It is shown to be consistent that countable, Fréchet,α 1-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of lengthω 2.

Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even ℝ may fail to be a sequential space.Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ℝ, are classes of Fréchet-Urysohn or sequential spaces.In this context, it is seen that there are metric spaces which are not (...) sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion.Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ℝ if and only if the axiom of countable choice holds for families of subsets of ℝ, and every metric space has a unique equation image-completion. (shrink)

In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between metric spaces, and the second declares that sequentially compact pseudometric spaces are \—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.

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 introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

In this paper I compare two well studied approaches to topological semantics – the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ and Typ Two Effectivity, exemplified by the category of Baire space representations, Rep . These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is how they are related.First, we show that Rep is equivalent to a full coreflective subcategory of Equ, consisting of the (...) so-called 0-equilogical spaces. This establishes a pair of adjoint functors between Rep and Equ. The inclusion Rep → Equ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of Rep and Equ are essentially different. How ever, in a second comparison we show that Rep and Equ do share a common cartesian closed subcategory that contains all countably based-T0 spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based T0-spaces. We consider several examples involving the natural numbers and the real numbers to demonstrate how these comparisons make it possible to transfer results from one setting to another. (shrink)

We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to any (...) finitely accessible category, any first-order definable class of structures, and any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: [email protected]: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf. (shrink)

Let X be a Polish space and a separable compact subset of the first Baire class on X. For every sequence dense in , the descriptive set-theoretic properties of the set are analyzed. It is shown that if is not first countable, then is -complete. This can also happen even if is a pre-metric compactum of degree at most two, in the sense of S. Todorčević. However, if is of degree exactly two, then is always Borel. (...) A deep result of G. Debs implies that contains a Borel cofinal set and this gives a tree-representation of . We show that classical ordinal assignments of Baire-1 functions are actually -ranks on . We also provide an example of a Ramsey-null subset A of for which there does not exist a Borel set BA such that the difference BA is Ramsey-null. (shrink)

-/- Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? -/- The (...) primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧φ (if and) only if Σ⊢φ ∸2−n for all n < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ. -/- Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ, the value φ T is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ, where φ T o is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable. (shrink)

In the second installment to Gruszczyński and Pietruszczak we carry out an analysis of spaces of points of Grzegorczyk structures. At the outset we introduce notions of a concentric and \-concentric topological space and we recollect some facts proven in the first part which are important for the sequel. Theorem 2.9 is a strengthening of Theorem 5.13, as we obtain stronger conclusion weakening Tychonoff separation axiom to mere regularity. This leads to a stronger version of Theorem 6.10. Further, (...) we show that Grzegorczyk points are maximal contracting filters in the sense of De Vries, but the converse inclusion is not necessarily true. We also compare the notions of a Grzegorczyk point and an ultrafilter, and establish several properties of topological spaces based on Grzegorczyk structures. The main results of the paper are representation and completion theorems for G-structures. We prove both set-theoretical and topological representation theorems for various classes of G-structures. We also present topological object duality theorem for the class of complete G-structures and the class of concentric spaces, both restricted to structures which satisfy countable chain condition. We conclude the paper with proving equivalence of the original Grzegorczyk axiom with the one accepted by us as axiom. (shrink)

The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space.For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model (...) theory, such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models.This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that theBorel cardinalitiesof the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action.One of the first things to be addressed concerns the complexity of these equivalence relations. This question forappears in [1]. (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 (...) class='Hi'>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)

If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] which is the least ordinal with no presentation on [Formula: see text] computable from (...) [Formula: see text]. If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text]. (shrink)

We define a notion of reducibility for subsets of a second countable T 0 topological space based on relatively continuous relations and admissible representations. This notion of reducibility induces a hierarchy that refines the Baire classes and the Hausdorff–Kuratowski classes of differences. It coincides with Wadge reducibility on zero dimensional spaces. However in virtually every second countable T 0 space, it yields a hierarchy on Borel sets, namely it is well founded and antichains are of length (...) at most 2. It thus differs from the Wadge reducibility in many important cases, for example on the real line $${\mathbb{R}}$$ or the Scott Domain $${\mathcal{P}\omega}$$. (shrink)

This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk :228–235, 1960. https://doi.org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation. Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze separation structures and (...) Grzegorczyk structures, and establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk’s corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces. (shrink)

Reviewed Works:Tomek Bartoszynski, Marion Scheepers, Set Theory, Annual Boise Extravaganza in Set Theory Conference, March 13-15, 1992, April 10-11, 1993, March 25-27, 1994, Boise State University, Boise, Idaho.R. Aharoni, A. Hajnal, E. C. Milner, Interval Covers of a Linearly Ordered Set.Eyal Amir, Haim Judah, Souslin Absoluteness, Uniformization and Regularity Properties of Projective Sets.Tomek Bartoszynski, Ireneusz Reclaw, Not Every $\gamma$-Set is Strongly Meager.Andreas Blass, Reductions Between Cardinal Characteristics of the Continuum.Claude Laflamme, Filter Games and Combinatorial Properties of Strategies.R. Daniel Mauldin, Analytic (...) non-Borel Sets Modulo Null Sets.Janusz Pawlikowski, Laver's Forcing and Outer Measure.Marion Scheepers, Meager Sets and Infinite Games.Saharon Shelah, On Some Problems in General Topology.Saharon Shelah, Remarks on $\aleph_1$-CWH not CWH First Countable Spaces.Juris Steprans, Cardinal Invariants Associated with Hausdorff Capacities. (shrink)

I criticise a paper by Peter Forrest in which he argues that a principle of unrestricted countable fusion has paradoxical consequences. I argue that the paradoxical consequences that he exhibits may be due to his Whiteheadean assumptions about the nature of spacetime rather than to the principle of unrestricted countable fusion.

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.

Given a space in an elementary submodel M of H, define XM to be X∩M with the topology generated by . It is established, using anti-large-cardinals assumptions, that if XM is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X=XM. Assuming in addition, the result holds for any compact XM satisfying the countable chain condition.

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)

This essay is a conversation between two South Asian Muslim sisters both of whom are feminist academics of color, where one immigrated to the United States and the other, a decade apart, to United Kingdom. The aim of this essay is to examine the ways in which white supremacist structures influenced and molded our personal journeys as well as how our narratives are deeply entangled within broader conversations around patriarchy, neo-liberal feminism, and anti-Muslim racism.

This book is designed to make accessible to nonspecialists the still evolving concepts of quantum mechanics and the terminology in which these are expressed. The opening chapters summarize elementary concepts of twentieth century quantum mechanics and describe the mathematical methods employed in the field, with clear explanation of, for example, Hilbert space, complex variables, complex vector spaces and Dirac notation, and the Heisenberg uncertainty principle. After detailed discussion of the Schrödinger equation, subsequent chapters focus on isotropic vectors, used to (...) construct spinors, and on conceptual problems associated with measurement, superposition, and decoherence in quantum systems. Here, due attention is paid to Bell's inequality and the possible existence of hidden variables. Finally, progression toward quantum computation is examined in detail: if quantum computers can be made practicable, enormous enhancements in computing power, artificial intelligence, and secure communication will result. This book will be of interest to a wide readership seeking to understand modern quantum mechanics and its potential applications. (shrink)

In this paper, I reconstruct an argument of Aristidis Arageorgis against empirical underdetermination of the state of a physical system in a C*-algebraic setting and explore its soundness. The argument, aiming against algebraic imperialism, the operationalist attitude which characterized the first steps of Algebraic Quantum Field Theory, is based on two topological properties of the state space: being T1 and being first countable in the weak*-topology. The first property is possessed trivially by the state (...) class='Hi'>space while the latter is highly non-trivial, and it can be derived from the assumption of the algebra of observables’ separability. I present some cases of classical and of quantum systems which satisfy the separability condition, and others which do not, and relate these facts to the dimension of the algebra and to whether it is a von Neumann algebra. Namely, I show that while in the case of finite-dimensional algebras of observables the argument is conclusive, in the case of infinite-dimensional von Neumann algebras it is not. In addition, there are cases of infinite-dimensional quasilocal algebras in which the argument is conclusive. Finally, I discuss Porrmann's construction of a net of local separable algebras in Minkowski spacetime which satisfies the basic postulates of Algebraic Quantum Field Theory. (shrink)

Based on the concepts of pseudocomplement of L -subsets and the implication operator where L is a completely distributive lattice with order-reversing involution, the definition of countable RL -fuzzy compactness degree and the Lindelöf property degree of an L -subset in RL -fuzzy topology are introduced and characterized. Since L -fuzzy topology in the sense of Kubiak and Šostak is a special case of RL -fuzzy topology, the degrees of RL -fuzzy compactness and the Lindelöf property are generalizations of (...) the corresponding degrees in L -fuzzy topology. (shrink)

We study the class of Sperner spaces, a generalized version of affine spaces, as defined in the language of pointline incidence and line parallelity. We show that, although the class of Sperner spaces is a pseudo-elementary class, it is not elementary nor even ℒ∞ω-axiomatizable. We also axiomatize the first-order theory of this class.

We show that some well known theorems in topology may not be true without the axiom of choice.

Closure spaces are generalizations of topological spaces, in which the Intersection of two open sets need not be open. The considered logic is related to closure spaces just as the standard logic S4 to topological ones. After describing basic properties of the logic we consider problems of representation of Lindenbaum algebras with some uncountable sets of infinite joins and meets, a notion of equality and a meaning of quantifiers. Results are extended onto the standard logic S4 and they are valid (...) also for some other standard modal logics. (shrink)

We consider the complexity of the isomorphism relation on countable first-order structures with transitive automorphism groups. We use the theory of Borel reducibility of equivalence relations to show that the isomorphism problem for vertex-transitive graphs is as complicated as the isomorphism problem for arbitrary graphs and determine for which first-order languages the isomorphism problem for transitive countable structures is as complicated as it is for arbitrary countable structures. We then use these results to characterize the (...) complexity of the isometry relation for certain classes of homogeneous and ultrahomogeneous metric spaces. (shrink)

In this paper we give a positive answer to a problem posed by Hofer-Szabó and Rédei (Int. J. Theor. Phys. 43:1819–1826, 2004) regarding the existence of infinite Reichenbachian common cause systems (RCCSs). An example of a countably infinite RCCS is presented. It is also determined that no RCCSs of greater cardinality exist.