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On the Topology $τ_R^{\diamond}$ of Primal Topological Spaces

Murad Özkoç, Büşra Köstel

Abstract

The main purpose of this paper is to introduce and study two new operators $(\cdot)_R^{\diamond}$ and $cl_R^{\diamond}(\cdot)$ via primal which is a new notion. We also show that the operator $cl_R^{\diamond}(\cdot)$ is a Kuratowski closure operator, while the operator $(\cdot)_R^{\diamond}$ is not. In addition, we prove that the topology on $X$, shown as $τ_R^{\diamond},$ obtained by means of the operator $cl_R^{\diamond}(\cdot)$ is finer than $τ_δ,$ where $τ_δ$ is the family of $δ$-open subsets of a space $(X,τ).$ Moreover, we not only obtain a base for the topology $τ_R^{\diamond}$ but also prove many fundamental results concerning this new structure. Furthermore, we give many counterexamples related to our results.

On the Topology $τ_R^{\diamond}$ of Primal Topological Spaces

Abstract

The main purpose of this paper is to introduce and study two new operators and via primal which is a new notion. We also show that the operator is a Kuratowski closure operator, while the operator is not. In addition, we prove that the topology on , shown as obtained by means of the operator is finer than where is the family of -open subsets of a space Moreover, we not only obtain a base for the topology but also prove many fundamental results concerning this new structure. Furthermore, we give many counterexamples related to our results.
Paper Structure (5 sections, 23 theorems, 6 equations)

This paper contains 5 sections, 23 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $(\cdot)^{\diamond}_R$ operator and its fundamental properties
  4. The $cl^{\diamond}_R$ operator and $\tau^{\diamond}_R$ topology
  5. Conclusion

Key Result

Theorem 2.1

vel Let $X$ be a topological space and $A\subseteq X.$ Then the followings hold. $(a)$$\delta\text{-}int(A)=\{x|(\exists U\in RO(X,x))(U\subseteq A)\}=\{x|(\exists U\in O(X,x))(int(cl(U))\subseteq A)\},$$(b)$$\delta\text{-}cl(A)=\{x|(\forall U\in RO(X,x))(U\cap A\neq\emptyset)\}=\{x|(\forall U\in O(

Theorems & Definitions (59)

  • Theorem 2.1
  • Definition 2.1
  • Corollary 2.1.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Corollary 3.0.1
  • Corollary 3.0.2
  • proof
  • Remark 3.1
  • ...and 49 more