Softly $\pi g\hat{D}$-Normal Spaces

TL;DR

This paper introduces the concept of softly $ ext{πg} ext{D}$-normal spaces, explores their properties, relationships with other spaces, and establishes their topological and hereditary nature, along with new characterizations and theorems.

Contribution

It defines and studies softly $ ext{πg} ext{D}$-normal spaces, establishing their properties, relationships, and demonstrating their topological and hereditary characteristics, which is a novel contribution.

Findings

Softly $ ext{πg} ext{D}$-normality is a topological property.

It is hereditary with respect to closed subspaces.

New characterizations and preservation theorems are provided.

Abstract

The aim of this paper is to introduce a new class of softly normal called softly $\pi g\widehat{D}$ -normality by using $\pi g\widehat{D}$ -open sets and obtained several properties of such a space. We discuss many properties of this new space and we give some properties that connect this new spaces with some other topological spaces, also we present some examples and counter examples that show the relationships between softly $\pi g\widehat{D}$ -normal spaces and some other topological spaces, also we introduced the concept of $\pi g\widehat{D}$ -normal, almost $\pi g\widehat{D}$ -normal, quasi $\pi g\widehat{D}$ -normal, mildly $\pi g\widehat{D}$ -normal. The main result of this paper is that softly $\pi g\widehat{D}$ -normality is a topological property and it is a hereditary property with respect to closed domain subspaces. Moreover, we obtain some new characterizations and preservation theorems of softly $\pi g\widehat{D}$ -normal spaces. We insure that existence of utility for new results of softly $\pi g\widehat{D}$ -normality using separation axioms in topological spaces which is separate on a known separation axioms in topological spaces.