We find some characterizations of the Axiom of Choice in terms of certain families of open sets in T1 spaces.

In this paper we develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. (...) In the succeeding sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel. (shrink)

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice.

For a free filter F on $$\omega $$ ω, endow the space $$N_F=\omega \cup \{p_F\}$$ N F = ω ∪ { p F }, where $$p_F\not \in \omega $$ p F ∉ ω, with the topology in which every element of $$\omega $$ ω is isolated whereas all open neighborhoods of $$p_F$$ p F are of the form $$A\cup \{p_F\}$$ A ∪ { p F } for $$A\in F$$ A ∈ F. Spaces of the form $$N_F$$ N F constitute (...) the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space $$N_F$$ N F carries a sequence $$\langle \mu _n:n\in \omega \rangle $$ ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that $$\mu _n(f)\rightarrow 0$$ μ n ( f ) → 0 for every bounded continuous real-valued function f on $$N_F$$ N F if and only if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, that is, the dual ideal $$F^*$$ F ∗ is Katětov below the asymptotic density ideal $${\mathcal {Z}}$$ Z. Consequently, we get that if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, then: (1) if X is a Tychonoff space and $$N_F$$ N F is homeomorphic to a subspace of X, then the space $$C_p^*(X)$$ C p ∗ ( X ) of bounded continuous real-valued functions on X contains a complemented copy of the space $$c_0$$ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and $$N_F$$ N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck. (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)

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show (...) that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space. (shrink)

This paper continues our study of computable point-free topological spaces and the metamathematical points in them. For us, a point is the intersection of a sequence of basic open sets with compact and nested closures. We call such a sequence a sharp filter. A function fF from points to points is generated by a function F from basic open sets to basic open sets such that sharp filters map to sharp filters. We restrict our study to functions that (...) have at least all computable points in their domains.We follow Turing's approach in stating that a point is computable if it is the limit of a computable sharp filter; we then define the Turing degree Deg of a general point x in an analogous way. Because of the vagaries of the definition, a result of J. Miller applies and we note that not all points in all our spaces have Turing degrees; but we also show a certain class of points do. We further show that in ℝn all points have Turing degrees and that these degrees are the same as the classical Turing degrees of points defined by other researchers.We also prove the following: For a point x that has a Turing degree and lies either on a computable tree T or in the domain of a computable function fF, there is a sharp filter on T or in dom converging to x and with the same Turing degree as x. Furthermore, all possible Turing degrees occur among the degrees of such points for a given computable function fF or a complete, computable, binary tree T. For each x ∈ dom for which x and fF have Turing degrees, Deg) ≤ Deg. Finally, the Turing degrees of the sharp filters convergent to a given x are closed upward in the partial order of all Turing degrees. (shrink)

In the first section we briefly describe methodological assumptions of point-free geometry and topology. We also outline history of geometrical theories based on the notion of emph{region}. The second section is devoted to concise presentation of the content of the LLP special issue on point-free theories of space.

Abstract. Traditional epistemology of knowledge and belief can be succinctly characterized as JTB-epistemology, i.e., it is characterized by the thesis that knowledge is justified true belief. Since Gettier’s trail-blazing paper of 1963 this account has become under heavy attack. The aim of is paper is to study the Gettier problem and related issues in the framework of topological epistemic logic. It is shown that in the framework of topological epistemic logic Gettier situations necessarily occur for most topological (...) models of knowledge and belief. On the other hand, there exists a special class of topological models (based on so called nodec spaces) for which traditional JTB-epistemology is valid. Further, it is shown that for each topological model of Stalnaker’s combined logic KB of knowledge and belief a canonical JTB-model (its JTB-doppelganger) can be constructed that shares many structural properties with the original model but is free of Gettier situations. The topological model and its JTB-doppelganger both share the same justified belief operator and have very similar knowledge operators. Seen from a somewhat different perspective, the JTB-account of epistemology amounts to a simplification of a more general epistemological account of knowledge and belief that assumes that these two concepts may differ in some cases. The JTB-account of knowledge and belief assumes that the epistemic agent’s cognitive powers are rather large. Thereby in the JTB-epistemology Gettier cases do not occur. Eventually, it is shown that for all topological models of Stalnaker’s KB-logic Gettier situations are topologically characterized as nowhere dense situations. This entails that Gettier situations are epistemologically invisible in the sense that they can neither be known nor believed with justification with respect to the knowledge operator and the belief operator of the models involved. (shrink)

Traditional epistemology of knowledge and belief can be succinctly characterized as JTB-epistemology, i.e., it is characterized by the thesis that knowledge is justified true belief. Since Gettier’s trail-blazing paper of 1963 this account has become under heavy attack. The aim of is paper is to study the Gettier problem and related issues in the framework of topological epistemic logic. It is shown that in the framework of topological epistemic logic Gettier situations necessarily occur for most topological models (...) of knowledge and belief. On the other hand, there exists a special class of topological models (based on so called nodec spaces) for which traditional JTB-epistemology is valid. Further, it is shown that for each topological model of Stalnaker’s combined logic KB of knowledge and belief a canonical JTB-model (its JTB-doppelganger) can be constructed that shares many structural properties with the original model but is free of Gettier situations. The topological model and its JTB-doppelganger both share the same justified belief operator and have very similar knowledge operators. Seen from a somewhat different perspective, the JTB-account of epistemology amounts to a simplification of a more general epistemological account of knowledge and belief that assumes that these two concepts may differ in some cases. The JTB-account of knowledge and belief assumes that the epistemic agent’s cognitive powers are rather large. Thereby in the JTB-epistemology Gettier cases do not occur. Eventually, it is shown that for all topological models of Stalnaker’s KB- logic Gettier situations are topologically characterized as nowhere dense situations. This entails that Gettier situations are epistemologically invisible in the sense that they can neither be known nor believed with justification with respect to the knowledge operator and the belief operator of the models involved. Keywords. Stalnaker’s logic KB of knowledge and belief; Topological epistemology; Epistemic Invisibility; Doxastic invisibility; Gettier cases; . (shrink)

We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions.

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

The general framework of this paper is a reformulation of Hilbert’s program using the theory of locales, also known as formal or point-free topology [P.T. Johnstone, Stone Spaces, in: Cambridge Studies in Advanced Mathematics, vol. 3, 1982; Th. Coquand, G. Sambin, J. Smith, S. Valentini, Inductively generated formal topologies, Ann. Pure Appl. Logic 124 71–106; G. Sambin, Intuitionistic formal spaces–a first communication, in: D. Skordev , Mathematical Logic and its Applications, Plenum, New York, 1987, pp. 187–204]. Formal topology presents (...) a topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. The application to Hilbert’s program is then the following. Hilbert’s ideal objects are represented by points of such a formal space. There are general methods to “eliminate” the use of points, close to the notion of forcing and to the “elimination of choice sequences” in intuitionist mathematics, which correspond to Hilbert’s required elimination of ideal objects. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [R. Dedekind, H. Weber, Theorie des algebraischen Funktionen einer Veränderlichen, J. de Crelle t. XCII 181–290] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [N. Bourbaki, Eléments de Mathématique. Algèbre commutative, Hermann, Paris, 1965, Chapitre 7]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology. (shrink)

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

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

It is shown that in ZF set theory the axiom of choice holds iff every non empty topological space has a maximal closed filter.

The purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last but not least, we do not refrain (...) from a philosophical interpretation of the results from the paper. (shrink)

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any (...) class='Hi'>topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras. (shrink)

This paper is an investigation of definability hierarchies on effective topological spaces. An open subset U of an effective space X is definable iff there is a parameter free definition φ of U so that the atomic predicate symbols of φ are recursively open relations on X . The complexity of a definable open set may be identified with the quantifier complexity of its definition. For example, a set U is an ∃∃∀∃-set if it has an ∃∃∀∃ parameter (...) free definition using only recursively open predicate symbols. Since X is not equipped with a natural pairing apparatus such a U need not be an ∃∀∃-set. Let Σ denote the class of all ∃-sets, ∃∃-sets, ∃∃∃-sets etc. We show that an open set is in Σ iff it is equivalent modulo a nowhere dense set to a recursively enumerable open set . Thus Σ = e.r.e. Indeed we show the existence of a universal Σ -set as well as the existence of universal sets for higher levels of the definability hierarchy. (shrink)

We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev’s (...) logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces). In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic. (shrink)

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

This article is devoted to the problem of ontological foundations of three-dimensional Euclidean geometry. Starting from Bertrand Russell’s intuitions concerning the sensual world we try to show that it is possible to build a foundation for pure geometry by means of the so called regions of space. It is not our intention to present mathematically developed theory, but rather demonstrate basic assumptions, tools and techniques that are used in construction of systems of point-free geometry and topology by means of (...) mereology and Whitehead-like connection structures. We list and briefly analyze axioms for mereological structures, as well as those for connection structures. We argue that mereology is a good tool to model so called spatial relations. We also try to justify our choice of axioms for connection relation. Finally, we briefly discuss two theories: Grzegorczyk’s point-free topology and Tarski’s geometry of solids. (shrink)

In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T 0-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a generalized Alexandroff's cube that is universal for T 0-closure spaces. By this theorem we obtain the following characterization of the (...) consequence operator of the classical logic: If is a countable set and C: P() P() is a closure operator on X, then C satisfies the compactness theorem iff the closure space ,C is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2 there exists a subset X of irrationals and a subset X of the Cantor's set such that X is both a continuous image of X and a continuous image of X. (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)

We offer a topological treatment of scattered theories intended to help to explain the parallelism between, on the one hand, the theorems provable using Descriptive Set Theory by analysis of the space of countable models and, on the other, those provable by studying a tree of theories in a hierarchy of fragments of infinintary logic. We state some theorems which are, we hope, a step on the road to fully understanding counterexamples to Vaught's Conjecture. This framework is in the (...) early stages of development, and one area for future exploration is the possibility of extending it to a setting in which the spaces of types of a theory are uncountable. (shrink)

We give an idea of uniform approach to the problem of characterization of absolute extensors for categories of topological spaces [21], closure spaces [15], Boolean algebras [22], and distributive lattices [4]. In this characterization we use the notion of retract of the closure space of filters in the lattice of all subsets.

We study topological constructions in the recursion theoretic framework of the lattice of recursively enumerable open subsets of a topological space X. Various constructions produce complemented recursively enumerable open sets with additional recursion theoretic properties, as well as noncomplemented open sets. In contrast to techniques in classical topology, we construct a disjoint recursively enumerable collection of basic open sets which cannot be extended to a recursively enumerable disjoint collection of basic open sets whose union is dense in (...) X. (shrink)

The first aim of this paper is to prove a topological completeness theorem for a weak version of Stalnaker’s logic KB of knowledge and belief. The weak version of KB is characterized by the assumption that the axioms and rules of KB have to be satisfied with the exception of the axiom (NI) of negative introspection. The proof of a topological completeness theorem for weak KB is based on the fact that nuclei (as defined in the framework of (...) point-free topology) give rise to a profusion of topological belief operators that are compatible with the familiar topological knowledge operator. Thereby a canonical topological model for weak KB can be constructed. For this canonical model a truth lemma for the K and B holds such that a completeness theorem for KB can be proved in the familiar way. The second aim of this paper is to show that the topological interpretation of knowledge K comes along with a complete Heyting algebra of belief operators B that all fit the knowledge operator K in the sense that the pairs (K, B) satisfy all axioms of weak KB. This amounts to a pluralistic relation between knowledge and belief: Knowledge does not fully determine belief, rather it designs a conceptual space for belief operators where different (competing) belief operators coexist. (shrink)

Noetherian type and Noetherian π-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the cellularity, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian π-type of κ-Suslin Lines, and we are able to determine it for every κ up to the (...) first singular cardinal. We then prove a consequence of Changʼs Conjecture for ℵω regarding the Noetherian type of countably supported box products which generalizes a result of Lajos Soukup. We finish with a connection between PCF theory and the Noetherian type of certain Pixley–Roy hyperspaces. (shrink)

In this paper, we introduce the concept of ""neutrosophic crisp neighborhoods system for the neutrosophic crisp point ". Added to, we introduce and study the concept of neutrosophic crisp local function, and construct a new type of neutrosophic crisp topological space via neutrosophic crisp ideals. Possible application to GIS topology rules are touched upon.

We study the flow of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f‐generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We shall give a description of f‐generic types of trigonalizable algebraic groups, and prove that every f‐generic type is almost periodic.

We study the topological models of a logic of knowledge for topological reasoning, introduced by Larry Moss and Rohit Parikh (1992). Among our results is the confirmation of a conjecture by Moss and Parikh, as well as the finite satisfiability property and decidability for the theory of topological models.

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)

To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ 1 is meagre yet there (...) are ℵ 1 rectifiable curves in R 3 whose union is not meagre. The consistency of this statement when the phrase "rectifiable curves" is replaced by "straight lines" remains open. (shrink)

The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the (...) former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions. (shrink)

In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that (...) of finite products, while FSpi is complete , but does not contain some naturally occurring classes of formal spaces. Moreover, completeness of FSpi only holds under the assumption of strong principles for the existence of inductively defined sets.In this paper, working in the context of constructive set theory, the category ImLoc of imaginary locales is introduced. ImLoc is complete already over weak formulations of constructive set theory, contains FSp and FSpi as full subcategories, and, as FSp and FSpi, is equivalent to the category of locales in IZF.Applications of the concept of imaginary locale to a proof of a point-free version of Tychonoff embedding theorem, and to set-representation theorems, are then presented. (shrink)

In this note a T1 formal space is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of a formal space, and prove that the class of points of a weakly set-presentable formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulate separation properties for constructive topological spaces , strengthening separation properties discussed elsewhere. Finally we (...) relate the properties for ct-spaces with corresponding properties of formal spaces. (shrink)

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF | p ∈ F}. We define a countably based MF space to be a space of the form MF for some (...) countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text]. (shrink)

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a (...) limit point”. implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω). (shrink)

We construct a collection of new topological Ramsey spaces of trees. It is based on the Halpern-Läuchli theorem, but different from the Milliken space of strong subtrees. We give an example of its application by proving a partition theorem for profinite graphs.

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

The political integration of the European Union is fragile for many reasons, not least the reassertion of nationalism. That said, if we examine specific practices and infrastructures, a more complicated story emerges. We juxtapose the political fragility of the EU in relation to the ongoing formation of data infrastructures in official statistics that take part in postnational enactments of Europe’s populations and territories. We develop this argument by analyzing transformations in how European populations are enacted through new technological infrastructures (...) that seek to integrate national census data in “cubes” of cross-tabulated social topics and spatial “grids” of maps. In doing so, these infrastructures give meaning to what “is” Europe in ways that are both old and new. Through standardization and harmonization of social and geographical spaces, “old” geometries of organizing and mapping populations are deployed along with “new” topological arrangements that mix and fold categories of population. Furthermore, we consider how grids and cubes are generative of methodological topologies by closing the distances or differences between methods and making their data equivalent. By paying attention to these practices and infrastructures, we examine how they enable reconfiguring what is known and imagined as Europe and how it is governed. (shrink)

This paper is a follow up of the author’s programme of characterizing Ramsey classes of structures by a combination of model theory and combinatorics. This relates the classification programme for countable homogeneous structures to the proof techniques of the structural Ramsey theory. Here we consider the classes of topological and metric spaces which recently were studied in the context of extremally amenable groups and of the Urysohn space. We show that Ramsey classes are essentially classes of finite objects only. (...) While for Ramsey classes of topological spaces we achieve a full characterization, for metric spaces this seems to be at present an intractable problem. (shrink)

D′ ⊆ D is a normal totality on a Scott domain D if it is upward closed and x ⊓ y ∈ D′ is an equivalence relation on D′. We prove that every topological space can be represented by a domain with norma totality.

The recursion-theoretic study of mathematical structures other thanωis now a field of much activity. Analysis and algebra are two such structures which have been studied for their effective contents. Studies in analysis began with the work of Specker on nonconstructive proofs in analysis [16] and in Lacombe's inspiring notes on relevant notions of recursive analysis [8]. Studies in algebra originated in the work of Frolich and Shepherdson on effective extensions of explicit fields [1] and in Rabin's study of computable fields (...) [15]. Equipped with the richness of modern techniques in recursion theory, Metakides and Nerode [11]–[13] began investigating the effective content of vector spaces and fields; these studies have been extended by Kalantari, Remmel, Retzlaff, Shore and others.Kalantari and Retzlaff [5] began a foundational inquiry into effectiveness in topological spaces. They consider a topological spaceXwith a countable basis ⊿ for the topology. The space isfully effective, that is, the basis elements are coded intoωand the operation of intersection of basis elements and the relation of inclusion among them are both computable. Similar to, the lattice of recursively enumerable subsets ofω, the collection of r.e. open subsets ofXforms a latticeℒunder the usual operations of union and intersection. (shrink)

While physical objects exist, they occupy space. We have developed a representation for the spaces that objects can occupy that meets the following criteria.