Some Variants of Strong Normality in Closure Spaces Generated via Relations

Ria Gupta,Ananga Kumar Das

doi:10.1155/2021/6917297

Abstract

In this paper, some variants of strongly normal closure spaces obtained by using binary relation are introduced, and examples in support of existence of the variants are provided by using graphs. The relationships that exist between variants of strongly normal closure spaces and covering axioms in absence/presence of lower separation axioms are investigated. Further, closure subspaces and preservation of the properties studied under mapping are also discussed.

Highlights

Introduction and PreliminariesIn 1960, the theory of digital topology arose for the study of geometric and topological properties which in turn can be used in computer graphics, image processing, etc
To compare digital topology and general topology, the generalized topological structure was introduced in [2] by Smyth. e topological structure that arises from the directed graph is used in digital topology, while Slapal [3] in 2003 studied closure operations for digital topology
He studied the closure space that arises from α-ary relation and studied connectedness in digital spaces via closure operators on graphs [4]

Summary

Introduction and Preliminaries

In 1960, the theory of digital topology arose for the study of geometric and topological properties which in turn can be used in computer graphics, image processing, etc. A closure space (X, clR) generated from a binary relation R is said to be T1 if and only if for every two distinct points x and y ∈ X, both x ∉ 〈y〉R and y ∉ 〈x〉R hold. Let R be a binary relation on X; the closure space (X, clR) is said to be strongly normal if for two disjoint closed sets A clR(A) and B clR(B), there exist distinct x and y such that A ⊆ (〈x〉R), B ⊆ (〈y〉R), and 〈x〉R ∩ 〈y〉R ∅. Let R be a binary relation on X; the closure space (X, clR) is said to be strongly regular if for any closed set clR(A) A and a point x ∉ clR(A), there exist disjoint 〈u〉R and 〈v〉R such that x ∈ 〈u〉R and clR(A) ⊆ 〈v〉R

Variants of Strongly Normal Closure Space

Subspace

Preservation under Mapping