Some Results on Rough k-Space

Mudheher Abdul Jabbar Hasan Albayati; Ahmed Lafta Mosa  Al-Hindawe; Heyam Khazaal  Hassan

Mudheher Abdul Jabbar Hasan Albayati Department of Economics, Faculty of Administration and Economics, University of Misan, Misan 64001, Iraq
Ahmed Lafta Mosa Al-Hindawe Department of Mathematics, Faculty of Computer Sciences and Mathematics, University of Kufa, Kufa 540011, Iraq
Heyam Khazaal Hassan Department of Mathematics, Faculty of Computer Sciences and Mathematics, University of Kufa, Kufa 540011, Iraq

Keywords: Hausdorff property, Kelley-space, rough set, rough topological space, rough Kelley space, rough open map, rough closed map

Abstract

This paper introduces the concept of rough k-space within the framework of rough set theory. The primary aim of this work is to define rough k-space and explore its properties, including rough continuity, homeomorphisms, and topological characteristics. Specifically, it is shown that the restriction of a rough continuous function to any rough compact subset of a rough space remains rough continuous. Additionally, the cross product of a rough k-space with a rough compact T₂-space results in a rough k-space. The study also highlights key hereditary and topological properties of rough k-spaces. The novelty of this research lies in its extension of rough set theory to include the concept of rough k-spaces, which integrates topological and rough set properties, and introduces a new approach to understanding the interaction between rough sets and continuous functions. Furthermore, the paper provides detailed results on the continuity and homeomorphism properties of rough k-spaces, offering a fresh perspective on their application in mathematical and computational contexts. The implications of these findings are significant for further research in rough topology, particularly in the development of robust mathematical models for rough set theory and its applications in areas such as decision-making, knowledge discovery, and artificial intelligence.

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