西 南 交 通 大 学 学 报
第 55 卷 第 2 期
2020 年 4 月
JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY
Vol. 55 No. 2
Apr. 2020
ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.2.25
Research article Mathematics
I
NTUITIONISTIC
F
UZZY
T
OPOLOGICAL
S
PACES
M
ODEL
直觉模糊拓扑空间模型
Hind Fadhil Abbas
Slah Al Deen, SAMMARA, Directorate of Education Salah Eddin, Khaled Ibn Al Walid School Tikrit City, Iraq, [email protected]
Received: January 14, 2020 ▪ Review: February 28, 2020 ▪ Accepted: April 11, 2020
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
Abstract
The fusion of technology and science is a very complex and scientific phenomenon that still carries mysteries that need to be understood. To unravel these phenomena, mathematical models are beneficial to treat different systems with unpredictable system elements. Here, the generalized intuitionistic fuzzy ideal is studied with topological space. These concepts are useful to analyze new generalized intuitionistic models. The basic structure is studied here with various relations between the generalized intuitionistic fuzzy ideals and the generalized intuitionistic fuzzy topologies. This study includes intuitionistic fuzzy topological spaces (IFS); the fundamental definitions of intuitionistic fuzzy Hausdorff space; intuitionistic fuzzy regular space; intuitionistic fuzzy normal space; intuitionistic fuzzy continuity; operations on IFS, the compactness and separation axioms.
Keywords:Topological Space, Intuitionistic Fuzzy Ideal, Intuitionistic Fuzzy Local Function
摘要 技术与科学的融合是一个非常复杂和科学的现象，仍然带有需要理解的谜团。为了解开这些 现象，数学模型有利于用不可预测的系统元素来处理不同的系统。在此，利用拓扑空间研究广义 直觉模糊理想。这些概念对于分析新的广义直觉模型很有用。这里研究了广义直觉模糊理想与广 义直觉模糊拓扑之间各种关系的基本结构。本研究包括直觉模糊拓扑空间；直觉模糊豪斯多夫空 间的基本定义；直觉模糊规则空间；直觉模糊法线空间此外，还详细研究了直觉模糊紧性，直觉 模糊连续性和分离公理的概念。 关键词: 拓扑空间，直觉模糊理想，直觉模糊局部函数
I. INTRODUCTION
Currently, the fusion of technology and science is a complex process. Additionally, this
scientific phenomenon still carries a few mysteries that need to be understood. To unravel this phenomenon, mathematical models are
beneficial to treat different systems with unpredictable system elements. Intuitionistic fuzzy sets can be used to develop maximum mathematical models that are based on a basic set theory extension. The concept was introduced by Zadeh with set operations [1]. In the same way, Chang introduced fuzzy topological spaces [2]. Ample literature is available regarding fuzzy sets [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Atanassov [3] reported intuitionistic fuzzy set and presented intuitionistic fuzzy sets based on the degrees of membership and of non-membership subject to the form fuzzy sets where sum of membership and of non-membership subject should not exceed 1. Tapas and Samanta [13] reported the basic relationship between generalized intuitionistic fuzzy topology and set notion. In the present study, the concept of generalized intuitionistic fuzzy topological space is presented [14]. The basic concept with mathematical expressions includes preliminaries considering generalized intuitionistic fuzzy ideals for a set, intuitionistic fuzzy sets (IFSs), and basic operations on IFSs. The generalized intuitionistic fuzzy local function regarding generalized intuitionistic fuzzy topological spaces generalized intuitionistic fuzzy topological spaces (GIFTS) and generalized intuitionistic fuzzy ideals is also included. The intuitionistic fuzzy topological space (IFTS), i.e., basis and sub-basis; the closure and interior of IFSs; intuitionistic fuzzy neighborhoods and intuitionistic fuzzy continuity; and compactness and separation axioms (intuitionistic fuzzy compactness) were studied. So, the aim of our study is to analyze generalized intuitionistic fuzzy topological space with general topologies and GIFTS.
II. BASIC CONCEPTS
A. Intuitionistic Fuzzy SetZadeh introduced fuzzy sets. Set X includes a fuzzy set A, which maps from X to interval (0, 1). Generalization of fuzzy sets is nothing but the Intuitionistic fuzzy sets which better model imperfect information with decision making ability. Degrees of membership and non-membership are considered under intuitionistic fuzzy sets. However, the condition is that the their sum should be less than 1. Consider one example that E is the collection of countries with elective democracy. Consider that country x belongs to E, i.e., x ∈ E the percentage voting for the respective government. It is denoted with A(x). Suppose, µ(x) = A(x)/100 is the validity degree (also known as membership degree). Let
B(x) = 1 − µ(x). B(x) indicates voting rate (rate at which votes were not cast). Here k(x) is the non-membership degree (non-validity degree). These votes are not given to the government. Here, we sort out voting rate as C(x) = 1 − µ(x) − B(x). We have a set: {(x, µ(x), B(x)): x ∈ E}.
Intuitionistic fuzzy set (IFS) is applicable in various areas of artificial intelligence. IF approaches with AI have the capacity to take decisions and perform machine learning and reasoning, pattern recognition, intelligent database systems, and logical programming. In the medical field, IFS is useful for diagnosis purpose, such as time to time update of patient’s temperature, heart rate, and / or their health condition [15]. Coker introduces that IFSs are studied in the topological framework. There are many general IF models in fields such as musicology, gravitational field, biology, astronomy, and controller’s sociology (a control system with inner and outer controls which works against tendencies to deviate).
B. Basic Operations on IFSs
Consider a set X has elements.
The IFSs P={x, µP(x), γ P (x): x ∈ X} and
Q={x, µ Q (x), γ Q (x) : x ∈ X}.
Some operations are listed here.
P ⊆ Q if µ P (x) ≤ µ Q (x) and γ P (x) ≥ γ Q (x) for values of x ∈ X. P = Q if P ⊆ Q Similarly, P = Q if Q ⊆ P. = {x, γ P (x), µ P (x) : x ∈ X}. P ∩ Q = {(x, µ P (x) ˄ µ Q (x), γ P (x) ˅ γ Q (x) )} and x ∈ X P U Q = {(x, µ P (x) ˅ µ Q (x), γ P (x) ˄ γ Q (x) )} and x ∈ X [ ] P = {x, µ P (x), 1 − µ P (x) : x ∈ X} P = {x, 1 − γ P (x), γ P (x) : x ∈ X} C. Intuitionistic Fuzzy Topological (IFT)
Space
An IFT with set X having elements is a household τ of an IFS with the following axioms
[t1] 0 ∼ and 1 ∼ ∈ τ
[t2] C1∩C2 belongs to τ (for any value C1, C 2∈τ)
[t3] U Ci belongs to τ, (for arbitrary family
{Ci:Ci ∈ τ, i ∈ I}).
Here, (X,τ) is known as an intuitionistic fuzzy topological space. IFS in household τ is called “intuitionistic fuzzy open in X set.”
D. Generalized Intuitionistic Fuzzy Local Functions and GIFTS
1) Properties of Generalized Intuitionistic Fuzzy Ideals
Consider that X is set with elements; F is a family of GIFTSs. Now F is a generalized intuitionistic fuzzy ideal if
1. P belongs to F, i.e., P∈ F and Q⊆ P ⇒ Q belongs to F i.e. Q ∈ F
2. P belongs to F, i.e., P∈ F and Q belongs to F i.e. Q ∈F⇒ P ˅ Q ∈ F .
A generalized intuitionistic fuzzy (GIF) L is known as σ GIF if ≤ F implies
Let (X,τ) be the general intuitionistic fuzzy topological spaces and F be the generalized intuitionistic fuzzy ideal. P is any one GIFTS of set X. A union of all generalized intuitionistic fuzzy points C(α, β) is known as a generalized intuitionistic fuzzy local function [ τ) ] in such a way that if U ∈ N(C(α , β )) and τ) = ∨{C(α,β) ∈ X: P ∧ U ∉ F U nbd of C(α,β)}. Then τ) is termed as the GIF local function of Set A. Here, F is denoted in a simple form: P*(F).
Theorem: Let’s consider (X, τ) to be the
GIFTS and L1, L2 to be generalized intuitionistic
fuzzy ideals at Set X. Therefore, for generalized intuitionistic fuzzy sets P, Q of Set X, the following statements are verified:
a. P ⊆ Q ⇒ τ} and τ} ⊆ τ} b. L1 ⊆ L2 ⇒ τ} ⊆ τ} c. P* = cl(P*) ⊆ cl(P) d. ⊆ P* e. (P ˅ Q)* = P* ˅ Q* f. (P ˄ Q)*(L) ≤ A*(L) ˄ B*(L) g. l ∈ L ⇒ (P ˅ l)* = P* Proof: a. As P ⊆ Q, let’s consider k = C(α, β) ∈ P*(L1); in this case, P ˄ U ∉ L for every U, which
belongs to N(k). Assuming B ˄ U ∉ L now, k = C(α,β) ∈ Q*
(L1).
b. L1 ⊆ L2 suggests τ) ⊆ τ). As
another IFSs belongs to L2, generalized
intuitionistic fuzzy k = C(α,β)∈P*
,but C(α,β) is not contained within .
c. As {O~} ⊆ L for any GIFL on Set X, (b) ) ⊆ P*
[{O~}] = cl(P) for any GIFS P on X. Consider k1 = C1(α,β) ∈ cl(P
*
(L1)). In this case,
for every U ∈ N(k1), P* ˄ U ≠ O~, there exists k 2
= C2(α,β) ∈ P *
(L1) ˄ U such that for every V nbd
of k2 ∈ N(k2), P ˄U does not belong to L. As U ˄
V ∈ N(k2), P ˄ (U ∩ V) does not belong to L,
which results in P ˄ U L for each U ∈ N (C(α,β)). Therefore, k1 = C(α,β) ∈ (P
*
(L)) and cl(P*) ≤ P*. While, the other inclusion obtained for follows directly the statement P*=cl(P*).
Thus, P* = cl(P*) but with the inequality where P* ≤ cl(P*
).
d. The inclusion of P* ˅ Q* ≤ (P ˅ Q)* can be directly followed through point a. To obtain another implication, let’s consider k = C(α,β) ∈ (P ˅ Q)*
;in such as case, for every U ∈ N(k), (P ˅ Q) ˄ U L, that is, (P ˄ U) ˅ (Q ˄ U) L. Now, we have the following two cases: (P ˄ U) L and (Q ˄ U) L or the converse. This implies that U1,U2 ∈ N (C(α,β)) exists in such a way that P ˄ U1 L, Q ˄ U1 L, P ˄ U2 L, and
Q ˄ U2 L. We can then write P ˄ (U1 ˄U2) ∈ L
and Q ˄ (U1 ˄ U2) ∈ L gives (P ˅ Q) ˄ (U1 ˄U2)
∈ L, (U1 ˄ U2) ∈ N(C(α,β)), which contradicts
hypothesis and consequently, the equality holds in various cases.
e. From point c, we have = cl(P*)* ≤ cl(P*) = P* Let us consider (X, τ) is a GIFTS and L be GIFL on set X. Let us define the generalized intuitionistic fuzzy closure operator cl*(P) = P P* for any GIFS P of set X. Clearly, let cl*(P) is a generalized intuitionistic fuzzy operator. Let τ*
(L) be GIFT generated by cl*, that is, τ*(L) = {P: cl*( ) = }. Now L = {O~} ⇒ cl*(P) = P P* = P cl(P) for every generalized intuitionistic fuzzy set P. So, τ*({O~}) = τ.
Again, L = {all GIFSs on set X} ⇒ cl*(P) = P, because P* = O~ for every generalized intuitionistic fuzzy set P, so the τ*
(L) is the generalized intuitionistic fuzzy discrete topology on X. So, we can conclude by this theorem, τ*({O~}) = τ*(L), that is, τ ⊆ τ*
, for any generalized intuitionistic fuzzy ideal (GIFI) on X. In particular, it has two generalized intuitionistic fuzzy ideals L1 and L2 on set X is L1 ⊆ L2 ⇒
τ*
(L1) ⊆ τ *
(L2).
E. Some Generalized Mappings in Intuitionistic Topological Spaces
Here, we introduce some definitions:
i. A map f: (P, ) → (Q, ) is known as an intuitionistic regular generalized α-closed map if the image of every intuitionistic closed set in (P, ) is Irgα closed in (Q, ). It is also known as, in brief, Irgα closed.
ii. A map f: (P, ) → (Q, ) is known as
Irgα continuous: when inverse image of each intuitionistic closed set in set R of (Q, ) is Irgα closed set in (P, ),
Irgα irresolute map: when inverse image of each Irga closed set in (Q, ) is Irga closed in (P, ),
strongly Irga continuous when inverse image of each Irga open set in (Q, ) is open in (P, ).
Irw closed: if every image is Irw closed in (Q, ) for each intuitionistic, regular, semi-open set of (P, )
Iw-closed: if every image is Iw in (Q, ) for each intuitionistic closed set of (P, )
Iwg closed: if every image is Iwg closed in (Q, ) for each intuitionistic closed set of (P, )
Irwg closed: if every image is Irwg closed in (Q, ) for each intuitionistic closed set of (P, )
Irg closed: if every image is Irg closed in (Q, ) for each intuitionistic closed set of (P, ).
Igpr closed: if every image is Igpr closed in (Q, ) for each intuitionistic closed set of (P,
),
Ig* closed if every image is Ig* closed in (Q, ) for each intuitionistic closed set of (P, ),
Irga closed if every image is Irga closed in (Q, ) for each intuitionistic regular α-closed in (P, ),
Iswg closed if every image is Iswg closed in (Q, ) for each intuitionistic semi closed in (P,
),
Ipwg closed if every image is Ipwg closed in (Q, ) for each intuitionistic pre closed in (P, )
Let f: (P, ) → (Q, ) be a mapping, with the following valid implications:
Figure 1. Generalized mapping intuitionistic topological spaces
F. Compactness and Separation Axioms
There are lots of restrictions to make various kinds of topological spaces that are given in the form of separation axioms. The separation axioms are those axioms that only exist when we define the notion of topological space, denoted by the letter ‘T.’ On the basis of axioms, we define the various kinds of topological spaces. An IFTS (P, τ) is called a fuzzy compact when X has a finite value for every fuzzy open cover.
III. CONCLUSION
The concept of a fuzzy set has wide applications in GIS fields, medical diagnosis, and microelectronic fault analysis. It was easy to understand the notion of IFTS. This finding was
explained by various concepts in the generalized IFTS.
