Search

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]

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]

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