The significance fore supra topological spaces as a subject of study cannot be overstated, as they represent a broader framework than traditional topological spaces. Numerous scholars have proposed extension to supra open sets, including supra semi open sets, supra per open and others. In this research, a notion for ⱨ-supra open created within the generalizations of the supra topology of sets. Our investigation involves harnessing this style of sets to introduce modern notions in these spaces, specifically supra ⱨ - interior, supra ⱨ - closure, supra ⱨ - limit points, supra ⱨ - boundary points and supra ⱨ - exterior of sets. It has been examining the relationship with supra open. The research was also enriched with many of characteristics of each concept. Building upon this set classification, we introduced several kinds of maps like supra ⱨ - continuous, supra ⱨ - open, supra ⱨ - tentative, supra ⱨ -globally and supra ⱨ - homeomorphism. Additionally, we have proven a collection of useful relationships for the aforementioned of functions. Furthermore, the research was enhanced with illustrative and refuting examples.
