New Operators Inspired by Ideal Topological Spaces and Their Application to Establishing Topology and Compatibility Property

In this article at hand, we employ the sharp operator to set up two novel operators via ideal topological spaces, namely, Δ-operator and clΔ-operators, and demonstrate how they interact with other ideas and properties. To prove some invalid relationships and further illustrate our discussion of some of their properties, we provide some elucidative examples. Then, we show that the operator clΔ turns out to be a Kuratowski closure operator; therefore, we use clΔ-operator to create a new topology that is incomparable with the old one. Furthermore, we investigate the core properties of the operator of ∇-sharp local function and make use of it to describe a topology TΔ established by clΔ-operator. Finally, we present the concept of ♯-compatibility and explore its main characterizations. We also reveal the relationship between compatibility and ♯-compatibility supported with some counterexamples.