Fuzzy Ideal Supra Topological Spaces

Fadhil Abbas,Antonin Dvorák

doi:10.1155/2020/8824296

Abstract

In this paper, we introduce the notion of fuzzy ideals in fuzzy supra topological spaces. The concept of a fuzzy s-local function is also introduced here by utilizing the s-neighbourhood structure for a fuzzy supra topological space. These concepts are discussed with a view to find new fuzzy supra topologies from the original one. The basic structure, especially a basis for such generated fuzzy supra topologies, and several relations between different fuzzy ideals and fuzzy supra topologies are also studied here. Moreover, we introduce a fuzzy set operatorΨSand study its properties. Finally, we introduce some sets of fuzzy ideal supra topological spaces (fuzzy∗-supra dense-in-itself sets, fuzzyS∗-supra closedsets, fuzzy∗-supra perfect sets, fuzzy regular-I-supra closedsets, fuzzy-I-supra opensets, fuzzy semi-I-supra opensets, fuzzy pre-I-supra opensets, fuzzyα-I-supra opensets, and fuzzyβ-I-supra opensets) and study some characteristics of these sets, and then, we introduce some fuzzy ideal supra continuous functions.

Highlights

E concept of supra topology was introduced by Mashhour et al in 1983 [7]
Let (X, S, I) be a fuzzy ideal supra topology and let A be any fuzzy set in X. en, the fuzzy s-local function A∗S(I, S) of A is the union of all fuzzy points xα such that if M ∈ NS(xα) and E ∈ I, there is at least one y ∈ X for which M(y) + A(y) − 1x > E(y)
Let (X, S, I) be a fuzzy ideal supra topological space, and let A be any fuzzy set in X. en, (A ∪ A∗S) ∗ S⊆A∗S

Summary

Preliminaries

Definition 1 (see [1]). Let X be a nonempty set. A subclass S⊆P(x) (P(X) is the collection of all fuzzy sets on X and is called a fuzzy supra topology on X if 0x, 1x ∈ S, and S is closed under arbitrary union. Let (X, S) be a fuzzy supra topological space. A fuzzy set A in a fuzzy supra topological space (X, S) is an s-neighbourhood of a fuzzy point xα if there is M ∈ S with xα ∈ M⊆A. e collection NS(xα) of all s-neighbourhoods of xα is called the s-neighbourhood system of xα. Let S1 and S2 be two fuzzy supra topologies on a set X such that S1⊆S2. Let (X, S) be a fuzzy supra topological space and β⊆S. en, β is called a base for the fuzzy supra topology S if every fuzzy supra openset U ∈ S is a union of members of β. A nonempty collection of fuzzy sets I of a set X is called a fuzzy ideal on X if and only if (1) A ∈ I and B⊆A⇒B ∈ I(heredity) (2) A ∈ I and B ∈ I⇒A ∪ B ∈ I(finite additivity)

Fuzzy S-Local Function

S-Compatible of Fuzzy Ideals with Fuzzy Supra Topology

Fuzzy Set Operator ΨS

Some Fuzzy Ideal Supra Continuous Functions