In this thesis we present two logical systems, $\bf MP$ and $\MP$, for the purpose of reasoning about knowledge and effort. These logical systems will be interpreted in a spatial context and therefore, the abstract concepts of knowledge and effort will be defined by concrete mathematical concepts.

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)

Let L be one of the topological languages L t , (L ∞ω ) t and (L κω ) t . We characterize the topological spaces which are models of the L-theory of the class of ordinals equipped with the order topology. The results show that the role played in classical model theory by the property of being well-ordered is taken over in the topological context by the property of being locally compact and scattered.

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.

We propose a multi-agent logic of knowledge, public announcements and arbitrary announcements, interpreted on topological spaces in the style of subset space semantics. The arbitrary announcement modality functions similarly to the effort modality in subset space logics, however, it comes with intuitive and semantic differences. We provide axiomatizations for three logics based on this setting, with S5 knowledge modality, and demonstrate their completeness. We moreover consider the weaker axiomatizations of three logics with S4 type of knowledge and (...) prove soundness and completeness results for these systems. (shrink)

We propose a multi-agent logic of knowledge, public announcements and arbitrary announcements, interpreted on topological spaces in the style of subset space semantics. The arbitrary announcement modality functions similarly to the effort modality in subset space logics, however, it comes with intuitive and semantic differences. We provide axiomatizations for three logics based on this setting, with S5 knowledge modality, and demonstrate their completeness. We moreover consider the weaker axiomatizations of three logics with S4 type of knowledge and (...) prove soundness and completeness results for these systems. (shrink)

Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan `open sets are semidecidable properties'. But whereas on effectively given Scott domains all such properties are also open, this is no longer true in general. In this paper a characterization of effectively given topological spaces is presented that says which semidecidable sets are open. This result has important consequences. Not only follows the classical (...) Rice-Shapiro Theorem and its generalization to effectively given Scott domains, but also a recursion theoretic characterization of the canonical topology of effectively given metric spaces. Moreover, it implies some well known theorems on the effective continuity of effective operators such as P. Young and the author's general result which in its turn entails the theorems by Myhill-Shepherdson, Kreisel-Lacombe-Shoenfield and Ceĭtin-Moschovakis, and a result by Eršov and Berger which says that the hereditarily effective operations coincide with the hereditarily effective total continuous functionals on the natural numbers. (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.

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.

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)

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)

In [5], Metakides and Nerode introduced the study of recursively enumerable substructures of a recursively presented structure. The main line of study presented in [5] is to examine the effective content of certain algebraic structures. In [6], Metakides and Nerode studied the lattice of r.e. subspaces of a recursively presented vector space. This lattice was later studied by Kalantari, Remmel, Retzlaff and Shore. Similar studies have been done by Metakides and Nerode [7] for algebraically closed fields, by Remmel [10] (...) for Boolean algebras and by Metakides and Remmel [8] and [9] for orderings. Kalantari and Retzlaff [4] introduced and studied the lattice of r.e. subsets of a recursively presented topological space. Kalantari and Retzlaff consideredX, a topological space with ⊿, a countable basis. This basis is coded into integers and with the help of this coding, r.e. subsets ofωgive rise to r.e. subsets ofX. The notion of “recursiveness” of a topological space is the natural next step which gives rise to the question of what should be the “degree” of an r.e. open subset ofX? It turns out that any r.e. open set partitions ⊿; into four sets whose Turing degrees become central in answering the question raised above.In this paper we show that the degrees of the elements of the partition of ⊿ imposed by an r.e. open set can be “controlled independently” in a sense to be made precise in the body of the paper. In [4], Kalantari and Retzlaff showed that givenAany r.e. set andany r.e. open subset ofX, there exists an r.e. open set ℋ which is a subset ofand is dense in and in whichAis coded. This shows that modulo a nowhere dense set, an r.e. open set can become as complicated as desired. After giving the general technical and notational machinery in §1, and giving the particulars of our needs in §2, in §3 we prove that the set ℋ described above could be made to be precisely of degree ofA. We then go on and establish various results on the mentioned partitioning of ⊿. One of the surprising results is that there are r.e. open sets such that every element of partitioning of ⊿ is of a different degree. Since the exact wording of the results uses the technical definitions of these partitioning elements, we do not summarize the results here and ask the reader to examine §3 after browsing through §§1 and 2. (shrink)

The aim of this paper is to elucidate the relationship between Aristotelian conceptual oppositions, commutative diagrams of relational structures, and Galois connections.This is done by investigating in detail some examples of Aristotelian conceptual oppositions arising from topological spaces and similarity structures. The main technical device for this endeavor is the notion of Galois connections of order structures.

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)

In this work, we present a multi-agent logic of knowledge and change of knowledge interpreted on topological structures. Our dynamics are of the so-called semi-private character where a group G of agents is informed of some piece of information $$\varphi $$ φ, while all the other agents observe that group G is informed, but are uncertain whether the information provided is $$\varphi $$ φ or $$\lnot \varphi $$ ¬φ. This article follows up on our prior work where the dynamics (...) were public events. We provide a complete axiomatization of our logic, and give two detailed examples of situations with agents learning information through semi-private announcements. (shrink)

In this work, we present a multi-agent logic of knowledge and change of knowledge interpreted on topological structures. Our dynamics are of the so-called semi-private character where a group G of agents is informed of some piece of information \, while all the other agents observe that group G is informed, but are uncertain whether the information provided is \ or \. This article follows up on our prior work where the dynamics were public events. We provide a complete (...) axiomatization of our logic, and give two detailed examples of situations with agents learning information through semi-private announcements. (shrink)

In this paper we introduce the concept of interval valued neutrosophic soft topological space together with interval valued neutrosophic soft finer and interval valued neutrosophic soft coarser topology. We also define interval valued neutrosophic interior and closer of an interval valued neutrosophic soft set. Some theorems and examples are cites. Interval valued neutrosophic soft subspace topology are studied. Some examples and theorems regarding this concept are presented.

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 analyse the category-theoretical structures involved with the notion of continuity within the framework of formal topology. We compare the category of basic pairs to other categories of “spaces” by means of canonically determined functors and show how the definition of continuity is determined in a certain, canonical sense. Finally, we prove a standard adjunction between the algebraic approach to spaces and the category of topological spaces.

We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the (...) six classes of spaces considered in the paper are pairwise distinct, while the C-logics of some of them coincide. (shrink)

In this paper, we generalize the crisp topological spaces to the notion of neutrosophic crisp topological space, and we construct the basic concepts of the neutrosophic crisp topology.

I argue that relations between non-collocated spatial entities, between non-identical topological spaces, and between non-identical basic building blocks of space, do not exist. If any spatially located entities are not at the same spatial location, or if any topological spaces or basic building blocks of space are non-identical, I will argue that there are no relations between or among them. The arguments I present are arguments that I have not seen in the literature.

We examine conditions under which, in a computable topological space, a semicomputable set is computable. It is known that in a computable metric space a semicomputable set S is computable if S is a continuum chainable from a to b, where a and b are computable points, or S is a circularly chainable continuum which is not chainable. We prove that this result holds in any computable topological space.

In this paper, I draw a parallel between aspects of René Thom’s topological program understood as semiophysics, and Jakob von Uexküll’s theory of meaning. Through the use of Thom’s semiophysics, I believe that it is possible to interpret Uexküll’s intuition that meaning unfolds a kind of transformation in an organism’s transactions with the environment: that is, meaning incorporates topological spaces. The central idea in question is that beyond the semantic, syntactical and pragmatic human use of language, meaning incorporates (...) specific topologies by expressing the spatial interaction between organism and environment. This is what I mean by René Thom’s reading of Jakob von Uexküll. (shrink)

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

The first-order theory of the lattice of recursively enumerable closed subsets of an effective topological space is proved undecidable using the undecidability of the first-order theory of the lattice of recursively enumerable sets. In particular, the first-order theory of the lattice of recursively enumerable closed subsets of Euclidean n -space, for all n , is undecidable. A more direct proof of the undecidability of the lattice of recursively enumerable closed subsets of Euclidean n -space, n ⩾ (...) 2, is provided using the method of reduction and the recursive inseparability of the set of all formulae satisfiable in every model of the theory of SIBs and the set of all formulae refutable in some finite model of the theory of SIBs. (shrink)

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

We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from Brunner in ``The axiom of choice in topology'':(i) For every T2 topological space (X, T) if X is well-ordered, then X has a well-ordered base,(ii) For every T2 topological space (X, T), if X is well-ordered, then there exists a function f : X × W T such that W is a well-ordered set and f (...) ({x} × W) is a neighborhood base at x for each x X,(iii) For every T2 topological space (X, T), if X has a well-ordered dense subset, then there exists a function f : X × W T such that W is a well-ordered set and {x} = f ({x} × W) for each x X. (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)

A central concept for information retrieval is that of similarity. Although an information retrieval system is expected to return a set of documents most relevant to the query word(s), it is often described as returning a set of documents most similar to the query. The authors argue that in order to reason with similarity we need to model the concept of discriminating power. They offer a simple topological notion called resolution space that provides a rich mathematical framework for (...) reasoning with limited discriminating power, avoiding the vagueness paradox. (shrink)

We reformulate a key definition given by Wáng and Ågotnes to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ???? to be the “local truth” of ????, semantically rendered via an interior operator. This is closely related to the notion of ???? being “knowable”. We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial (...) role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness. (shrink)

We look at bimodal logics interpreted by cartesian products of topological spaces and discuss the validity of certain bimodal formulae in products of so-called cardinal spaces. This solves an open problem of van Benthem et al.

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)

Must space be a unity? This question, which exercised Aristotle, Descartes and Kant, is a specific instance of a more general one; namely, can the topology of physical space change with time? In this paper we show how the discussion of the unity of space has been altered but survives in contemporary research in theoretical physics. With a pedagogical review of the role played by the Euler characteristic in the mathematics of relativistic spacetimes, we explain how classical (...) general relativity (modulo considerations about energy conditions) allows virtually unrestrained spatial topology change in four dimensions. We also survey the situation in many other dimensions of interest. However, topology change comes with a cost: a famous theorem by Robert Geroch shows that, for many interesting types of such change, transitions of spatial topology imply the existence of closed timelike curves or temporal non-orientability. Ways of living with this theorem and of evading it are discussed. (shrink)