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Meekly SC*-Normal Spaces in Topological Spaces

Neeraj Kumar Tomar, Saroj Rani

Abstract

This chapter develops the concept of \textbf{meekly $SC^*$-normality}, a novel generalization of the classical notion of normality in topology. The proposed framework simultaneously broadens $SC^*$-normality and other established forms of normality, offering a unified perspective on generalized separation axioms. Fundamental properties are systematically derived, several equivalent characterizations are obtained, and the relationships between meekly $SC^*$-normal spaces and a range of existing normal-type spaces are rigorously analyzed. By establishing these structural connections, the chapter not only enriches the theory of generalized closed sets and separation axioms but also opens new directions for further research in advanced topological studies.

Meekly SC*-Normal Spaces in Topological Spaces

Abstract

This chapter develops the concept of \textbf{meekly -normality}, a novel generalization of the classical notion of normality in topology. The proposed framework simultaneously broadens -normality and other established forms of normality, offering a unified perspective on generalized separation axioms. Fundamental properties are systematically derived, several equivalent characterizations are obtained, and the relationships between meekly -normal spaces and a range of existing normal-type spaces are rigorously analyzed. By establishing these structural connections, the chapter not only enriches the theory of generalized closed sets and separation axioms but also opens new directions for further research in advanced topological studies.
Paper Structure (13 sections, 7 theorems, 36 equations)

This paper contains 13 sections, 7 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Basic Types of Closed and Open Sets
  4. Relationship Table of Closed Sets
  5. Normality and Its Generalizations
  6. Meekly SC*-Normal Spaces
  7. Hierarchy of Normality Concepts
  8. Relationship of Normality Concepts with Examples in Common Topologies
  9. Finite Topology
  10. Co-finite Topology
  11. Usual Topology on R
  12. Relationship Table in Different Topologies
  13. Remarks

Key Result

Theorem 5.3

Every $SC^*$-normal space is meekly $SC^*$-normal.

Theorems & Definitions (40)

  • Definition 3.1: SC$^*$-closed set chandrakala2022sc
  • Definition 3.2
  • Example 3.3
  • Definition 4.1: Normal Space bourbaki1951topologiedugundji1966topologyengelking1977general
  • Definition 4.2: k-Normal schepin1972real (Mildly Normal singal1973mildly) Space
  • Definition 4.3: Almost Normal Space singal1969almost
  • Definition 4.4: $\pi$-Normal Space kalantan2008pinormal
  • Definition 4.5: Almost Regular Space singal1969almost
  • Definition 4.6: Softly Regular Space kumar2018softly
  • Definition 4.7: $\alpha$-Normal Space arhangelskii2001alpha
  • ...and 30 more