A Topological Space Defined On The Group Of Unites Modulo p. - A. Daouba, Hamza; A. Shafahb, Osama; A. Bribesha, Fathi (Libyan Journal of Science &Technology, 2026)
Highlights A novel non-discrete topology is defined on the unit group using conjugate pairs as basic open sets, where every open set is also closed (clopen). Highlights A novel non-discrete topology is defined on the unit group using conjugate pairs as basic open sets, where every open set is also closed (clopen). The function mapping to is continuous, open, and closed, establishing a strong topological link between the unit group and quadratic residues. The quotient space , under the equivalence , is homeomorphic to the discrete space of quadratic residues . The topology is disconnected and fails the separation axiom, whereas the discrete topology on satisfies stronger separation properties such as . Key topological operators (interior, closure, boundary, limit points) are explicitly characterized for arbitrary subsets of , revealing how structure depends on conjugate-pair symmetryfunction mapping to is continuous, open, and closed, establishing a strong topological link between the unit group and quadratic residues. The quotient space , under the equivalence , is homeomorphic to the discrete space of quadratic residues . The topology is disconnected and fails the separation axiom, whereas the discrete topology on satisfies stronger separation properties such as . Key topological operators (interior, closure, boundary, limit points) are explicitly characterized for arbitrary subsets of , revealing how structure depends on conjugate-pair symmetry This paper introduces a finite topological space on the group of units modulo a prime , defined by its basis of conjugate residue pairs for all units . We investigate the fundamental topological concepts such as point-set topology, separation axioms, and characterise the structure and behaviour of this topology. Additionally, we examine a function from to the topology of quadratic residues , mapping each unit to its square modulo . We analyse the continuity, openness of , and explore its implications for separation properties. Furthermore, we define a quotient topology on based on the equivalence relation if and only if, showing that the resulting quotient space is homeomorphic to .