A Topological Power Space

Abstract

A Power Set is not only a container of all family of subsets of a set and the set itself, but , in topology, it is also a generator of all topologies on the defined set. So, there is a topological existence of power set, being the strongest topology ever defined on a set, there are some properties of it's topological existence. In this paper, such properties are being proved and concluded. The following theorems stated are on the basis of the topological properties and separated axioms, which by satisfying, moves to a conclusion that, not only a power set is just a topology on the given defined set, but also it can be considered as a "Universal Topology" or a "Universal Topological Space" , that is the container of all topological spaces. This paper gives a general understanding about what a power set is, topologically and gives us a new perceptive from a "power set" to a "topological power space".