$f$ topology

This paper presents and investigates the concept of soft optimal maps within the framework of soft topological spaces. A soft map is called soft optimal if it preserves soft disjointness for soft open sets in the soft structure. The link between soft injectivity and soft optimality is investigated, revealing that every soft injective map is soft optimal, but the converse is not true. Some correlations between this class of soft optimal maps and its analogs in general topology are given. The paper further shows that soft optimality is independent of all forms of soft continuity, soft openness, and soft closedness. Furthermore, the impact of soft optimality on soft semi-Hausdorff and soft Hausdorff properties is considered with concrete theorems showing how soft optimality affects the soft semi-Hausdorff state in soft topological spaces under various soft continuity assumptions. Finally, we show that the soft optimal map, which is soft s-open and soft pre-continuous, will preserve soft disjointness for the class of soft semi-open sets.

In this paper, we introduce the notion of primal topological group, which is a generalization of topological group by a primal. We discuss the characterizations of primal topological groups with illustrative examples.

The intention of this paper is to bring forth new maps namely ${\alpha g}_{(\gamma,\delta )}$- irresolute and ${\alpha g}_{(\gamma,\delta )}$-homeomorphism using ${\alpha g}_{\gamma }$-open sets and study their properties. This new concept enables us to determine the conditions under which maps and inverse maps, preserve ${\alpha g}_{\gamma }$-open sets. We derive the relation between ${\alpha g}_{(\gamma,\delta )}$- irresolute, $\alpha g$-$(\gamma,\delta)$-open, $\alpha g$-$(\gamma,\delta)$-closed maps and  ${\alpha g}_{(\gamma,\delta )}$-homeomorp-hism in topological spaces.

This paper introduces a notion of generalised geometric logic. Connections of generalised geometric logic with the L-topological system and L-topological space are established.

The fuzzy set generally identifies the topological relationship between two vague spatial objects. Indeterminacy can arise at any point in the modelling process, and the fuzzy model is unable to deal with this. Given that the Pythagorean fuzzy is better equipped to deal with indeterminacies than the fuzzy set, we advocated using Pythagorean fuzzy modelling to determine the topological relation between two spatial objects. This paper contributes to the expanding study of Pythagorean fuzzy topological spaces by introducing core, fringe, outer, regular open set, regular closed set, double connectedness and homeomorphism in Pythagorean fuzzy environment. Using these definitions, the paper proposes an algebraic model, namely the pythagorean fuzzy 9-intersection matrix, to find topological relations between any two Pythagorean fuzzy sets in a Pythagorean topological space. The inbuilt capability of the PFS to handle indeterminacy establishes the proposed model as the potential tool to find topological relations between two indeterminate spatial objects. Ample instances are discussed to nourish the existence of indeterminate spatial objects. Finally, a simple Pythagorean fuzzy region is defined and all possible relations between such two Pythagorean fuzzy regions are examined.

The prime goal of this article is to initiate the notion of fuzzy    $\mu^*$-open(closed) sets and fuzzy $\mu^*$-continuous functions and characterize  them. These concepts are defined in a fuzzy topological space in presence of a  generalized fuzzy topology, which becomes a new tool to study fuzzy topological spaces. It is observed that this class of fuzzy sets fail to form a fuzzy topology but it form a generalized fuzzy topology. Furthermore, the relationship of these fuzzy sets and fuzzy continuity with some existing fuzzy notions are established. Also the notion of fuzzy $(\tau, \mu^*)$-open(closed) functions is introduced and their equivalent conditions with  fuzzy $\mu^*$-continuous functions are established.

An interval set (or an interval-valued set), is a special set, which is an effective tool for illustrating and describing obscure information systems and partially known problems. Recently, Kim et al.\cite{r5} defined the topological structure for interval-value sets and studied many properties of them. In this work, we discuss some characteristics and relations of continuity in interval-valued topological spaces with some necessary illustrative examples. Then we provide a categorical framework for interval-valued topological spaces $\mathcal{IV}$-$\mathcal{TOP}$. Many functors and subcategories of $\mathcal{IV}$-$\mathcal{TOP}$ are defined and studied. Furthermore, the relationships between the $\mathcal{IV}$-$\mathcal{TOP}$ and its subcategories are investigated. We show that the category $\mathcal{TOP}$ is isomorphic to the category $\mathcal{IV}$-${\mathcal{TOP}_{1}}.$ Moreover, we demonstrate that $\mathcal{TOP}$ and $\mathcal{IV}$-$\mathcal{TOP}_{1}$ are bireflective full subcategories of $\mathcal{IV}$-$\mathcal{TOP}$.

The aim of this paper is to introduce a concept of type-2 $L$-preorders for $L$ being a complete residuated lattice. It can be considered as a common framework of $L$-preorders and hemimetrics, and also contains various kinds of fuzzy metrics, including Morsi fuzzy metrics, KM-fuzzy metrics and modular metrics, as natural examples. It is shown that the category of $L$-preordered sets can be reflectively and coreflectively embedded in that of type-2 $L$-preordered sets. A type-2 $L$-preorder can be supplied as different models for further study.

In this paper, we define and study the concept of $\mathcal{I}_{3}$-limit points and $\mathcal{I}_{3}$-cluster points of triple sequences in the topology induced by random $2$-normed spaces. We discuss the relationship between $\mathcal{I}_{3}$-cluster points and limit points and prove some important results.

Let $C_c(X)$ (resp., $C_c^*(X)$) denote the functionallycountable subalgebra of $C(X)$ (resp., $C^*(X)$),consisting of all functions (resp., bounded functions) with countable image.$C_c(X)$ (resp., $C_c^*(X)$) as a topological ring via $m_c$-topology (resp., $m^*_c$-topology) and $u_c$-topology (resp., $u^*_c$-topology) is investigated and the equality of the latter two topologies is characterized. Topological spaces which are called $N$-spaces are introduced and studied.It is shown that the $m_c$-topology on $C_c(X)$ and its relative topology as a subspace of $C(X)$ (with $m$-topology) coincide if and only if $X$ is an $N$-space. We also show that $X$ is pseudocompact if and only if it is both a countably pseudocompact, and an $N$-space.

We construct some types of universal closure operations induced by certain collection of morphisms. For this purpose, we use Lawvere-Tierney topologies and universal closure operations that correspond to each other to establish the equivalent conditions over the collection of morphisms. In this way we use multiple sieves instead of principal sieves for constructing results. Examples are also given to illustrate the established results.

‎For a positive integer $m$‎, ‎a subset of divisors of $m$ is called a textit{divisor topology on $m$} if it contains $1 $ and $m$ and it is closed under taking $gcd$ and $rm lcm$‎. ‎If $m=p_1dots p_n$ is a square free positive integer‎, ‎then a divisor topology $m$ corresponds to a topology on the set $[n]={1,2,ldots,n}$‎. ‎Giving some facts about divisor topologies‎, ‎we give a recursive formula for the number of divisor topologies on a positive integer‎.

‎For a given topological space $(X‎, ‎~Im)$,‎ ‎there is a coarser topology on $X$ which is called the semi-regular topology on $X$ (generated by regularly open subsets) and it is denoted by $Im^{delta}$‎. ‎In this paper‎, ‎we study the continuity of the group operation and the inversion mapping ($varsigmalongmapstovarsigma^{-1}$) as regards the semi-regular topology $Im^{delta}$ (not necessarily with the given topology)‎. ‎Then we study the said mappings with the blend of the given topology $Im$ and the semi-regular topology $Im^{delta}$‎. ‎In the twilight of this note‎, ‎we pose some questions which are noteworthy‎.

In this paper, we introduce and study three notions of  property in fuzzy topological spaces using quasi-coincidence sense, and we relate to other such notions. Then, we show that all these notions satisfy good extension property. These concepts also satisfy hereditary, productive and projective properties. We note that all these concepts are preserved under one-one, onto, fuzzy regular open and fuzzy regular continuous mappings. Finally, we discuss initial and final fuzzy topological spaces on our concepts.

This paper investigates how the extensions of a double fuzzy topological space affect its regular fuzzy closed sets. While some type of extensions leave all the regular fuzzy closed sets intact, there are regular fuzzy closed sets which remain so under all its extensions.

‎‎Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎We define the primary spectrum of $M$‎, ‎denoted by $mathcal{PS}(M)$‎, ‎to be the set of all primary submodules $Q$ of $M$ such that $(operatorname{rad}Q:M)=sqrt{(Q:M)}$‎. ‎In this paper‎, ‎we topologize $mathcal{PS}(M)$ with a topology having the Zariski topology on the prime spectrum $operatorname{Spec}(M)$ as a subspace topology‎. ‎We investigate compactness and irreducibility of this topological space and provide some conditions under which $mathcal{PS}(M)$ is a spectral space‎.

In this paper, pretopological and topological operators are introduced based on partially continuous linear transformations of the membership and non-membership functions for intuitionistic fuzzy sets. They turn out to be a generalization of the topological operators for intuitionistic fuzzy sets.On the other hand it is a generalization of the fuzzy set pretopological operators introduced by Wenzhong and Kimfung.