as a platform for exchange and experimentation and strives to stimulate an engaging dialogue that is inclusive of all voices. (shrink)

We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.

Introduction -- Henri Lefebvre : the production of theory -- Research : from practices of dwelling to the production of space -- Critique : space as concrete abstraction -- Project : urban society and its architecture -- Afterword : toward an architecture of jouissance.

Recent discussions of human categories have suffered from an over emphasis on intention and language, and have not paid enough attention to the role of material conditions, and, specifically, of social space in the construction of human categories. The relationship between human categories and social spaces is vital, especially with the categories of class, race, and gender. This paper argues that social space is not merely the consequent of the division of the world into social categories; it is (...) constitutive of social categories. To put it more bluntly, if who we are is bound up with place, then not only do we inhabit a divided America; divided America inhabits us. The second, and equally dramatic, conclusion is that attempts to transform social categories must involve the transformation of social space. When we sort people by categories, we do so spatially: with race come racialized spaces. And because our place comes to inhabit us, when we divide spatially we cannot help but to inscribe and produce categories and identities associated with our spatial divisions: with racialized spaces come race. Recognition of this dialectic is a direct challenge to the one-way considerations of social identity and social space that occurs in much urban sociology and history. Moreover, it demonstrates that there is an internal contradiction in policies--often based in urban sociology and history--that assume that integration can be accomplished along with the conservation of ethnic and racial identity. (shrink)

We investigate in ZF conditions that are necessary and sufficient for countable products ∏m∈ℕXm of finite Hausdorff spaces Xm resp. Hausdorff spaces Xm with at most n points to be compact resp. Baire. Typica results: Countable products of finite Hausdorff spaces are compact if and only if countable products of non-empty finite sets are non-empty. Countable products of discrete spaces with at most n + 1 points are compact if and only if countable products of non-empty sets with at most (...) n points are non-empty. (shrink)

In this paper, we argue that nurses need to be aware of how the production of space in specific contexts – including health care systems and research institutions – perpetuates marginalized populations' state of social otherness. Lefebvre's idea regarding spatial triad is mobilized in this paper, as it pertains to two‐spirited, lesbian, gay, bisexual, trans and queer populations (2SLGBTQ*). We believe that nurses can create counter‐spaces within health care systems and research institutions that challenge normative discourses. Lefebvre's work provides (...) us the necessary tools to understand how various places or environments produce identities. In understanding Lefebvre's principles, we believe that nurses can play an essential role in creating counter‐spaces, thereby instigating counter‐institutional practices, for those who experience otherness. (shrink)

Techniques for constructing the tensor product of two generalized sample spaces which admit unital sets of dispersion-free weights are discussed. A duality theory is developed, based on the 1-cuts of the dispersion-free weights, and used to produce a candidate for the tensor product. This construction is verified for Dacification manuals, a conjecture is given for other reflexive cases, and some adjustments for nonreflexive cases are considered. An alternate approach, using graphs of interpretation morphisms on the duals, is also (...) presented. (shrink)

How can performance transform places of urban renewal and regeneration? What does performance contribute to the creation of community? These are some of the questions addressed in this study of the relationship of performance to urban space. Marrying theory with a series of international case studies of performance practice and interviews with practitioners, this interdisciplinary study examines how space is performatively produced to create a sense of 'placeness'. Case studies are drawn from Canada, Brazil, the Czech Republic, the (...) UK and Spain. (shrink)

This paper argues against evolutionary accounts of aesthetics by defending the idea that our fundamental aesthetic categories have undergone great changes in the last two millennia, in particular, during an “artistic revolution” that lasted from 1680 to 1830. This revolution was made possible by the development of a number of technologies of art that created a separate cultural space for this new invention. The attempt to extend this revolution to include the aesthetic appreciation of the natural environment is aided (...) by a new set of technologies that help make an aesthetic object out of natural environments. This even morerecent development is further evidence against an evolutionary explanation of art. (shrink)