[Paper Review] Primal-Proximity Spaces

Ahmad Al-Omari, Murad Ozcog+1 more|arXiv (Cornell University)|May 26, 2023

Fuzzy and Soft Set TheoryDecision Sciences3 citations

TL;DR

This paper introduces primal-proximity spaces, a novel type of proximity structure built on the duality of grills and ideals in topology. It defines a point-primal proximity operator and proves it acts as a Kuratowski closure operator under specific conditions, while also constructing a weaker topology via new closure-like operators, establishing foundational properties and relationships through examples and theorems.

ABSTRACT

The main purpose of this paper is to introduce and study the primal-proximity spaces. Also, we define two new operators via primal proximity spaces and investigate some of their fundamental properties. In addition, we obtain a new topology, which is weaker than old one, via these new operators. Moreover, we not only discuss some of their properties but also enrich with some examples.

Motivation & Objective

• To introduce and formalize the concept of primal-proximity spaces as a new class of topological structures.
• To define a point-primal proximity operator and investigate its properties, particularly its behavior as a Kuratowski closure operator under specific conditions.
• To construct a new topology that is weaker than the original, using operators derived from primal-proximity.
• To establish relationships between primal-proximity, closure operators, and topological properties, supported by examples and theorems.
• To extend existing frameworks of proximity and ideal topologies by introducing primal duality as a foundational mechanism.

Proposed method

• Define a primal-proximity relation $\hookrightarrow$ on a set $X$ equipped with a primal $\mathcal{P}$, satisfying five axioms including symmetry, union decomposition, and a condition linking complement sets to non-proximity.
• Introduce the primal local function $A^\diamond(\delta,\mathcal{P})$ to characterize points where $A$ is not locally primal, forming the basis for the point-primal proximity operator.
• Prove that the point-primal proximity operator satisfies the Kuratowski closure axioms if $\mathcal{P}$ satisfies a specific condition involving $A^c \notin \mathcal{P}$.
• Define a second operator via the point-primal proximity that is a Kuratowski closure operator without additional assumptions.
• Construct a topology $\overset{\hookrightarrow}{\tau}$ from the primal-proximity relation and prove that a set $A$ is open in this topology iff $\{x\} \centernot\hookrightarrow A^c$ for all $x \in A$.
• Use the closure and interior operators derived from the topology to prove that $A \hookrightarrow B$ if and only if $cl_{\overset{\hookrightarrow}{\tau}}(A) \hookrightarrow cl_{\overset{\hookrightarrow}{\tau}}(B)$, establishing a key topological equivalence.

Experimental results

Research questions

• RQ1 How can a new type of proximity relation be defined using the duality of grills and ideals, specifically via a primal structure?
• RQ2 Under what conditions does the point-primal proximity operator satisfy the axioms of a Kuratowski closure operator?
• RQ3 What is the relationship between the primal-proximity topology and the closure properties of sets under this new structure?
• RQ4 How does the new topology $\overset{\hookrightarrow}{\tau}$ compare to the original topology in terms of openness and closedness?
• RQ5 Can the primal-proximity framework be used to derive weaker topologies while preserving key topological invariants?

Key findings

• The primal-proximity relation $\hookrightarrow$ satisfies all axioms of an Efremovič proximity, making $(X, \hookrightarrow, \mathcal{P})$ a valid proximity space with a primal structure.
• The point-primal proximity operator $A^\diamond$ is a Kuratowski closure operator if $A^c \notin \mathcal{P}$, establishing a link between primal duality and closure properties.
• A second operator derived from the point-primal proximity is a Kuratowski closure operator without any condition, providing a robust construction for closure in primal-proximity spaces.
• The topology $\overset{\hookrightarrow}{\tau}$ is strictly weaker than the original topology, as shown by the fact that $A \in \overset{\hookrightarrow}{\tau}$ iff $\{x\} \centernot\hookrightarrow A^c$ for all $x \in A$.
• The closure of a set $A$ in $\overset{\hookrightarrow}{\tau}$ satisfies $cl_{\overset{\hookrightarrow}{\tau}}(A) = A^\diamond$, meaning the closure is precisely the primal local function.
• The fundamental equivalence $A \hookrightarrow B$ if and only if $cl_{\overset{\hookrightarrow}{\tau}}(A) \hookrightarrow cl_{\overset{\hookrightarrow}{\tau}}(B)$ holds, demonstrating that the topological closure preserves the primal-proximity relation.