$f$-topology

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 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.

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.

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.

The purpose of this paper is to construct a weak hyper semi-quantale as a generalization of the concept of semi-quantale and used it as an appropriate hyperlattice-theoretic basis to formulate new lattice-valued topological theories. Based on such weak hyper semi-quantale, we aim to construct the notion of a weak hypervalued-topology as a generalized form of the so-called lattice-valued topology (or many-valued topology). Some properties of weak hyper semi-quantales and weak hypervalued-topologies will be studied. An adjunction between the category of weak hyper semi-quantales and the category of weak hypervalued quasi-topological spaces will be established.