New pairwise fuzzy topology through fuzzy ideal

The aim of this paper is introducing pairwise fuzzy topology on fuzzy bitopological spaces in Lowen's sense. we define the concept of fuzzy local function in pairwise fuzzy topological spaces and study some of its properties. Finally we find a new pairwise fuzzy topology from old one through fuzzy ideals.

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Introduction

Zadeh [9] introduced the concept of fuzzy sets and operations on fuzzy sets. Zadeh defined as follows: let Y be a non-empty set. A fuzzy set on Y is a function with domain Y and values in I = [0, 1]. Lowen [4] defined the notion of fuzzy topology. Mahomoud [5] introduced and studies about the notions of fuzzy ideal and fuzzy local functions. Nouh [7] started the study of fuzzy bitopological spaces. A fuzzy bitopological space [2] (shortly fbts) is a triple (𝑋, 𝜏 1, 𝜏 2 ) where X is a non-empty set, 𝜏 1 and 𝜏 2 are any two fuzzy topologies on X. In this paper we follow the definition of fuzzy topology in Lowen's [4] sense.

Preliminaries

Before entering into our work we need the following definitions and examples. [8] If 𝐴 is a fuzzy set of 𝑋, then the support of 𝐴 is defined as 𝑆(𝐴) = {𝑥 ∈ 𝑋  𝐴(𝑥) > 0}. Let 𝑋 be a non-empty set. A fuzzy set 𝐵 is said to be finite fuzzy set of X if and only if 𝑆(𝐵)𝑖s a finite set. [8] A non-empty collection ℐ of fuzzy sets of 𝑋 𝑖𝑠 𝑠aid to be a fuzzy ideal on X,if 𝐴, 𝐵 ∈ ℐ ⇒ 𝐴⋁𝐵 ∈ ℐ and A ∈ ℐ, B ≤ A => B ∈ ℐ. [8] Let A and B are two fuzzy sets of X. Then A 'intersection' B is defined as (𝐴 ∩ 𝐵)(𝑥) = 𝑚𝑎𝑥 {0, 𝐴(𝑥) + 𝐵(𝑥) -1} for all 𝑥 ∈ 𝑋. [8] Let (X, 𝜏) be a fuzzy topological space. The cl(A), the closure of a fuzzy set A is a fuzzy set defined by -cl(A)(x) = ⋁{⋋∕ 𝐵 ∈ 𝜏, 𝐵(𝑥) > 1 -⋋⇒ 𝐴 ∩ 𝐵 ≠ 0 ̅ } for all x ∈ X. [4] Let (X,𝜏) be a fuzzy topological space. The interior 𝐴 0 of a fuzzy set A of X is defined as 𝐴 0 = ⋁{𝐵: 𝐵 ≤ 𝐴, 𝐵 ∈ 𝜏}. Now we write the example 2.3.3 in [8] with small modification as follows:

Definition

Definition

Definition

Definition

Definition

Definition

Definition

Definition

Definition

is also a fuzzy topology on X. [8] Let (X,𝜏) be a fuzzy topological space with fuzzy ideal ℐ on X. Let A be a fuzzy set on X. The fuzzy local function A* is defined by 𝐴 * (𝑥) = ⋁ {⋋ 𝐵 ∈ 𝜏, 𝐵(𝑥) > 1 -⋋⇒ 𝐴 ∩ 𝐵 ∉ ℐ} for all 𝑥 ∈ 𝑋. [8] Let ( X, 𝜏 ) be a fuzzy topological space with fuzzy ideal ℐ on X. We define a fuzzy topology 𝜏 * (ℐ) induced by 𝜏 and ℐ as 𝜏 * (ℐ)= {𝑈 ∈ 𝐼 𝑋 : 𝜓(1 -𝑈) = 1 -𝑈} where 𝜓(𝐴) = 𝐴⋁𝐴 * : 𝜓(𝐴)is denoted by cl* (A).

Definition

Definition

Definition

2.9. Remark [1] Let (𝑋, 𝜏 1 , 𝜏 2 ) be a fuzzy bitopological space and  be a fuzzy set on X. 𝜏 𝑖 -cl(),𝜏 𝑖 -int(),i∈{1, 2} denote closure and interior of . In this paper we refer 𝜏 𝑖 -cl() by cl() 𝑖 and 𝜏 𝑖 -int() by  𝑖 0 .

2.10. Lemma [8] If A, B and C are fuzzy sets on X then C∩ (𝐴⋁𝐵) = (𝐶 ∩ 𝐴)⋁(𝐶 ∩ 𝐵). In this paper the collection of all pairwise fuzzy open sets on X is denoted by 𝜏 1,2 . We refer the collection 𝜏 1,2 as pairwise fuzzy topology on X and the pair (X, 𝜏 1,2 ) is called as pairwise fuzzy topological space (shortly pfts).

Pairwise fuzzy topological space

Pairwise fuzzy topological space

Definition

The following example asserts that 𝜏 1,2 need not be a fuzzy topology on X. and 𝑓 2 = 𝑓3 ̅ ⋁ ℎ) ∉ 𝜏 1,2 . Hence 𝜏 1,2 is not a fuzzy topology on X.

Example

Example

Remark

3. For a given pfts (X, 𝜏 1,2 ) there must be a fbts (𝑋, 𝜏 1 , 𝜏 2 ).

Definitions

Let (𝑋, 𝜏 1 , 𝜏 2 ) be a fbts with fuzzy ideal ℐ on X. Then for each 𝑖 ∈ {1,2} 1.𝑐𝑙(𝐴) 𝑖 (𝑥) = ⋁{𝜆 ∕ 𝐵 ∈ 𝜏 𝑖, 𝐵(𝑥) > 1 -𝜆 ⇒ 𝐴 ∩ 𝐵 ≠ 0} ̅ for all 𝑥 ∈ 𝑋.

2.𝑐𝑙(𝐴

Consider the fuzzy topology 𝜏 2 on ℕ and the fuzzy ideal ℐ 𝑓 of finite fuzzy sets in ℕ, then we show that for any 𝛼 2 ∈ ℕ, (𝛼 ̅ 2 ) * (ℐ 𝑓 ) = 0 ̅ . Let x ∈ ℕ. If (𝛼 ̅ 2 ) * (x) ≠ 0, then (𝛼 ̅ 2 ) * (x) = 𝜇 for some 0 < 𝜇 ≤ 1. By definition of (𝛼 ̅ 2 ) * , If 𝐵 ∈ 𝜏 2 with 𝐵(x) > 1 -𝜇 then 𝛼 ̅ 2 ∩ 𝐵 ∉ ℐ 𝑓 . That is S (𝛼 ̅ 2 ∩ 𝐵) is not a finite subset of ℕ. Hence { y ∈ ℕ ∕ 𝛼 ̅ 2 (y) + B(y) > 1} is not a finite subset of ℕ. But this is not possible, because x ∈ {2n-1,2n} for exactly one n ∈ ℕ. Let e be the other element of {2n-1,2n}. Define a fuzzy set h such that h (x) = h(e) = 1 and h(y) = 0 for all y ≠ x, e. Clearly h ∈ 𝜏 2 and { y ∈ ℕ ∕ 𝛼 ̅ 2 (y) + B(y) > 1} is a finite subset of ℕ. Hence (𝛼 ̅ 2 ) * (x) = 0 for allx∈ ℕ.Therefore (𝛼 ̅ 1,2 ) * ≤ (𝛼 ̅ 1 ) * ⋁(𝛼 ̅ 2 ) * = 𝛼 ̅ ⋁ 0 ̅ = 𝛼 ̅.That is (𝛼 ̅ 1,2 ) * ≤ 𝛼. ̅

Acknowledgement

AcknowledgementsThe authors express their sincere thanks to the reviewers for their careful checking of the details and for helpful comments that improved this paper. The authors are also thankful to the editor-in-chief and managing editors for their important comments which helped to improve the presentation of the paper. The authors declare that there is no conflict of interest regarding the publication of this paper.

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