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A (generalized) topological space is called an iso-dense space if the set of all its isolated points is dense in the space. The main aim of the article is to show in ZF a new characterization of iso-dense spaces in terms of special quasiorders. For a non-empty family A of subsets of a set X, a quasiorder ≲ A on X determined by A is defined. Necessary and sufficient conditions for A are given to have the property that the topology consisting of all ≲ A -increasing sets coincides with the generalized topology on X consisting of the empty set and all supersets of non-empty members of A . The results obtained, applied to the quasiorder ≲ D determined by the family D of all dense sets of a given (generalized) topological space, lead to a new characterization of non-trivial iso-dense spaces. Independence results concerning resolvable spaces are also obtained. [ABSTRACT FROM AUTHOR]

In this work we introduce the concept of bornology on the theory of Frölicher spaces. A canonical bornology, called the functional bornology, is induced from structure functions, and bounded maps under this bornology are determined. Based on the structure of structure curves, we deduce that we cannot induce, canonically, a bornology from structure curves. For each of Frölicher subspace and product, an initial and functional bornology are induced and compared. Similarly, for each of Frölicher quotient and coproduct, a final and functional bornology are induced and compared. [ABSTRACT FROM AUTHOR]

We introduce the classes of (strongly) Θ-discrete homogeneous spaces. We discuss the relationships of these classes to other classes of spaces possessing homogeneity-related properties, such as (strongly) n-homogeneous spaces. Many examples are given distinguishing discrete homogeneity and other types of homogeneity. [ABSTRACT FROM AUTHOR]

The rapid evolution of rough-set theory has prompted the need for enhanced methodologies in medical diagnostics, particularly regarding COVID-19 variant detection. This study introduces refined mathematical techniques based on topological structures (called nearly initial-rough sets) derived directly from initial-rough sets. Four categories of rough-set methodologies are presented, demonstrating heightened accuracy through comprehensive comparisons against existing methods. By leveraging these techniques, a rule-based classification system for COVID-19 variants is established, achieving 100 % accuracy measures through rigorous testing against real-world and computer-generated data. The implications of these advancements in medical diagnosis hold promise for future research, offering accessible and precise tools for variant identification and prediction. Using a medical application as a case study, we demonstrate superiority through comparative analyses, aligning mathematical results with medical data and showcasing the potential for broader applications beyond experts in topology. Furthermore, the study outlines an algorithm simplifying implementation, particularly in MATLAB, and suggests future explorations in medical, economic, and diverse theoretical frameworks to enhance applicability.

Let 𝒞ℕ be a monoid which is generated by the partial shift α : n↦n +1 of the set of positive integers ℕ and its inverse partial shift β : n + 1 ↦n. In this paper we prove that if S is a submonoid of the monoid Iℕ∞ of all partial cofinite isometries of positive integers which contains Cscr;ℕ as a submonoid then every Hausdorff locally compact shift-continuous topology on S with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup S with an adjoined compact ideal.

In this paper our main interest is to introduce a new topology on an ideal topological space.We have studied some properties of this newly defined topology. It is shown that X is Lindelöf with respect to this new topology if and only if it is Lindelöf with respect to the old one.We have introduced and studied ω*-continuity and some related notions. Finally we have given decomposition theorem for continuity and ω*-continuity.

Sarsak [M. S. Sarsak, On some properties of generalized open sets in generalized topological spaces, Demonstr. Math. 46 (2013), no. 2, 415–427] studied some properties of generalized open sets in generalized topological spaces (GTSs); the primary purpose of this article is to investigate more properties of generalized open sets in GTSs. We mainly study the behaviours of regular closed sets, semi-open sets, regular semi-open sets, preopen sets, and β\beta -open sets in GTSs analogous to their behaviours in topological spaces.

Quite recently, a new minimal structure mH⋆{m}_{H}^{\star } and an mnmn-ℋg{{\mathcal{ {\mathcal H} }}}_{g}-closed set have been introduced in a previous study [T. Noiri and V. Popa, Closed sets in hereditary bi m-spaces, Questions Answers General Topol. 38 (2020), 133–142] by using two minimal structures m, nn and a hereditary class ℋ{\mathcal{ {\mathcal H} }}. In this paper, we introduce and investigate the notions of (m,n)\left(m,n)-ℋg⋆{{\mathcal{ {\mathcal H} }}}_{g}^{\star }-regularity and (m,n)\left(m,n)-ℋg⋆{{\mathcal{ {\mathcal H} }}}_{g}^{\star }-normality in a hereditary bi m-space (X,m,n,ℋ)\left(X,m,n,{\mathcal{ {\mathcal H} }}).

In this paper by using hereditary classes [6], we define the notion of γ -Lindelöf modulo hereditary classes called γℋ-Lindelöf and obtain several properties of γℋ-Lindelöf spaces.

In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.

This work studies a new version of normality called epi-quasi normality, which lies between epi-normality and epi-mild normality. In this paper, we investigate this property and present some examples that illustrate the relationships between epi-quasi normality and other kinds of both normality and regularity.

We obtain several characterizations of the existence of greatest elements of a total preorder. The characterizations pertain to the existence of unconstrained greatest elements of a total preorder and to the existence of constrained greatest elements of a total preorder on every nonempty compact subset of its ground set. The necessary and sufficient conditions are purely topological and, in the case of constrained greatest elements, are formulated by making use of a preorder relation on the set of all topologies that can be defined on the ground set of the objective relation. Observing that every function into a totally ordered set can be naturally conceived as a total preorder, we then reformulate the mentioned characterizations in the more restrictive case of an objective function with a totally ordered codomain. The reformulations are expressed in terms of upper semi- and pseudo-continuity by showing a topological connection between the two notions of generalized continuity. [ABSTRACT FROM AUTHOR]

The space-time geometry is rooted in the Minkowski 4-manifold. Minkowski and Euclidean topological 4-manifolds behave differently in view of compactness and local homogeneity. As a result, Zeno sequences are selectively admitted in such topological spaces. In this paper, the generalizations of topologically fibered n-spaces are proposed to formulate topological decomposition and the formation of projective fibered n-subspaces. The concept of quasi-compact fibering is introduced to analyze the formation of Zeno sequences in topological n-spaces (i.e., n-manifolds), where a quasi-compact fiber relaxes the Minkowski-type (algebraically) strict ordering relation under topological projections. The topological analyses of fibered Minkowski as well as Euclidean 4-spaces are presented under quasi-compact fibering and topological projections. The topological n-spaces endowed with quasi-compact fibers facilitated the detection of local as well as global compactness and the non-analytic behavior of a continuous function. It is illustrated that the 5-manifold with boundary embedding Minkowski 4-space transformed a quasi-compact fiber into a compact fiber maintaining generality. [ABSTRACT FROM AUTHOR]

Some sufficient conditions are presented for the continuity of the mapping f : ⟨ X , τ X * ⟩ → ⟨ Y , τ Y * ⟩ , where τ X * and τ Y * are the topologies induced by the local function on 푋 and 푌, resp. under the assumption that the mapping from ⟨ X , τ X ⟩ to ⟨ Y , τ Y ⟩ is continuous. Further, we consider open and closed functions in the cases in which the open (or closed) mapping is preserved through the domain and codomain "idealization". Through several examples, we illustrate that the considered conditions cannot be weakened. [ABSTRACT FROM AUTHOR]

Mixed fuzzy topological space is one of the major developments in topological spaces. Tripathy and Ray (Soft Comput 16(10):1691–1695, 2012) developed the concept of mixed fuzzy topological space in a different manner. In this article we try to investigate countable axioms and covering properties in the mixed fuzzy topological space developed by Tripathy and Ray. [ABSTRACT FROM AUTHOR]

Sarsak [On μ\mu -compact sets in μ\mu -spaces, Questions Answers Gen. Topology 31 (2013), no. 1, 49–57] introduced and studied the class of μ\mu -Lindelöf sets in μ\mu -spaces. Mustafa [μ\mu -semi compactness and μ\mu -semi Lindelöfness in generalized topological spaces, Int. J. Pure Appl. Math. 78 (2012), no. 4, 535–541] introduced and studied the class of μ\mu -semi-Lindelöf sets in generalized topological spaces (GTSs); the primary purpose of this paper is to investigate more properties and mapping properties of μ\mu -semi-Lindelöf sets in μ\mu -spaces. The class of μ\mu -semi-Lindelöf sets in μ\mu -spaces is a proper subclass of the class of μ\mu -Lindelöf sets in μ\mu -spaces. It is shown that the property of being μ\mu -semi-Lindelöf is not monotonic, that is, if (X,μ)\left(X,\mu ) is a μ\mu -space and A⊂B⊂XA\subset B\subset X, where AA is μB{\mu }_{B}-semi-Lindelöf, then AA need not be μ\mu -semi-Lindelöf. We also introduce and study a new type of generalized open sets in GTSs, called ωμ{\omega }_{\mu }-semi-open sets, and investigate them to obtain new properties and characterizations of μ\mu -semi-Lindelöf sets in μ\mu -spaces.

Very recently the concept of separation axioms in multiset topological space and the concept of mixed multiset topological space were developed. In this article we redefine the definition of mixed multiset topological space.We introduce the separation axioms in it and investigate some of their properties.We also show that the properties of being T 0 , T 1 , T 2 spaces are hereditary. [ABSTRACT FROM AUTHOR]

In this paper we study dimensions of the type dim for the class of ideal topological spaces. We shall start from the classical definition of the dimension functions dim and dim*, continue with the dimension -dim and finally, give two different kinds of functions of covering dimensional type. [ABSTRACT FROM AUTHOR]

In this article, we use quite elementary and simple ideas to rebuild and study the patch and flat topologies on the prime spectrum from a natural point of view (this new approach is based on the significant applications of the power set ring). Especially, the proof of a major result in the literature on the comparison of topologies greatly simplified and shortened. Also a new characterization for the finiteness of the minimal primes of a ring is given. Then as an application, all of the related results of Kaplansky, Anderson, Gilmer-Heinzer, Bahmanpour-Khojali-Naghipour and Naghipour on the finiteness of the minimal primes are easily deduced as special cases of this result. Another finiteness result due to Matlis is also easily obtained which states that a given ring has finitely many minimal primes if and only if no minimal prime is contained in the union of the remaining minimal primes. [ABSTRACT FROM AUTHOR]

Many mathematicians defined and studied soft separation axioms and soft continuity in soft spaces by using ordinary points of a topological space X. Also, some of them studied the same concepts by using soft points. In this paper, we introduce the concepts of soft pc−Ti{p}_{c}-{T}_{i} and soft pc−Ti⁎{p}_{c}-{T}_{i}^{\ast }, i=0,1,2i=0,1,2 by using the concept of soft pc{p}_{c}-open sets in soft topological spaces. We explore several properties of such spaces. We also investigate the relationship among these spaces and provide a counter example when it is needed.

In this paper, a new class of generalized closure and some properties are introduced in topological spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a new category of similarity measures, we begin to introduce the concept of effect matrix T ∧ (R 1 × R 2 × R 3 , p m , q n , f v) of type 3-tuple and study some of their properties. Moreover, from the soft set we find the pictures of type regular and irregular using effect matrix T ∧ (R 1 × R 2 × R 3 , p m , q n , f v) of type 3-tuple are found. An effect matrix T ∧ (R 1 × R 2 × R 3 , p m , q n , f v) of type 3-tuple is better than effect matrix T ∧ (ℜ 1 × ℜ 2 , p n , q m) of type 2-tuple, because we can deal with three different sets R 1 (a family of objectives), R 2 (a family of parameters), R 3 (a family of second parameters) in the same problem. This burden can be alleviated by application of type 3-tuple. Some applications of soft effect matrix of type 3-tuple T ∧ (R 1 × R 2 × R 3 , p m , q n , f v) in decision making problems are studied and explained. In this work we deal with pictures. Moreover, the similarity measure between two different soft sets under the same universal soft set (R 1 × R 2 × R 3) can be studied and explained its applications in medical diagnosis problems. [ABSTRACT FROM AUTHOR]

In this work, some new connotations of soft open sets and soft contra continuous mapping are investigated, they are called soft α* - open sets and soft contra α* - continuous mapping and their prosperities are studied and discussed. [ABSTRACT FROM AUTHOR]

We study properties of the Golomb topology on polynomial rings over fields, in particular trying to determine conditions under which two such spaces are not homeomorphic. We show that if K is an algebraic extension of a finite field and K′ is a field of the same characteristic, then the Golomb spaces of K[X] and K′[X] are homeomorphic if and only if K and K′ are isomorphic. [ABSTRACT FROM AUTHOR]

This paper constructs the compactification of rough pseudo metric spaces using the concept of Čech rough proximity spaces, in terms of rough linked grills which are generalizations of Cauchy sequences. [ABSTRACT FROM AUTHOR]

This paper concerns a generalization of approximately continuous functions, namely S -approximately continuous functions. This notion is associated with S -density points, where S is a sequence of measurable sets tending to 0. Moreover, we present some properties of these functions and show their connection with measurable functions, functions from the first Baire class, and Darboux functions. [ABSTRACT FROM AUTHOR]

The rapid evolution of rough-set theory has prompted the need for enhanced methodologies in medical diagnostics, particularly regarding COVID-19 variant detection. This study introduces refined mathematical techniques based on topological structures (called nearly initial-rough sets) derived directly from initial-rough sets. Four categories of rough-set methodologies are presented, demonstrating heightened accuracy through comprehensive comparisons against existing methods. By leveraging these techniques, a rule-based classification system for COVID-19 variants is established, achieving 100 % accuracy measures through rigorous testing against real-world and computer-generated data. The implications of these advancements in medical diagnosis hold promise for future research, offering accessible and precise tools for variant identification and prediction. Using a medical application as a case study, we demonstrate superiority through comparative analyses, aligning mathematical results with medical data and showcasing the potential for broader applications beyond experts in topology. Furthermore, the study outlines an algorithm simplifying implementation, particularly in MATLAB, and suggests future explorations in medical, economic, and diverse theoretical frameworks to enhance applicability., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)

Given a countable set of families { D k : k ∈ N } of pseudometrics over the same set D , we study the power-aggregations of this class, that are defined as convex combinations of integral averages of powers of the elements of ∪ k D k . We prove that a Lipschitz function f is dominated by such a power-aggregation if and only if a certain property of super-additivity involving the powers of the elements of ∪ k D k is fulfilled by f . In particular, we show that a pseudo-metric is equivalent to a power-aggregation of other pseudometrics if this kind of domination holds. When the super-additivity property involves a p -power domination, we say that the elements of D k are p -concave. As an application of our results, we prove under this requirement a new extension result of McShane-Whitney type for Lipschitz p -concave real valued maps. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish a relation between fuzzy soft lattices and fuzzy soft Priestley spaces. We have proved that each bounded distributive lattice of fuzzy soft sets is isomorphic to the lattice of all clopen upsets of a Priestley space. For this reason, we have defined fuzzy soft upsets, fuzzy soft downsets, fuzzy soft filters, fuzzy soft ideals and fuzzy soft Priestley space. To endorse the above relation, we have proved some related results. [ABSTRACT FROM AUTHOR]

In this paper we will define and discuss semi-Hurewicz type covering properties in ditopological texture spaces. We consider the behaviour of semi-Hurewicz and co-semi-Hurewicz selection properties under direlation and difunction between ditopological texture spaces.

A space (X, τ) is called epi-mildly normal if there exists a coarser topology τ′ on X such that (X, τ′) is Hausdorff (T2) mildly normal. We investigate this property and present some examples to illustrate the relationships between epi-mild normality and other weaker kinds of normality.

We describe the structure of Hausdorff locally compact semitopological O-bisimple inverse ω- semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological O-bisimple inverse ω-semigroup with a compact maximal subgroup is either compact or it is a topological sum of its H-classes. We describe the structure of Hausdorff locally compact semitopological O-bisimple inverse ω-semigroups with a monothetic maximal subgroups. We show the following dichotomy: a T1 locally compact semitopological Reilly semigroup (B(Z+, θ)0, τ) over the additive group of integers Z+, with adjoined zero and with a non-annihilating homomorphism is either compact or discrete. At the end we establish some properties of the remainder of the closure of the discrete Reilly semigroup B(Z+, θ) in a semitopological semigroup.

Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set X endowed with n pairwise comparable topologies Ʈ1 ⊂ · · · ⊂ Ʈn, by repeated application of the operations of complement and closure in the topologies Ʈ1, . . . , Ʈn to a subset A ⊂ X we can obtain at most

In this article, we discuss the relation of soft ω -open sets and soft ω -closed sets with other soft sets. We further define the concept of soft ω -interior and soft ω -closure and study their relation with soft interior and soft closure sets respectively. We also study some properties of soft ω -interior and soft ω-closure with the help of some illustrations. [ABSTRACT FROM AUTHOR]

Borsík and Doboš studied the problem of how to merge a family of metric spaces into a single one through a function. They called such functions metric preserving and provided a characterization of them in terms of the so-called triangle triplets. Since then, different papers have extended their study to the case of generalized metric spaces. Concretely, Mayor and Valero (Inf Sci 180:803–812, 2010) provided two characterizations of those functions, called quasi-metric aggregation functions, that allows us to merge a collection of quasi-metric spaces into a new one. In Massanet and Valero (in: Sainz-Palmero et al (eds) Proceedings of the 16th Spanish conference on fuzzy technology and fuzzy logic, European Society for Fuzzy Logic and Techonology, Valladolid, 2012) gave a characterization of the functions, called partial metric aggregation function, that are useful for merging a collection of partial metric spaces into single one as final output. Inspired by the preceding work, Martín et al. (in: Bustince et al (eds) Aggregation functions in theory and in practice. Advances in intelligent systems and computing, vol 228, Springer, Berlin, 2013) addressed the problem of constructing metrics from quasi-metrics, in a general way, using a class of functions that they called metric generating functions. In particular, they solved the posed problem providing a characterization of such functions and, thus, all ways under which a metric can be induced from a quasi-metric from an aggregation viewpoint. Following this idea, we propose the same problem in the framework of partial metric spaces. So, we characterize those functions that are able to generate a quasi-metric from a partial metric, and conversely, in such a way that Matthews' relationship between both type of generalized metrics is retrieved as a particular case. Moreover, we study if both, the partial order and the topology induced by a partial metric or a quasi-metric, respectively, are preserved by the new method in the spirit of Matthews. Furthermore, we discuss the relationship between the new functions and those families introduced in the literature, i.e., metric preserving functions, quasi-metric aggregation functions, partial metric aggregation functions and metric generating functions. [ABSTRACT FROM AUTHOR]

We give various relations between lower (upper) semi-continuous functions and cliquish functions in a generalized topological space. Also, we answer the open problems raised by Korczak-Kubiak et al. in [5]. [ABSTRACT FROM AUTHOR]

In this paper, we present the concept of generalized closed fuzzy soft sets in the Čech fuzzy soft closure spaces, termed g-closed fss's, and investigate their basic properties. By using g-closed fss's, we define a new fuzzy soft closure operator which is generates a new fuzzy soft topology finer than that induced from the Čech fuzzy soft closure operator. In addition, we define generalized Čech fuzzy soft continuous (respectively, irresolute) mapping by means of g-closed fss's. As application of these sets, -Čech fuzzy soft closure spaces are introduced and discuses. [ABSTRACT FROM AUTHOR]

A topological space is said to be resolvable if it has two disjoint dense subsets. If ℵ is a cardinal number(finite or infinite), a topological space is said to be ℵ -resolvable if it has a paiwise disjoint family of ℵ many dense subsets. Illanes [3] showed that a topological space which is κ -resolvable for every finite integer κ is necessarily ℵ 0 -resolvable. We generalize this result to infinite cardinals. We show that if a topological space X is κ -resolvable for every κ < ℵ and if cofinality of ℵ is ℵ 0 , then, X is ℵ -resolvable. [ABSTRACT FROM AUTHOR]