Some new results of difference perfect functions in topological spaces

Everyday problems are characterized by voluminous data and varying levels of ambiguity. Thereupon, it is critical to develop new mathematical approaches to dealing with them. In this context, the perfect functions are anticipated to be the best instrument for this purpose. Therefore, we investigate in this paper how to generate perfect functions using a variety of set operators. Symmetry is related to the interactions among specific types of perfect functions and their classical topologies. We can explore the properties and behaviors of classical topological concepts through the study of sets, thanks to symmetry. In this paper, we introduce a novel class of perfect functions in topological spaces that we term D-perfect functions and analyze them. Additionally, we establish the links between this new class of perfect functions and classes of generalized functions. Furthermore, while introducing the herein proposed D-perfect functions and analyzing them, we illustrate this new idea, explicate the associated relationships, determine the conditions necessary for their successful application, and give examples and counter-examples. Alternative proofs for the Hausdorff topological spaces and the D-compact topological spaces are also provided. For each of these functions, we examine the images and inverse images of specific topological features. Lastly, product theorems relating to these concepts have been discovered.