Some Studies on the Topology of Power Set

Abstract

This paper investigates the topological structure of the power set of an infinite set XX, focusing on properties such as extreme and total disconnectedness, as well as the hierarchy of separation axioms τ0,τ1,τ2,τ3,τ4,τ5,τ6\tau_0, \tau_1, \tau_2, \tau_3, \tau_4, \tau_5, \tau_6. By defining a topology on the power set, the study explores how classical topological concepts manifest in this context and introduces a novel perspective that bridges the power set with a Universal Topological Space. The analysis contributes to a deeper understanding of how separation properties and disconnectedness behave in non-traditional topological constructions, offering a foundational approach for further exploration in abstract topological frameworks.