Journal of Linear and Topological Algebra
Volume:12 Issue: 2, Spring 2023

‎The Mackey topology in the context of Hausdorff linearly topologized modules over a complete discrete valuation ring is introduced and characterizations of this concept are established‎. ‎Moreover the interplay between the concept of Mackey topology and two special classes of linearly topologized modules is discussed‎.

In topology, we found enough literature on topological operators but in generalized topology, there is only $\mu$-interior, $\mu$-closure, and $\mu$-boundary operator. In this article, we explore different types of operators like $\mu$-derived set operator, $\mu$-exterior operator, $\mu$-preboundary operator in generalized topology. We have shown that any operator can be developed as the above operators impose certain conditions, giving a unique generalized topology in each case.

‎Some curvature functionals which are defined according to the quadratic curvature invariants were studied on a special class of space-times‎. ‎We exactly determine metrics that are critical for those considering curvature functionals‎, ‎through homogeneous classes‎.

‎The vertex cover problem is a famous combinatorial problem‎, ‎and its complexity has been heavily studied‎. ‎It is known that it is hard to approximate to within any constant factor better than $2$‎, ‎while a $2$-approximation for it can be trivially obtained‎. ‎In this paper‎, ‎new properties and new techniques are introduced which lead to approximation ratios smaller than $2$ on special graphs;‎ ‎e.g‎. ‎graphs for which their maximum cut values are less than $0.999|E|$‎. ‎In fact‎, ‎we produce a ($1.9997$)-approximation ratio on graph $G$‎, ‎where the ($0.878$)-approximation algorithm of the Goemans-Williamson for the maximum cut problem on $G$ produces a value smaller than $0.877122|E|$‎.

‎The main purpose of this paper is to introduce and investigate a new class of open sets called $\delta_e$-open sets‎. For this aim‎, ‎first we define and study the notion of $e$-regular open set via $e$-closure operator‎. ‎Then‎, ‎we introduce the notion of $\delta_e$-open set via $e$-regular open set‎. ‎Several fundamental properties of the notion of $\delta_e$-open set have been revealed‎. ‎Also‎, ‎we show that the family of $\delta_e$-open sets is a topology strictly weaker than $\tau^{\delta}$ and stronger than $\tau$‎. ‎In addition‎, ‎we investigate relationships between the notion of $\delta_e$-open set and other existing notions in topology such as open sets‎, ‎regular open sets and $\delta$-open sets‎. ‎Furthermore‎, ‎we give not only various properties and characterizations but also examples and counterexamples‎. ‎Finally‎, ‎some properties related to separation axioms are revealed‎.

‎This research paper introduces the concept of $m$-topological transformation semigroup spaces and explores their fundamental set operations‎. ‎Additionally‎, ‎the study explores the properties of vector spaces defined on $m$-topological transformation semigroup spaces‎, ‎examining how algebraic structures interact with the underlying spaces‎.

Following [6], we define Grothendieck topologies on a small category and describe sheaves for these Grothendieck topologies. This generalizes, in a natural way, the theory of sheaves on a topological space.