A Topological Space Defined On The Group Of Unites Modulo p.

Abstract

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 .