Topologies derived from the old one via ideals

TL;DR

This paper introduces new topologies derived from existing ones using ideals, exploring their properties, relationships, and applications in ideal topological spaces.

Contribution

It defines and studies minimal and maximal ideals, ideal quotients, annihilators, and introduces the sharp topology, expanding the theoretical framework of ideal topological spaces.

Findings

Defined minimal and maximal ideals on ideal topological spaces

Introduced the sharp topology, finer than the original topology

Provided applications demonstrating the concepts

Abstract

The main purpose of this paper is to introduce and study minimal and maximal ideals defined on ideal topological spaces. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of $2^X$, where $2^X$ is the power set of $X.$ We obtain some of their fundamental properties. In addition, several relationships among the above notions have been discussed. Moreover, we get a new topology, called sharp topology via the sharp operator defined in the scope of this study, finer than the old one. Furthermore, a decomposition of the notion of open set has been obtained. Finally, we conclude our work with some interesting applications.