Some Studies on the Topology of Power Set

Morawo, Monsuru A
Kiltho, Ahmadu
Y, Azeez, K.
O, Shobanke, E.
O, Okoro N.

Journal of Multidisciplinary Science: MIKAILALSYS
Vol 3 No 2 (2025): Journal of Multidisciplinary Science: MIKAILALSYS

This paper examines the topological structure of the power set of an infinite set X, with a focus on properties such as extreme and total disconnectedness, as well as the hierarchy of separation axioms T₀, T₁, T₂, T₃, T₄, T₅, and T₆. By defining a topology τ on the power set ????(X), the study explores the manifestation of classical topological properties within this framework. The investigation introduces a novel approach that connects ????(X) to a universal topological space, providing new insights into the behavior of separation axioms and disconnectedness in non-standard topological constructions. The results offer a foundational perspective for further study in abstract and generalized topology, particularly in contexts where conventional space constructions do not apply.