Countable Paracompactness of Nano Topological Space Modeled by Pre-open Sets Structures

Abstract

A new class of paracompactness in the nano topological spaces, termed nano pre open sets, are presented and explored in this article. Since the Nano topology contains at most five sets this leads us to establish new structures of nano open sets, nano open covers, and nano locally refinement coves   new requirement concepts are discussed like Nano pre open covers, Nano pre , , , and Nano pre- regular, Nano pre-normal  spaces, and Countable Nano Pre-open Paracompact. Many new theorems are stated and proved.