Ordinary Topological Spaces Research Articles

Intuitionistic S4 as a logic of topological spaces

Abstract We design and study various topological semantics for the diamond-free intuitionistic modal logic $\textsf{iS4}$, an intuitionistic analogue of $\textsf{S4}$. Ultimately we prove that ordinary topological spaces can be used as semantics, using the specialization order to interpret intuitionistic implication and the interior for the modality. Some of our soundness and completeness results are mechanised in Coq.

Novel Approaches Characterizing Separation in Fuzzy Topology with Quasi-Coincidence

Fuzzy sets were introduced by Zadeh in 1965, and three years later, Chang defined fuzzy topological spaces. Fuzzy topological spaces are families of fuzzy sets that satisfy the three classical axioms of topology. In this paper, we introduce and study some new notions of To separation axioms in fuzzy topological spaces using the quasi-coincident relation for fuzzy sets. Every ordinary (crisp) topological space vacuously satisfies the condition of being quasi-To. We define the quasi-separation axioms for fuzzy topological spaces, as quasi-To, quasi-T1, Quasi-T2. We have introduced and studied some new notions of To separation axioms in fuzzy topological spaces using the quasi-coincident relation for fuzzy sets. These quasi-separation axioms are weaker than the corresponding classical To separation axioms, but they are still useful for characterizing and studying fuzzy topological spaces.

New Concepts of To Separation Axioms in Fuzzy Topology

Fuzzy set was introduced by Zadeh in his classical paper of 1965. Three years later, Chang gave the definition of fuzzy topology, which is a family of fuzzy set satisfying the three classical axioms. In this paper, we have introduced and studied some new notions of To separation axioms in fuzzy topological spaces by using quasi-coincident relation for fuzzy set. Every ordinary (crisp) topological space vacuously satisfies condition of being quasi-To. In this paper concept of quasi coincident relation used to introduce and investigate some quasi separation axioms such as To, T1, & T2. Concerning quasi-To space in the general frame work of fuzzy topological spaces.

Stone dualities from opfibrations

Stone dualities are dual equivalences between certain categories of algebras and those of topological spaces. A Stone duality is often derived from a dual adjunction between such categories by cutting down unnecessary objects. This dual adjunction is called the fundamental adjunction of the duality, but building it often requires concrete topological arguments. The aim of this paper is to construct fundamental adjunctions generically using (op)fibered category theory. This paper defines an abstract notion of formal spaces (including ordinary topological spaces as the leading example), and gives a construction of a fundamental adjunction between the category of algebras and the category of corresponding formal spaces. Moreover, prove an Adjoint Lifting Theorem in the setting of near 2-fibrations, and construct the fundamental adjunction of Priestley duality.

Theory of Generalized Compactness in Generalized Topological Spaces: Part I. Basic Properties

In this paper, a novel class of generalized compact sets (briefly, g-Tg-compact sets ) in generalized topological spaces (briefly, 
 Tg-spaces ) is studied. The study reveals that g-Tg -compactness implies ordinary compactness (briefly, Tg -compactness ) in 
 Tg-spaces, and such statement implies its analogue in ordinary topological spaces (briefly, T -spaces ). Diagrams establish the various relationships amongst these types of g-Tg -compactness presented here and in relation to other types of g-T -compactness
 in T-spaces presented in the literature of Tg -spaces, and a nice application supports the overall theory.

Digital Hopf Spaces and Their Duals

In this article, we study the fundamental notions of digital Hopf and co‐Hopf spaces based on pointed digital images. We show that a digital Hopf space, a digital associative Hopf space, a digital Hopf group, and a digital commutative Hopf space are unique up to digital homotopy type; that is, there is only one possible digital Hopf structure up to digital homotopy type on the underlying digital image. We also establish an equivalent condition for a digital image to be a digital Hopf space and investigate the difference between ordinary topological co‐Hopf spaces and their digital counterparts by showing that any digital co‐Hopf space is a digitally contractible space focusing on deep‐learning methods in imaging science.

Fundamental Group of Rough Topological Spaces

In this paper we study and define the concept of the fundamental group of rough topological spaces (RTSs), which deeply depends on the concepts of rough sets (RSs) and rough topology (RT). Working towards this stated objective, we define the concept of rough path (RPt) which gives room for the introduction of rough loop (RL). We also define the concepts of rough homotopy (RH) and later shows that it is indeed an equivalence relation. We introduce the fundamental group of rough topological spaces by showing that all the group axioms satisfied. Also, this paper establish the fact that most of the results in fundamental group of ordinary topological spaces are also hold for the fundamental group of rough topological spaces.

On Two Classes of Soft Sets in Soft Topological Spaces

In this paper, we define soft ω -open sets and strongly soft ω -open sets as two new classes of soft sets. We study the natural properties of these types of soft sets and we study the validity of the exact versions of some known results in ordinary topological spaces regarding ω -open sets in soft topological spaces. Also, we study the relationships between the ω -open sets of a given indexed family of topological spaces and the soft ω -open sets (resp. strongly soft ω -open sets) of their generated soft topological space. These relationships form a biconditional logical connective which is a symmetry. As an application of strongly soft ω -open sets, we characterize soft Lindelof (resp. soft weakly Lindelof) soft topological spaces.

On some generated soft topological spaces and soft homogeneity

We introduce soft homogeneity as an extension of homogeneity in ordinary topological spaces. Based on the generated soft topology of a given indexed family of classical topologies inspite of a one topology given by Terepeta in [16], we investigate soft minimal open set and homogeneity relation between the generated soft topology and the given indexed family of topologies. We introduce several results, examples and counterexamples.

The Spectrum of a Separable Dynkin Algebra and the Topology Defined on It

The author continues his previous works on preparation to develop generalized axiomatics of the probability theory. The approach is based on the study of set systems of a more general form than the traditional set algebras and their Boolean versions. They are referred to as Dynkin algebras. The author introduces the spectrum of a separable Dynkin algebra and an appropriate Grothendieck topology on this spectrum. Separable Dynkin algebras constitute a natural class of abstract Dynkin algebras, previously distinguished by the author. For these algebras, one can define partial Boolean operations with appropriate properties. The previous work found a structural result: each separable Dynkin algebra is the union of its maximal Boolean subalgebras. In the present note, leaning upon this result, the spectrum of a separable Dynkin algebra is defined and an appropriate Grothendieck topology on this spectrum is introduced. The corresponding constructions somewhat resemble the constructions of a simple spectrum of a commutative ring and the Zariski topology on it. This analogy is not complete: the Zariski topology makes the spectrum of a commutative ring an ordinary topological space, while the Grothendieck topology, which, generally speaking, is not a topology in the usual sense, turns the spectrum of a Dynkin algebra into a more abstract object (site or situs, according to Grothendieck). This suffices for the purposes of the work.

On separating axioms and similarity of soft topological spaces

We will consider soft topologies defined on the same universe X with E as the set of parameters. It is shown that soft topologies are not equivalent to the general topologies defined on X. Moreover, some implications between soft separating axioms are different than those for ordinary topological spaces. The relation of similarity of soft topological spaces is also introduced and examined. The soft topologies {mathcal {T}}_{1} and {mathcal {T}}_{2} on X are called similar, if the families {mathcal {T}}_{1} and {mathcal {T}}_{2} are mutually coinitial. We study some basic properties of similar soft topological spaces: we check whether the family of all such spaces forms a lattice, we examine the relationship between similarity and being homeomorphic and consider the similarity of e-parameterized topologies.

Initial and Final Characterized Fuzzy 1 3 2 T and Finer Characterized Fuzzy 12 2 R -Spaces

Basic notions related to the characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces are introduced and studied. The metrizable characterized fuzzy spaces are classified by the characterized fuzzy 1 2 2 R and the characterized fuzzy T4-spaces in our sense. The induced characterized fuzzy space is characterized by the characterized fuzzy 1 3 2 T and characterized fuzzy 1 3 2 T -space if and only if the related ordinary topological space is 1 2, 2 R I• 12 -space and 1 3, 2 T I• 12 -space, respectively. Moreover, the α-level and the initial characterized spaces are characterized 1 2 2 R and characterized 1 3 2 T -spaces if the related characterized fuzzy space is characterized fuzzy 12 2 R and characterized fuzzy 1 3 2 T , respectively. The categories of all characterized fuzzy 1 2 2 R and of all characterized fuzzy 1 3 2 T -spaces will be denoted by CFR-Space and CRF-Tych and they are concrete categories. These categories are full subcategories of the category CF-Space of all characterized fuzzy spaces, which are topological over the category SET of all subsets and hence all the initial and final lifts exist uniquely in CFR-Space and CRF-Tych. That is, all the initial and final characterized fuzzy 1 2 2 R spaces and all the initial and final characterized fuzzy 1 3 2 T -spaces exist in CFR-Space and in CRF-Tych. The initial and final characterized fuzzy spaces of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy 1 3 2 T -space are characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces, respectively. As special cases, the characterized fuzzy subspace, characterized fuzzy product space, characterized fuzzy quotient space and characterized fuzzy sum space of a characterized fuzzy 1 2 2 R -space and of a characterized fuzzy 1 3 2 T -space are also characterized fuzzy 1 2 2 R and characterized fuzzy 1 3 2 T -spaces, respectively. Finally, three finer characterized fuzzy 1 2 2 R -spaces and three finer characterized fuzzy 1 3 2 T -spaces are introduced and studied.

Characterization of Fuzzy δg*-Closed Sets in Fuzzy Topological Spaces

The purpose of this research article is to explain the meaning of g-closed sets in fuzzy topological spaces, which is more understandable to the readers and we find some of its basic properties. The concept of fuzzy sets was introduced by Zadeh in his classical paper (1965). Thereafter many investigations have been carried out, in the general theoretical field and also in different applied areas, based on this concept. The idea of fuzzy topological space was introduced by Chang (1968). The idea is more or less a generalization of ordinary topological spaces. Different aspects of such spaces have been developed, by several investigators. This paper is also devoted to the development of the theory of fuzzy topological spaces.

On CDH fuzzy spaces

We extend the concept of being countable dense homogeneous to include fuzzy topological spaces. Our extension is proved to be a good extension in the sense of Lowen. We study the relation between fuzzy CDH spaces and some ordinary topological spaces generated by these fuzzy spaces.

Peritopological Spaces and Bisimulations

Generalizing ordinary topological and pretopological spaces, we introduce the notion of peritopology where neighborhoods of a point need not contain that point, and some points might even have an empty neighborhood. We briefly describe various intrinsic aspects of this notion. Applied to modal logic, it gives rise to peritopological models, a generalization of topological models, a spacial case of neighborhood semantics. (In a last section, the relation between the latter and the former is discussed, cursorily). A new cladding for bisimulation is presented. The concept of Alexandroff peritopology is used in order to determine the logic of all peritopological spaces, and we prove that the minimal logic K is strongly complete with respect to the class of all peritopological spaces. We also show that the classes of T0, T1 and T2-peritopological spaces are not modal definable, and that D is the logic of all proper peritopological spaces. AMS subject classification : 03B45, 54A05

Intuitionistic Interval-Valued Fuzzy Topological Spaces

By using the concept of intuitionistic interval-valued fuzzy sets, we introduce the notion of intuitionistic interval-valued fuzzy topology. And we study some fundamental properties of intuitionistic interval-valued fuzzy topological spaces: First, we obtain analogues[see Theorem 3.11 and 3.12] of neighborhood systems in ordinary topological spaces. Second, we obtain the result[see Theorem 4.9] corresponding to "the 14-set Theorem" in ordinary topological spaces. Finally, we give the initial structure on intuitionistic interval-valued fuzzy topologies[see Theorem 5.9].

Functorial relationships between lattice-valued topology and topological systems

This paper investigates functorial relationships between lattice-valued topology (arising from fuzzy sets and fuzzy logic) and topological systems (arising from topological and localic aspects of domains and finite observational logic in computer science). Two such relationships are embeddings from TopSys into Loc-Top, both having two fold significance: for computer science the significance is that TopSys is not topological over Set × Loc, yet Loc-Top is topological over Set × Loc; hence these embeddings can be used to construct in Loc-Top the unique initial [final] lifts of all forgetful functor structured sources [sinks] in TopSys; and for topology, the significance is that both embeddings generate anti-stratified topological spaces from ordinary topological spaces and spatial locales rewritten as topological systems, thus justifying the current structural axioms of Loc-Top and lattice-valued topology (which include all anti-stratified, non-stratified, and stratified spaces).

Local homogeneity in fuzzy topological spaces

Three types of local homogeneity in L‐topological spaces are introduced and studied, and each is characterized and proved to be a good extension of local homogeneity in ordinary topological spaces. Many implications concerning them are introduced. The study deals with the L‐topologically generated topological spaces.

Comonoids in chu: a large cartesian closed sibling of topological spaces

ComK may be defined as the (cartesian closed) category of comonoids in chuK, or equivalently as dictionaries D for which any crossword over D has its main diagonal in D. Com2 resembles Top, ordinary topological spaces. Common to both are the Alexandroff posets and the Scott DCPOs, while the topological space R and the dual DCPO {−∞ < … < −2 < −1 < 0} jointly witness the incomparability of Com2 and Top. Such comonoids support a notion of bitopology admitting limits simultaneously for convergence and divergence. We raise the questions of whether a comonoid in chu2 can be fully specified in terms of its specialization order and omitted cuts, and which cuts are optional. These questions have been actively pursued for four weeks as of this writing on the theory-edge mailing list in response to Puzzle 1.5 starting with http://groups.yahoo.com/group/theory-edge/messages/6957.

Completely induced L-fuzzy topological spaces

The first aim of this paper is to introduce and to study the concepts of ‘complete Scott continuity’ and ‘completely induced L-fuzzy topological space’. The second is to discuss the connections between some separation, countability and covering properties of an ordinary topological space and its corresponding completely induced L-fuzzy topological space.