New topologies derived from the old one via operators

Abstract The main purpose of this work is to study the ideal topology defined by the minimal and maximal ideals on a topological space. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>X</m:mi> </m:mrow> </m:msup> </m:math> {2}^{X} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>X</m:mi> </m:mrow> </m:msup> </m:math> {2}^{X} is the power set of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X . We obtain some of their fundamental properties. In addition, several relationships among the above notions have been discussed. Moreover, we define a new topology on an ideal topological space, called sharp topology, via the sharp operator defined in this study, which turns out to be finer than the original topology. Furthermore, a decomposition of open sets (in the original topology) has been obtained. Finally, we conclude our work with some interesting applications.