On Fuzzy Point Applications of Fuzzy Topological Spaces

Radwan Abu-Gdairi; Arafa A. Nasef; Mostafa A. El-Gayar; Mostafa K. El-Bably

doi:10.5391/IJFIS.2023.23.2.162

Original Article

Split Viewer

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 162-172

Published online June 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.2.162

On Fuzzy Point Applications of Fuzzy Topological Spaces

Radwan Abu-Gdairi¹, Arafa A. Nasef², Mostafa A. El-Gayar³, and Mostafa K. El-Bably⁴

¹Department of Mathematics, Faculty of Science, Zarqa University, Zarqa, Jordan
²Department of Physics and Engineering Mathematics, Faculty of Engineering, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt
³Department of Mathematics, Faculty of Science, Helwan University, Helwan, Egypt
⁴Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

Correspondence to :
Mostafa K. El-Bably (mkamel_bably@yahoo.com)

Received: January 31, 2023; Revised: March 21, 2023; Accepted: April 7, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

In this study, different kinds of fuzzy point applications in fuzzy topological spaces were examined. In addition, the characteristics of fuzzy β-closed sets via the contribution of fuzzy points, as well as new separation axioms, were investigated. Besides, some properties related to the β-closure of such points using the notions of weak and strong fuzzy points were examined with illustrated examples. New ideas regarding fuzzy β-upper limit, fuzzy β-lower limit, and fuzzy β-limit using the idea of fuzzy points were also discussed. Lastly, the fuzzy -continuous convergence on the set of fuzzy β-continuous functions was characterized with the help of the fuzzy β-upper limit.

Keywords: Fuzzy topological space, Fuzzy point, Fuzzy convergence, Fuzzy separation axioms, Fuzzy β-open sets

1. Introduction

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

Recently, topological structures have been used in many approaches and applications, such as rough sets and their extensions [1–6], decision-making problems [7–9], medical applications [10–14], topological reduction of attributes for predicting lung cancer and heart failure [15,16], biochemistry [17–20], fuzzy sets applications [21, 22], and bipolar hypersoft classes [23–26]. Thus, the main goal of this study is to discuss and study some topological structures in fuzzy topological spaces. First, we summarize some properties of fuzzy β-closed sets via the contribution of fuzzy points, and new separation axioms in fuzzy topological spaces are examined. In addition, based on the ideas of weak and strong fuzzy points, some properties related to the β-closure of such points are presented. Section 3 is describes the investigation of the concepts of fuzzy β-upper limit, fuzzy β-lower limit, and fuzzy β-limit. As a final point, the fuzzy β-continuous convergence on the set of fuzzy β-continuous functions is given. Accordingly, a characterization of the fuzzy β-continuous convergence with the assistance of fuzzy β-upper limit is discussed in Section 4. In this paper, we customize the symbol / to denote the unit interval [0, 1].

The main notion of a fuzzy set, which led to the expansion of fuzzy mathematics, was established in 1965 by Zadeh [27]. Let X be a nonempty set, and then a fuzzy set in X is defined by a function with domain X and values in I, i.e., an element of I^X.

Then, for any members M,N of I^X, we say that M is contained in N if M(s)≤N(s) for every s∈X, denoted by M≤N.

Definition 1.1 ([27])

For each M,N∈IX, we define the next fuzzy sets in I^X as follows:

(i) (M∧N) (s)=min{M(s),N(s)}, for each s∈X.

(ii) (M∨N) (s)=max{M(s),N(s)}, for each s∈X.

(iii) Mc(s)=1-M(s), for each s∈X.

(iv) Letting f:X→Y,M∈IX and N∈IY, then f(M) is a fuzzy set in Y, such that f(M) (t)=sup{M(s):s∈f-1(t)}, if f-1(t)≠Φ. Moreover, f(M) (t)=0 if f-1(t)=Φ. Also, f-1(N) is a fuzzy set in X, given by f-1(N) (s)=N(f(s)), s∈X.

Definition 1.2 ([27])

A fuzzy topology T on a set X is defined by a family of fuzzy sets (T⊆IX) such that T satisfies the following three conditions:

(i) If 0̄ and 1̄ are the characteristic functions of χ_Φ and χ_X, respectively, then 0̄, 1¯∈T.

(ii) M∧N∈T, for each M,N∈T.

(iii) If {Mj:j∈J}⊆T, then ∨{Mj:j∈J}∈T.

Therefore, the pair (X, T) is called a fuzzy topological space (and in briefly, we call X a fuzzy space).

Wong [28] used the concept of fuzzy set to present and examine the ideas of fuzzy points. In the current article, we assumed the definition of a fuzzy point in the sense of Pu and Liu [29,30] as follows:

Considering a set X, the point s∈X is called a fuzzy point if it takes the value 0 for all t∈X except one 1. Moreover, if its value at s is λ such that 0 < λ ≤ 1, then the fuzzy point is denoted by psλ, and the point s is called its support, denoted by supp(psλ), that is supp(psλ)=s. We symbolize the class of all fuzzy points in X by ℵ.

Definition 1.3 ([29, 30])

Suppose that psλ is a fuzzy point and M is a fuzzy set. We say that psλ is contained in M or belongs to M, indicated by psλ∈M, if λ≤M(s). Obviously, each fuzzy set M can be represented as a union of all fuzzy points that belong to M.

Definition 1.4 ([29, 30])

A fuzzy point psλ is called quasi-coincident with a fuzzy set M if and only if λ>Mc(s) or λ+M(s)>1 and denoted by psλqM. Furthermore, the fuzzy set M is said to be quasi-coincident with N, denoted by MqN, if and only if there exists s∈X such that M(s)>Nc(s) or M(s)+N(s)>1. On the contrary, we write Mq̸N if M is not quasi-coincident with N.

The following results were proved in [29–38].

For any function f from X to Y, the following properties are true:

i. f-1(Nc)=(f-1(N))c, for any fuzzy set N∈Y.
ii. f(f-1(N))≤N, for any fuzzy set N∈Y.
iii. M≤f-1(f(M)), for any fuzzy set M∈X.
iv. Let p be a fuzzy point of X and M,N be fuzzy sets in X. Then, we get
- (1) if f(p)qN, then pqf-1(N), and
- (2) if pqM, then f(p)qf(m).
v. Let M (resp. N) be a fuzzy set in X (resp. Y) and p be a fuzzy point in X. Then,
- (1) p∈f-1(N) if f(p)∈N, and
- (2) f(p)∈f(M) if p∈M.

Definition 1.5 ([28])

For a directed set Λ and ordinary set X, a function S:Λ→χ is called a fuzzy net in X and we denote S(λ), ∀λ ∈ Λ by Sλ. Accordingly, the net S is symbolized as {Sλ, λ ∈ Λ}.

Definition 1.6 ([29])

Consider the fuzzy space X and let {A_n, n ∈ N} represent a net of fuzzy sets in X. Thus, we denote the fuzzy upper limit of the net {A_n, n ∈ N} in I^X by F-limN¯(An). Thus, the fuzzy set that is a union of all fuzzy points pxλ in X such that for all n₀ ∈ N and each fuzzy open Q-neighborhood U of pxλ in X, there exists an element n ∈ N for which n ≥ n₀ and AnqU. Otherwise, we write F-limN¯(An)=0¯.

Definition 1.7 ([39–42])

Let X be a fuzzy space. Then, a fuzzy subset A of X is said to be

(i) fuzzy β-open (or fuzzy semi-preopen) if A ≤ Cl(Int(Cl(A))) and

(ii) fuzzy β-closed if Int(Cl(Int(A))) ≤ A.

The family of all fuzzy β-open (resp. fuzzy β-closed) sets of X is denoted by FβO(X) (resp. FβC(X)).

Definition 1.8 ([39–42])

Let X be a fuzzy space and A be a fuzzy subset of X. Then,

(i) the fuzzy β-closure of A is defined by the intersection of all fuzzy β-closed sets containing A and is symbolized by βCl(A), that is, βCl(A) = inf{K : A ≤ K, K ∈ FβC(X)}, and

(ii) the fuzzy β-interior of A, denoted by βInt(A), is defined as follows:

βInt(A)=sup{U:U≤A,U∈FβO(X)}.

Definition 1.9 ([39–42])

Let X be a fuzzy space and A ⊆ X. If A is fuzzy β-open (resp. β-closed), then βInt(A) = A (resp. βCl(A) = A), and we get βCl(A^c) = 1̄ − βInt(A) = 1̄ − A = A^c (resp. βInt(A^c) = 1̄ − βCl(A) = 1̄ − A = A^c). Hence, the fuzzy set A^c is fuzzy β-closed (resp. β-open).

2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

In this section, we introduce and study the concept of fuzzy points and β-closed sets. Moreover, we give and discuss separation axioms in fuzzy topological space.

Definition 2.1

Let X be a fuzzy space and A ⊆ X. Then, A is said to be a fuzzy β-neighborhood of a fuzzy point pxλ there exists a υ ∈ FβO(X) such that pxλ∈v≤A. Therefore, a fuzzy β-neighborhood is fuzzy β-open if A ∈ FβO(X).

Definition 2.2

Let X be a fuzzy space and A ⊆ X. Then, A is said to be a fuzzy Q-β-neighborhood of pxλ if there exists B∈FβO(X) such that pxλqB and B≤A.

Remark 2.3

In general, we note that any point does not belong to its fuzzy Q-β-neighborhood.

In the following, we denote a family of all fuzzy β-open Q-β-neighborhoods of the fuzzy point pxλ in X by NQ-p-n(pxλ). The set NQ-p-n(pxλ) with relation ≤^* (that is, U1≤*U2 if and only if U2≤U1) forms a directed set.

Proposition 2.4

Suppose that X is a fuzzy space and A ⊆ X. Then, a fuzzy point pxλ∈βCl(A) if and only if for all U∈FβO(X) for which pxλqU, we get UqA.

Proof

First, a fuzzy point pxλ∈βCl(A) if and only if pxλ∈F, for each fuzzy β-closed set F of X for which A ≤ F. Thus, this is equivalent to pxλ∈βCl(A) if and only if λ≤1-U(s), for all fuzzy β-open sets U for which A≤1¯-U. Consequently, pxλ∈βCl(A) if and only if U(s)≤1-λ, for each fuzzy β-open set U for which U≤1¯-A. Therefore, pxλ∈βCl(A) if and only if for each fuzzy β-open set U of X such that U(s)>1-λ,U≰1¯-A. Consequently, pxλ∈βCl(A) if and only if for each fuzzy β-open set U of X, U(s)+λ>1 and UqA. Thus, pxλ∈βCl(A) if and only if for each fuzzy β-open set U of X, pxλqU and UqA.

Definition 2.5

Let X be a fuzzy space and A ⊆ X. Then, the fuzzy point pxλ is called a β-boundary point of a fuzzy set A if and only if pxλ∈βCl(A)∧(1¯-βCl(A)). Moreover, the fuzzy set βCl(A) ∧ (1̄ − βCl(A)) is denoted by βBd(A).

Proposition 2.6

Suppose that X is a fuzzy space and A ⊆ X. Then, A ∨ βBd(A) ≤ βCl(A).

Proof

First, if pxλ∈A∨βBd(A), then pxλ∈A or pxλ∈βBd(A). Obviously, if pxλ∈βBd(A), then pxλ∈βCl(A). Now, let us suppose that pxλ∈A. Then, we get βCl(A) = ∧{F : F ∈ I^X, F is β-closed and A ≤ F}. Accordingly, if pxλ∈A, then pxλ∈F, for each fuzzy β-closed set F of X for which A ≤ F, and hence, pxλ∈βCl(A).

Example 2.7

Consider a fuzzy topological space (X, T), where X={s,t} and T={0¯,1¯,ps12}. Then, the fuzzy sets A of X contained in the family of all fuzzy β-closed sets of X are

(i) A ∈ I^X such that A(s)∈[0,12) and A(t)∈[0,1].

Consequently, Int(Cl(Int(A))) = Int(Cl(0̄)) = Cl(0̄) = 0̄ ≤ A.
(ii) A ∈ I^X such that A(s)∈[12,1] and A(t)=1.

Consequently, Int(Cl(Int(A)))=Cl(ps12)≤(ps12)c≤A.

Similarly, the family of all fuzzy β-open sets of X is given by the following fuzzy sets Uof X:

(i) U∈IX such that U(s)∈[0,12] and U(t)=0.

Consequently, Cl(Int(Cl(U)))=Int((ps12)c)≤ps12≥U.
(ii) U∈IX such that U(s)∈[12,1) and U(t)∈[0,1].

Consequently, Cl(Int(Cl(U)))=Int(1¯)=1¯≥U.

Now, take a fuzzy set B∈IX, where B=ps23. Accordingly, we get βCl(B)=(ps13)c such that (ps13)c(z)=23 if z=s, and (ps13)c(z)=1 if z=s.

Also, we get 1¯-βCl(B)=ps13 and βBd(B)=βCl(B)∧(1¯-βCl(B))=ps13. Thus, B ∨ βBd(B) = B ≠ βCl(B).

Definition 2.8

If for every two fuzzy points psλ and ptμ such that psλ≠ptμ, either psλ∉βCl(ptμ) or ptμ∉βCl(psλ), then a fuzzy space X is said to be β-T0.

Definition 2.9

If every fuzzy point is fuzzy β-closed, then the fuzzy space X is called β-T1.

Remark 2.10

Obviously, every β-T1 fuzzy space is β-T0.

Proposition 2.11

A fuzzy space X is β-T1 if and only if for each s∈X and each λ ∈ [0, 1], there exists a fuzzy β-open set A such that A(s)=1-λ and A(t)=1 for all t≠s.

Proof

(Necessity) Suppose that λ = 0. We set A = 1̄. Then, A is a fuzzy β-open set such that A(s)=1-0 and A(t)=1 for all t≠s. Now, let λ ∈ (0, 1] and s∈X. We set A=(psλ)c. The set A is fuzzy β-open such that A(s)=1-λ and A(t)=1 for all t≠s.

(Sufficiency) Let psλ be an arbitrary fuzzy point of X. We prove that the fuzzy point psλ is fuzzy β-closed. By assumption, there exists a fuzzy β-open set A such that A(s)=1-λ and A(t)=1 for all t≠s. Evidently, Ac=psλ. Accordingly, the fuzzy point psλ is fuzzy β-closed, and therefore, the fuzzy space X is β-T1.

Definition 2.12

A fuzzy space X is called a β-Hausdorff space if for any fuzzy points psλ and ptμ for which supp(psλ)=s≠supp(ptμ)=t, there exist two fuzzy β-open Q-β-neighborhoods U and υ of psλ and ptμ, respectively, such that U∧v=0¯.

Example 2.13

Consider the fuzzy space (X, T), where X={s,t} and T={0¯,1¯,ps12}.

The fuzzy point ps12 is not fuzzy β-closed. Certainly, we get Int(Cl(Int(ps12)))=Cl(ps12)c≰ps12. Hence, the fuzzy space X is not β-T1. Meanwhile, it is clear that X is β-T0.

Example 2.14

Consider the fuzzy space (X, T), where X={s,t} and T={0¯,1¯}. Therefore, every fuzzy point psλ is fuzzy β-closed. Certainly, we get Int(Cl(Int(psλ)))=0¯≤psλ. Consequently, the fuzzy space X is β-T1 and hence is β-T0. Meanwhile, it is clear that X is β-Hausdorff.

Note: According to the definitions of T0,T1, and Hausdorff spaces in [21], it is clear that the fuzzy space X is not T0,T1, and Hausdorff.

Definition 2.15

A fuzzy space X is called a β-regular space if for any fuzzy point pxλ and fuzzy β-closed set F not containing pxλ, there exist U, υ ∈ FβO(X) such that pxλ∈U, F ≤ υ and U∧v=0¯.

Example 2.16

Consider the fuzzy space (X, T), where X={s,t} and T={0¯,1¯}. Thus, X is β-Hausdorff but not β-regular, as illustrated as follows:

Suppose the fuzzy point ps13 and fuzzy set A of X such that A(s)=14 and A(t)=1.

For the fuzzy set A, we obtain Int(Cl(Int(A))) = 0̄ ≤ A. Thus, A is fuzzy β-closed. Also, we get ps13∉A. Now, let U and υ be two arbitrary fuzzy β-open sets such that ps13∈U and A ≤ υ. Then, (U∧v) (s)≤14, and hence, U∧v≠0¯. Consequently, X is not β-regular.

Definition 2.17

The fuzzy space X is called quasi β-T1 if for any fuzzy points psλ and pyμ for which supp(psλ)=s≠supp(ptμ)=t, there exists a fuzzy β-open set U such that psλ∈U,ptμ∉U and another υ such that pxλ∉v and ptμ∈v.

Example 2.18

Consider the fuzzy space (X, T), where X={s,t} and T={0¯,1¯,ps12}. Thus, X is quasi β-T1 but not β-T1.

Definition 2.19 ([43])

A fuzzy point psλ is called weak (resp. strong) if λ≤12 (resp. λ>12).

Definition 2.20

A fuzzy set A of a fuzzy space X is called fuzzy β-generalized closed (or fβg-closed) if βCl(A)≤U whenever A≤U and U is a fuzzy β-open set of X.

Proposition 2.21

Let X be a fuzzy space, and let pxλ and ptμ be weak and strong fuzzy points, respectively. If psλ is β-generalized closed, then

ptμ∈βCl(psλ)⇒psλ∈βCl(ptμ).

Proof

Let ptμ∈βCl(psλ) and psλ∉βCl(ptμ). Then, βCl(ptμ) (s)<λ. Also, λ≤12. Thus, βCl(ptμ) (s)≤1-λ, and then λ≤1-βCl(ptμ) (s). Therefore, psλ∈(βCl(ptμ))c, but psλ is β-generalized closed and (βCl(ptμ))c is fuzzy β-open. Thus, βCl(psλ)≤(βCl(ptμ))c. By assumption, we get ptμ∈βCl(psλ). Thus, ptμ∈(βCl(ptμ))c.

Now, we prove that this is a contradiction. Indeed, we have μ≤1-βCl(ptμ) (t) or βCl(ptμ) (t)≤1-μ. Also, ptμ∈βCl(ptμ). Thus, μ ≤ 1 − μ.

However, ptμ is a strong fuzzy point, that is, μ>12. Thus, the above relation μ ≤ 1 − μ is a contradiction. Hence, psλ∈βCl(ptμ).

Proposition 2.22

If X is a quasi β-T₁ fuzzy space and psλ a weak fuzzy point in X, then (pxλ)c is a fuzzy β-neighborhood of each fuzzy point ptμ with t≠s.

Proof

Let t≠s and ptμ be a fuzzy point of X. Because X is quasi β-T1, there exists a fuzzy β-open U of X such that ptμ∈U and pxλ∉U. This implies that λ>U(s). Also, λ≤12. Thus, U(s)≤1-λ. Hence, U(t)≤1=(psλ)c(t) for every t∈X\{s}. Thus, U≤(psλ)c. Therefore, the fuzzy point psλ is a β-neighborhood of (ptμ)c.

Proposition 2.23

If X is a β-regular fuzzy space, then for any strong fuzzy point psλ and any fuzzy β-open set U containing psλ, there exists a fuzzy β-open set W containing psλ such that βCl(W)≤U.

Proof

Suppose that psλ is any strong fuzzy point contained in U∈FβO(X). Then, 12<λ≤U(s). Thus, the complement of U, that is, the fuzzy set Uc, is a fuzzy β-closed set that does not contain the fuzzy point psλ. Thus, there exist W, V ∈ FβO(X) such that psλ∈W

and Uc<v with W ∧ υ = 0̄. Hence, we have W ≤ υ^c and βCl(W) ≤ βCl(υ^c) = υ^c. Now, Uc≤v implies vc≤U. This means that βCl(W)≤U, which completes the proof.

Proposition 2.24

If X is a fuzzy β-regular space, then the strong fuzzy points in X are fβg-closed.

Proof

Let psλ be any strong fuzzy point in X and U be a fuzzy open set such that psλ∈U. By Proposition 2.23, there exists a W ∈ FβO(X) such that psλ∈W and βCl(W)≤U. Now, we have βCl(psλ)≤βCl(W)≤U. Thus, the fuzzy point psλ is fβg-closed.

Definition 2.25

A fuzzy space X is called a weakly β-regular space if for any weak fuzzy point psλ and fuzzy β-closed set F not containing psλ, there exist U, υ ∈ FβO(X) such that psλ∈U, F ≤ υ and U ∧ υ = 0̄. Observe that every β-regular fuzzy space is weakly β-regular.

Definition 2.26

Suppose that X is a fuzzy space. A fuzzy set U in X is said to be fuzzy β-nearly crisp if βCl(U)∧(βCl(U))c=0¯.

Proposition 2.27

Suppose that X is a fuzzy space. If for any weak fuzzy point psλ and any U∈FβO(X) containing psλ, there exists a fuzzy β-open and β-nearly crisp fuzzy set W containing psλ such that βCl(W)≤U, then X is fuzzy weakly β-regular.

Proof

Assume that F is a fuzzy β-closed set not containing the weak fuzzy point psλ. Then, F^c is a fuzzy β-open set containing psλ. By hypothesis, there exists a fuzzy β-open and β-nearly crisp fuzzy set W such that psλ∈W and βCl(W) ≤ F^c. We set N = βInt(βCl(W)) and M = 1 − βCl(W). Then, N is fuzzy β-open, psλ∈N, and F ≤ M. We are going to prove that M ∧ N = 0̄. Now, assume that there exists t∈X such that (N ∧ M)(y) = μ ≠ 0̄. Then, ptμ∈N∧M. Hence, ptμ∈βCl(W) and ptμ∈(βCl(W))c. This is a contradiction because W is β-nearly crisp. Thus, the fuzzy space X is weakly β-regular.

Definition 2.28

Let X be a fuzzy space. A fuzzy point psλ in X is said to be well-β-closed if there exists ptμ∈βCl(psλ) such that supppsλ≠supp(ptμ).

Proposition 2.29

If X is a fuzzy space and psλ is a fβg-closed and well-β-closed fuzzy point, then X is not a quasi β-T1 space.

Proof

Let X be a fuzzy quasi β-T1 space. By the fact that psλ is well-β-closed, there exists a fuzzy point ptμ with supppsλ≠supp(ptμ) such that ptμ∈βCl(psλ). Then, there exists U∈FβO(X) such that psλ∈U and ptμ∉U. Therefore, βCl(psλ)≤U and ptμ∈U. However, this is a contradiction, and hence, X cannot be a quasi β-T1 space.

Definition 2.30

Let X be a fuzzy space. A fuzzy point psλ is said to be just-β-closed if the fuzzy set βCl(psλ) is again a fuzzy point. Clearly, in a fuzzy β-T1 space every fuzzy point is just-β-closed.

Proposition 2.31

Let X be a fuzzy space. If psλ and ptμ are two fuzzy points such that λ < μ and ptμ is fuzzy β-open, then psλ is just-β-closed if it is fβg-closed.

Proof

We prove that the fuzzy set βCl(psλ) is again fuzzy point. We have psλ∈psμ and the fuzzy set psμ is fuzzy β-open. Since psλ is fβg-closed we have βCl(psλ)≤psμ. Thus βCl(psλ) (s)≤μ and βCl(psλ) (z)≤0, for every z∈X\{s}. So, the fuzzy set βCl(psλ) is a fuzzy point.

3. Fuzzy β-Convergence and Fuzzy Points

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

In this section, we discuss the fuzzy β-convergence and fuzzy points, and some of their properties are investigated.

Definition 3.1

Let {A_n, n ∈ N} be a net of fuzzy sets in a fuzzy space X. Then, by F-β-limN¯(An), we denote the fuzzy β-upper limit of the net {A_n, n ∈ N} in ℑX, that is, the fuzzy set which is the union of all fuzzy points psλ in X, such that for every n₀ ∈ N and every fuzzy β-open Q-β-neighborhood U of pxλ in X, there exists an element n ∈ N for which n ≥ n₀ and AnqU. In other cases, we set F-β-limN¯(An)=0¯.

Example 3.2

Let (X, T) be a fuzzy space such that X={s,t} and T={0,1¯,ps12}. Also, let {A_n, n ∈ N} be a net of fuzzy sets of X such that A_n(X) = {0.5} for every n ∈ N. The fuzzy point ps12∈F-β-limN¯(An). Indeed, for every n₀ ∈ N and for the only fuzzy open Q-neighborhood U=1¯ of ps12, there exists an element n ∈ N for which n ≥ n₀ and AnqU. The fuzzy point ps12∉F-β-limN¯(An). Indeed, for every n₀ ∈ N and for the fuzzy β-open Q-β-neighborhood U=px23 of ps12, there is no element n ∈ N such that n ≥ n₀ and AnqU. However, from the above, we have F-limN¯(An)≠F-β-limN¯(An).

Definition 3.3

Let {A_n, n ∈ N} be a net of fuzzy sets in a fuzzy space X. Then, by F-β-limN(An), we denote the fuzzy β-lower limit of the net {A_n, n ∈ N} in JX, that is, the fuzzy set which is the union of all fuzzy points psλ in X such that for every fuzzy β-open Q-neighborhood U of psλ in X, there exists an element n₀ ∈ N such that AnqU, for every n ∈ N and n ≥ n₀. In other cases, we set F-β-limN(An)=0¯.

Definition 3.4

A net {A_n, n ∈ N} of fuzzy sets in a fuzzy topological space X is said to be fuzzy β-convergent to the fuzzy set A if F-β-limN(An)=F-β-limN¯(An)=A. We then write F-β-limN(An)=A.

Proposition 3.5

Let {A_n, n ∈ N} and {B_n, n ∈ N} be two nets of fuzzy sets in X. Then, the following statements are true:

(i) The fuzzy β-upper limit is β-closed.
(ii) F-β-limN¯(An)=F-limN¯(βCl(A)).
(iii) If A_n = A for every n ∈ N, then F-β-limN¯(An)=βCl(A).
(iv) The fuzzy upper limit is not affected by changing a finite number of A_n.
(v) F-β-limN¯(An)≤βCl(∨{An:n∈N}).
(vi) If A_n ≤ B_n for every n ∈ N, then F-β-limN¯(An)≤F-β-limN¯(Bn).
(vii) F-β-limN¯(An∨Bn)=F-β-limN¯(An)∨F-β-limN¯(Bn).
(viii) F-β-limN¯(An∧Bn)≤F-β-limN¯(An)∧F-β-limN¯(Bn).

Proof

We prove only Statements (i) to (v).

(i) It is sufficient to prove that βCl(F-β-limN¯(An))≤F-β-limN¯(An).

Let psλ∈βCl(F-β-limN¯(An)), and let U be an arbitrary fuzzy β-open Q-β-neighborhood of ptr. Then, we have UqF-β-limN¯(An).

Hence, there exists an element s′∈X such that U(s′)+F-β-limN¯(An)(s′)>1.

Let F-β-limN¯(An)(t′)=k. Then, for the fuzzy point ps′k in X, we have ps′kqU and ps′k∈F-β-limN¯(An).

Thus, for every element n₀ ∈ N, there exists n ≥ n₀, n ∈ N such that AnqU. This means that psr∈F-β-limN¯(An).
(ii) Clearly, it is sufficient to prove that for every fuzzy β-open set U, the condition UqAn is equivalent to UqβCl(An).

Let UqAn. Then, there exists an element s∈X such that U(t)+An(s)>1. As we have U(s)+βCl(An)(s)>1, then UqβCl(An). Conversely, let UqβCl(An). Then, A_n ≤ βCl(A_n). Then, there exists an element s∈X such that U(s)+βCl(An)(s)>1. Let βCl(An)(s)=r. Then, psr∈βCl(An), and the fuzzy β-open set U is a fuzzy β-open Q-β-neighborhood of psr. Thus, UqAn.
(iii) This follows by Proposition 2.4 and the definition of the fuzzy β-upper limit.
(iv) This follows by definition of the fuzzy β-upper limit.
(v) Let psr∈F-β-limN¯(An) and U be a fuzzy β-open Q-β-neighborhood of psr in X. Then, for every n₀ ∈ N, there exists n ∈ N, n ≥ n₀ such that AnqU, and therefore, ∨{An,n∈N}qU. Thus, psr∈βCl(∨{An,n∈N}).

Proposition 3.6

Let {A_n, n ∈ N} and {B_n, n ∈ N} be two nets of fuzzy sets in Y. Then, the following statements are true:

(1) The fuzzy β-lower limit is β-closed.
(2) F-β-limN(An)=F-β-limN(βCl(An).
(3) If A_n = A for every n ∈ N, then F-β-limN(An)=βCl(A).
(4) The fuzzy upper limit is not affected by changing a finite number of A_n.
(5) ∧{An:n∈N}≤F-β-limN(An).
(6) F-β-limN(An)≤βCl(∨{An:n∈N}).
(7) If A_n ≤ B_n for every n ∈ N, then F-β-limN(An)≤F-β-limN(Bn).
(8) F-β-limN(An∨Bn)≥F-β-limN(An)∨F-β-limN(Bn).
(9) F-β-limN(An∧Bn)≤F-β-limN(An)∧F-β-limN(Bn).

Proof

The proof is similar to that of Proposition 3.5.

Proposition 3.7

For the fuzzy upper and lower limit, we have the relation F-β-limN(An)≤F-β-limN¯(An).

Proof

This is a consequence of the definitions of fuzzy β-upper and fuzzy β-lower limits.

Proposition 3.8

Let {A_n, n ∈ N} and {B_n, n ∈ N} be two nets of fuzzy sets in a fuzzy space Y. Then, the following propositions are true (in the following properties, the nets {A_n, n ∈ N} and {B_n, n ∈ N} are supposed to be fuzzy β-convergent):

(1) βCl(F-β-limN(An))=F-β-limN(An)=F-β-limN(βCl(An)).
(2) If A_n = A for every n ∈ N, then F-β-limN(An)=βCl(A).
(3) If A_n ≤ B_n for every n ∈ N, then F-β-limN(An)≤F-β-limN(Bn).
(4) F-β-limN(An∨Bn)=F-β-limN(An)∨F-β-limN(Bn).

Proof

The proof of this proposition follows by Propositions 3.5 and 3.6.

4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

The main goals of this part is to study and discuss the notions of fuzzy β-continuous functions, fuzzy β-continuous convergence, and fuzzy points.

Definition 4.1

A function f from a fuzzy space Y into a fuzzy space Z is called fuzzy β-continuous if for every fuzzy point psλ in Y and every fuzzy β-open Q-β-neighborhood V of f(psλ), there exists a fuzzy β-open Q-β-neighborhood U of psλ such that f(U)≤v.

Let Y and Z be two fuzzy spaces. Then, by FβC(Y, Z), we denote the set of all fuzzy β-continuous maps of Y into Z.

Example 4.2

Let (Y, T1) and (Y, T2) be two fuzzy spaces such that Y={s,t},T1={0¯,1¯} and T2={0¯,1¯,ps12}.

We consider the map i:(Y,T1)→(Y,T2) for which i(z) = z for every z ∈ Y. We prove that the map i is not fuzzy continuous at the fuzzy point ps0.8 but is fuzzy β-continuous at the fuzzy point ps0.8. Indeed, for the fuzzy open Q-neighborhood v=ps12 of i(ps0.8)=ps0.8, a fuzzy open Q-neighborhood U of ps0.8 such that i(U)≤v does not exist. The only fuzzy open Q-neighborhood U of ps0.8 in (Y, T1) is the fuzzy set 1̄, and i(1̄) ≰ υ.

Now, we prove that the map i is fuzzy β-continuous at the fuzzy point ps0.8. Let υ be an arbitrary fuzzy β-open Q-β-neighborhood υ of i(ps0.8)=ps0.8.

The family of all fuzzy β-open sets in (Y, T2) is given by the following fuzzy sets υ of Y:

(i) υ ∈ I^Y such that v(s)∈[0,12] and v(t)=0 and
(ii) υ ∈ I^Y such that v(s)∈(0,12] and v(t)=[0,1].

The above fuzzy sets V (cases (i) and (ii)) are also fuzzy β-open sets of (Y, T1). Thus, for every fuzzy β-open Q-β-neighborhood V of i(ps0.8) in (Y, T2), there exists the fuzzy β-open Q-β-neighborhood U = V of ps0.8 in (Y, T1) such that i(U) ≤ V.

Definition 4.3

A fuzzy net S = {s_λ, λ ∈ Λ} in a fuzzy space (X, T) is said to be β-convergent to a fuzzy point e in X relative to T and write β lim s_λ = e if for every fuzzy β-open Q-β-neighborhood U of e and for every λ ∈ Λ, there exists m ∈ Λ such that Uqs_m and m ≥ λ.

Proposition 4.4

Let f : Y → Z be a fuzzy β-continuous map. Then, p is a fuzzy point in U, and U, υ are fuzzy β-open Q-neighborhoods of p and f(p), respectively, such that f(U)≰v. Then, there exists a fuzzy point p1 in Y such that p1qU and f(p1)q̸v.

Proof

Because f(U)≰v, U≰f-1(v), and hence, s∈Y such that U(s)>f-1(v)(s) or U(s)-f-1(v)(s)>0, and therefore, U(s)+1-f-1(v)(s)>1 or U(s)+(f-1(v))c(s)>1. Let (f-1(v))c(s)=r. Clearly, for the fuzzy point psr, we have psrqU and psr∈(f-1(v))c. Hence, for the fuzzy point p1≡psr, then p1qU and f(p1)q̸v.

Definition 4.5

A net {f_μ, μ ∈ M} in FβC(Y, Z) fuzzy β-continuously converges to f ∈ FβC(Y, Z) if for every fuzzy net {pλ, λ ∈ Λ} in Y, which β-converges to a fuzzy point p in Y. Thus, we have that the fuzzy net {fμ(pλ), (λ, μ) ∈ Λ × M} β-converges to the fuzzy point f(p) in Z.

Proposition 4.6

A net {f_μ, μ ∈ M}in FβC(Y, Z) fuzzy β-continuously converges to f ∈ FβC(Y, Z) if and only if for every fuzzy point p in Y and for every fuzzy β-open Q-β-neighborhood υ of f(p) in Z, there exist an element μ₀ ∈ M and fuzzy β-open Q-β-neighborhood U of p in Y such that fμ(U)≤v for every μ ≥ μ₀, μ ∈ M.

Proof

Let p be a fuzzy point in Y and υ be a fuzzy β-open Q-β-neighborhood of f(p) in Z such that for every μ ∈ M and every fuzzy β-open Q-β-neighborhood U of p in Y, there exists μ′ ≥ μ such that fμ′(U)≰v. Then, for every fuzzy β-open Q-neighborhood U of p in Y, we can choose a fuzzy point pU in Y (Proposition 4.5) such that pUqU and fμ′(pU)q̸v.

Clearly, the fuzzy net pU,U∈NQ-p-n(P)}β-converges top, but the fuzzy net {fμ(pU),(U,μ)∈NQ-p-n(P)×M} does not β-converge to f(p) in Z.

Conversely, let {pλ, λ ∈ Λ} be a fuzzy net in FβC(Y, Z), which β-converges to the fuzzy point p in Y, and let υ be an arbitrary fuzzy β-open Q-β-neighborhood of f(p) in Z. By assumption, there exist a fuzzy β-open Q-β-neighborhood U of p in Y and an element μ₀ ∈ M such that fμ(U)≤v for every μ ≥ μ₀, μ ∈ M. Because the fuzzy net {p_λ, λ ∈ Λ} β-converges to p in Y, there exists λ₀ ∈ Λ such that pλqU for every λ ∈ Λ, λ ≥ λ₀. Let (λ₀, μ₀) ∈ Λ × M. Then, for every (λ, μ) ∈ Λ × M, (λ, μ) ≥ (λ₀, μ₀), we have fμ(pλ)qfμ(U)≤v, that is, fμ(pλ)qv. Thus, the net {fμ(pλ), (λ, μ) ∈ Λ × M} β-converges to f(p), and the net {f_μ, μ ∈ M} fuzzy β-continuously converges to f.

Proposition 4.7

A net {f_λ, λ ∈ Λ} in FβC(Y, Z) fuzzy β-continuously converges to f ∈ FβC(Y, Z) if and only if

(1)F-β-limΛ¯(fλ-1(K))≤f-1(K),

for every fuzzy β-closed subset K of Z.

Proof

Let {f_λ, λ ∈ Λ} be a net in FβC(Y, Z), which fuzzy β-continuously converges to f, and let K be an arbitrary fuzzy β-closed subset of Z. Let p∈F-β-limΛ¯(fλ-1(K))

and W be an arbitrary fuzzy β-open Q-β-neighborhood of f(p) in Z. Because the net {f_λ, λ ∈ Λ} fuzzy β-continuously converges to f, there exist a fuzzy β-open Q-β-neighborhood V of p in Y and an element λ₀ ∈ Λ such that f_λ(υ) ≤ W for every λ ∈ Λ, λ ≥ λ₀. Meanwhile, there exists an element λ ≥ λ₀ such that vqfλ-1(K). Hence, f_λ(υ)qK, and therefore, WqK. This means that f(p)∈βCl(K)=K. Thus, p∈f-1(K).

Conversely, let {f_λ, λ ∈ Λ} be a net in FβC(Y, Z) and f ∈ FβC(Y, Z) such that the relation (1) holds for every fuzzy β-closed subset K of Z. We prove that the net {f_λ, λ ∈ Λ} fuzzy continuously converges to f. Let p be a fuzzy point of Y and W be a fuzzy β-open Q-β-neighborhood of f(p) in Z. Because p∉f-1(K), where K = W^c, we have p∉F-β-limΛ¯(fλ-1(K)). This means that there exist an element λ₀ ∈ Λ and fuzzy β-open Q-β-neighborhood υ of p in Y such that fλ-1(K)q̸v for every λ ∈ Λ, λ ≥ λ₀. Then, we have v≤(fλ-1(K))c=fλ-1(Kc)=fλ-1(W). Therefore, f_λ(υ) ≤ W for every λ ∈ Λ, λ ≥ λ₀, that is, the net {f_λ, λ ∈ Λ} fuzzy β-continuously converges to f.

Proposition 4.8

The following statements are true:

(1) If {f_λ, λ ∈ Λ} is a net in FβC(Y, Z) such that f_λ = f, for every λ ∈ Λ, then {f_λ, λ ∈ Λ} fuzzy β-continuously converges to f ∈ FβC(Y, Z).
(2) If {f_λ, λ ∈ Λ} is a net in FβC(Y, Z), which fuzzy β-continuously converges to f ∈ FβC(Y, Z) and {g_μ, μ ∈ M} is a subnet of {f_λ, λ ∈ Λ}, then the net {g_μ, μ ∈ M} fuzzy β-continuously converges to f.

5. Conclusion

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

The notions of the sets, separation axioms, and functions in fuzzy topological spaces are highly developed and are used extensively in many practical and engineering problems, computational topology for geometric design, computer-aided geometric design, engineering design research, and mathematical science. In this paper, we discussed and studied some topological structures of fuzzy point applications in fuzzy topological spaces. Namely, we defined the fuzzy β-neighborhood, fuzzy Q-β-neighborhood, β-boundary point, fuzzy β-generalized closed, β-T0, β-T1, β-Hausdorff, β-regular-quasi β-T1, and other related topics. Moreover, many results and examples were investigated to illustrate the importance of our methods. In future works, we will study and apply these concepts in many fields, including bipolar hypersoft topological spaces [24], topological soft rough sets [44], covering rough sets [45], and neighborhood systems of rough sets [45].

Conflict of Interest

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

No potential conflict of interest relevant to this article was reported.

References

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

Abo-Tabl, ESA, and El-Bably, MK (2022). Rough topological structure based on reflexivity with some applications. AIMS Mathematics. 7, 9911-9925. http://dx.doi.org/10.3934/math.2022553
Abu-Gdairi, R, El-Gayar, MA, El-Bably, MK, and Fleifel, KK (2021). Two different views for generalized rough sets with applications. Mathematics. 9. article no. 2275
El-Bably, MK, and El Atik, AA (2021). Soft β-rough sets and its application to determine COVID-19. Turkish Journal of Mathematics. 45. article no. 4
Nawar, AS, El-Gayar, MA, El-Bably, MK, and Hosny, RA (2022). θβ-ideal approximation spaces and their applications. AIMS Mathematics. 7, 2479-2497. https://doi.org/10.3934/math.2022139
Abd El-Monsef, ME, El-Gayar, MA, and Aqeel, RM (2014). On relationships between revised rough fuzzy approximation operators and fuzzy topological spaces. International Journal of Granular Computing, Rough Sets and Intelligent Systems. 3, 257-271. https://doi.org/10.1504/IJGCRSIS.2014.068022
Abd El-Monsef, ME, El-Gayar, MA, and Aqeel, RM (2017). A comparison of three types of rough fuzzy sets based on two universal sets. International Journal of Machine Learning and Cybernetics. 8, 343-353. https://doi.org/10.1007/s13042-015-0327-8
Abu-Gdairi, R, El-Gayar, MA, Al-shami, TM, Nawar, AS, and El-Bably, MK (2022). Some topological approaches for generalized rough sets and their decision-making applications. Symmetry. 14. article no. 95
El-Gayar, MA, and El Atik, AA (2022). Topological models of rough sets and decision making of COVID-19. Complexity. 2022. article no. 2989236
Lu, H, Khalil, AM, Alharbi, W, and El-Gayar, MA (2021). A new type of generalized picture fuzzy soft set and its application in decision making. Journal of Intelligent & Fuzzy Systems. 40, 12459-12475. https://doi.org/10.3233/JIFS-201706
Azzam, AA, Khalil, AM, and Li, SG (2020). Medical applications via minimal topological structure. Journal of Intelligent & Fuzzy Systems. 39, 4723-4730. https://doi.org/10.3233/JIFS-200651
Nasef, AA, and El-Sayed, MK (2022). Properties of topological concepts associated with weak systems. Mathematical Statistician and Engineering Applications. 71, 6631-6641.
El Atik, AEFA, and Nasef, AA (2022). Topological kernel of sets and application on fractals. Journal of Contemporary Technology and Applied Engineering. 1, 1-8. https://doi.org/10.21608/jctae.2022.143392.1000
Nasef, AA, El Atik, AA, and Revathi, GK (2022). A model of hepatitis C and blood pressure via nano topological spaces. Thai Journal of Mathematics. 20, 669-677.
El-Bably, MK, Abu-Gdairi, R, and El-Gayar, MA (2023). Medical diagnosis for the problem of Chikungunya disease using soft rough sets. AIMS Mathematics. 8, 9082-9105. http://dx.doi.org/10.3934/math.2023455
El-Bably, MK, and Abo-Tabl, EA (2021). A topological reduction for predicting of a lung cancer disease based on generalized rough sets. Journal of Intelligent & Fuzzy Systems. 41, 3045-3060. https://doi.org/10.3233/JIFS-210167
El-Bably, MK, Ali, MI, and Abo-Tabl, EA (2021). New topological approaches to generalized soft rough approximations with medical applications. Journal of Mathematics. 2021. article no. 2559495
Stadler, BMR, and Stadler, PF (2002). Generalized topological spaces in evolutionary theory and combinatorial chemistry. Journal of Chemical Information and Modeling. 42, 577-585. https://doi.org/10.1021/ci0100898
Stadler, BM, and Stadler, PF (2004). The topology of evolutionary Biology. Modeling in Molecular Biology. Heidelberg, Germany: Springer, pp. 267-286 https://doi.org/10.1007/978-3-642-18734-6
El-Sharkasy, MM (2018). Topological model for recombination of DNA and RNA. International Journal of Biomathematics. 11. article no. 1850097
El-Sayed, MK, and Nasef, AA (2021). Aapplications of rough sets in plant sciences and disease diagnosis. Journal of Computational and Theoretical Nanoscience. 18, 1759-1762. https://doi.org/10.1166/jctn.2021.9720
Hosny, RA, Abu-Gdairi, R, and El-Bably, MK (2023). Approximations by ideal minimal structure with chemical application. Intelligent Automation and Soft Computing. 36, 3073-3085. http://dx.doi.org/10.32604/iasc.2023.034234
Alghazzwi, D, Ali, A, Almutlg, A, Abo-Tabl, EA, and Azzam, AA (2023). A novel structure of q-rung orthopair fuzzy sets in ring theory. AIMS Mathematics. 8, 8365-8385. http://dx.doi.org/10.3934/math.2023422
Ameen, ZA, Al-Shami, TM, Azzam, AA, and Mhemdi, A (2022). A novel fuzzy structure: infra-fuzzy topological spaces. Journal of Function Spaces. 2022. article no. 97778069
Musa, SY, and Asaad, BA (2023). Separation axioms on bipolar hypersoft topological spaces. International Journal of Neutrosophic Science. 20, 150-164. http://dx.doi.org/10.54216/IJNS.200112
Asaad, BA, and Musa, SY (2023). A novel class of bipolar soft separation axioms concerning crisp points. Demonstratio Mathematica. 56. article no. 20220189
Saleh, HY, Asaad, BA, and Mohammed, RA (2022). Bipolar soft limit points in bipolar soft generalized topological spaces. Mathematics and Statistics. 10, 1264-1274. https://doi.org/10.13189/ms.2022.100612
Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Wong, CK (1974). Fuzzy points and local properties of fuzzy topology. Journal of Mathematical Analysis and Applications. 46, 316-328. https://doi.org/10.1016/0022-247X(74)90242-X
Pu, PM, and Liu, YM (1980). Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence. Journal of Mathematical Analysis and Applications. 76, 571-599. https://doi.org/10.1016/0022-247X(80)90048-7
Pu, PM, and Liu, YM (1980). Fuzzy topology. II. Product and quotient spaces. Journal of Mathematical Analysis and Applications. 77, 20-37. https://doi.org/10.1016/0022-247X(80)90258-9
Azad, KK (1981). On fuzzy semi continuity, fuzzy almost continuity, and fuzzy weakly continuity. Journal of Mathematical Analysis and Applications. 82, 14-32. https://doi.org/10.1016/0022-247X(81)90222-5
Ajmal, N, and Tyagi, BK (1991). On fuzzy almost continuous functions. Fuzzy Sets and Systems. 41, 221-232. https://doi.org/10.1016/0165-0114(91)90226-G
Chang, CL (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 24, 182-190. https://doi.org/10.1016/0022-247X(68)90057-7
Hu, CM (1985). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 110, 141-178. https://doi.org/10.1016/0022-247X(85)90340-3
Saha, S (1987). Fuzzy δ-continuous mappings. Journal of Mathematical Analysis and Applications. 126, 130-142. https://doi.org/10.1016/0022-247X(87)90081-3
Wong, CK (1975). Fuzzy topology. Fuzzy Sets and their Applications to Cognitive and Decision Processes. Amsterdam, Netherlands: Elsevier, pp. 171-190
Yalvac, TH (1987). Fuzzy sets and functions on fuzzy spaces. Journal of Mathematical Analysis and Applications. 126, 409-423. https://doi.org/10.1016/0022-247X(87)90050-3
Georgiou, DN, and Papadopoulos, BK (1999). Convergences in fuzzy topological spaces. Fuzzy Sets and Systems. 101, 495-504. https://doi.org/10.1016/S0165-0114(97)00112-7
Fathalla, MA 1984. On fuzzy topological spaces. Ph.D. dissertation. Assuit University. Sohag, Egypt.
Hanafy, IM (2004). Fuzzy β-compactness and fuzzy β-closed spaces. Turkish Journal of Mathematics. 28, 281-293.
Thakur, SS, and Singh, S (1998). On fuzzy semi-preopen sets and fuzzy semi-precontinuity. Fuzzy Sets and Systems. 98, 383-391. https://doi.org/10.1016/s0165-0114(96)00363-6
Mukherjee, MN, and Sinha, SP (1990). On some near-fuzzy continuous functions between fuzzy topological spaces. Fuzzy Sets and Systems. 34, 245-254. https://doi.org/10.1016/0165-0114(90)90163-Z
Johnson, TP, and Mathew, SC (2000). On fuzzy point in topological spaces. Far East Journal of Mathematical Sciences, Part I (Geometry and Topology), 75-86.
Ali, MI, El-Bably, MK, and Abo-Tabl, EA (2022). Topological approach to generalized soft rough sets via near concepts. Soft Computing. 26, 499-509. https://doi.org/10.1007/s00500-021-06456-z
Nawar, AS, El-Bably, MK, and El-Atik, AA (2020). Certain types of coverings based rough sets with application. Journal of Intelligent & Fuzzy Systems. 39, 3085-3098. https://doi.org/10.3233/JIFS-191542
El-Bably, MK, Al-shami, TA, Nawar, AS, and Mhemdi, A (2021). Corrigendum to “Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space”. Journal of Intelligent & Fuzzy Systems. 41, 7353-7361. https://doi.org/10.3233/JIFS-211198

Biographies

Go to

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion
Conflict of Interest
References
Biographies

Radwan Abu-Gdairi is assistant professor of Mathematics Faculty of Science, Zarqa University, Jordan. He received Ph.D. degree in Mathematics from Tanta University, in 2011, Egypt. His research interests are in the areas of pure and applied mathematics including topology, fuzzy topology, rough sets. E-mail: rgdairi@zu.edu.jo

Arafa A. Nasef is professor of Engineering Mathematics, Faculty of Engineering, Kafer Elsheikh, Egypt. He received the B.Sc., M.S., and Ph.D. degrees in Pure Mathematics from Faculty of Science, Tanta University, Egypt. He published more than 300 papers in refereed journals and conference proceedings. His research interests are topology and its applications, rough sets, fuzzy sets, soft sets, graph theory and granular computing. His h-index is 17 in Google Scholar. E-mail: nasefa50@yahoo.com

Mostafa A. El-Gayar received Ph.D. degree in Mathematics from Tanta University, in 2009, Egypt. His research interests are topology, rough set, data mining, fuzzy sets and information systems. His h-index is 7 in Google Scholar. E-mail: m.elgayar@science.helwan.edu.eg

Mostafa K. El-Bably received the B.Sc., M.S., and Ph.D. degrees in Pure Mathematics from Faculty of Science, Tanta University, Egypt. He has authored/co-authored over 30 papers in top-ranked international journals and conference proceedings. His research interests are topology, rough sets, fuzzy sets, and soft sets. His h-index is 11 in Google Scholar. E-mail: mkamel bably@yahoo.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(2): 162-172

Published online June 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.2.162

On Fuzzy Point Applications of Fuzzy Topological Spaces

Radwan Abu-Gdairi¹, Arafa A. Nasef², Mostafa A. El-Gayar³, and Mostafa K. El-Bably⁴

Correspondence to:Mostafa K. El-Bably (mkamel_bably@yahoo.com)

Received: January 31, 2023; Revised: March 21, 2023; Accepted: April 7, 2023

Abstract

Keywords: Fuzzy topological space, Fuzzy point, Fuzzy convergence, Fuzzy separation axioms, Fuzzy &beta,-open sets

1. Introduction

Then, for any members $M, N$ of I^X, we say that $M$ is contained in $N$ if $M (s) \leq N (s)$ for every $s \in X$ , denoted by $M \leq N$ .

Definition 1.1 ([27])

For each $M, N \in I^{X}$ , we define the next fuzzy sets in I^X as follows:

(i) $(M \land N) (s) = \min {M (s), N (s)}$ , for each $s \in X$ .

(ii) $(M \lor N) (s) = \max {M (s), N (s)}$ , for each $s \in X$ .

(iii) $M^{c} (s) = 1 - M (s)$ , for each $s \in X$ .

(iv) Letting $f : X \to Y, M \in I^{X}$ and $N \in I^{Y}$ , then $f (M)$ is a fuzzy set in Y, such that $f (M) (t) = \sup {M (s) : s \in f^{- 1} (t)}$ , if $f^{- 1} (t) \neq Φ$ . Moreover, $f (M) (t) = 0$ if $f^{- 1} (t) = Φ$ . Also, $f^{- 1} (N)$ is a fuzzy set in X, given by $f^{- 1} (N) (s) = N (f (s)), s \in X$ .

Definition 1.2 ([27])

A fuzzy topology $T$ on a set X is defined by a family of fuzzy sets ( $T \subseteq I^{X}$ ) such that $T$ satisfies the following three conditions:

(i) If 0̄ and 1̄ are the characteristic functions of χ_Φ and χ_X, respectively, then 0̄, $\bar{1} \in T$ .

(ii) $M \land N \in T$ , for each $M, N \in T$ .

(iii) If ${M_{j} : j \in J} \subseteq T$ , then $\lor {M_{j} : j \in J} \in T$ .

Therefore, the pair (X, $T$ ) is called a fuzzy topological space (and in briefly, we call X a fuzzy space).

Considering a set X, the point $s \in X$ is called a fuzzy point if it takes the value 0 for all $t \in X$ except one 1. Moreover, if its value at $s$ is λ such that 0 < λ ≤ 1, then the fuzzy point is denoted by $p_{s}^{λ}$ , and the point $s$ is called its support, denoted by $supp (p_{s}^{λ})$ , that is $supp (p_{s}^{λ}) = s$ . We symbolize the class of all fuzzy points in X by ℵ.

Definition 1.3 ([29, 30])

Suppose that $p_{s}^{λ}$ is a fuzzy point and $M$ is a fuzzy set. We say that $p_{s}^{λ}$ is contained in $M$ or belongs to $M$ , indicated by $p_{s}^{λ} \in M$ , if $λ \leq M (s)$ . Obviously, each fuzzy set $M$ can be represented as a union of all fuzzy points that belong to $M$ .

Definition 1.4 ([29, 30])

A fuzzy point $p_{s}^{λ}$ is called quasi-coincident with a fuzzy set $M$ if and only if $λ > M^{c} (s)$ or $λ + M (s) > 1$ and denoted by $p_{s}^{λ} q M$ . Furthermore, the fuzzy set $M$ is said to be quasi-coincident with $N$ , denoted by $M q N$ , if and only if there exists $s \in X$ such that $M (s) > N^{c} (s)$ or $M (s) + N (s) > 1$ . On the contrary, we write $M q̸ N$ if $M$ is not quasi-coincident with $N$ .

The following results were proved in [29–38].

For any function $f$ from X to Y, the following properties are true:

i. $f^{- 1} (N^{c}) = {(f^{- 1} (N))}^{c}$ , for any fuzzy set $N \in Y$ .
ii. $f (f^{- 1} (N)) \leq N$ , for any fuzzy set $N \in Y$ .
iii. $M \leq f^{- 1} (f (M))$ , for any fuzzy set $M \in X$ .
iv. Let $p$ be a fuzzy point of X and $M, N$ be fuzzy sets in X. Then, we get
- (1) if $f (p) q N$ , then $p q f^{- 1} (N)$ , and
- (2) if $p q M$ , then $f (p) q f (m)$ .
v. Let $M$ (resp. $N$ ) be a fuzzy set in X (resp. Y) and $p$ be a fuzzy point in X. Then,
- (1) $p \in f^{- 1} (N)$ if $f (p) \in N$ , and
- (2) $f (p) \in f (M)$ if $p \in M$ .

Definition 1.5 ([28])

For a directed set Λ and ordinary set X, a function $S : Λ \to χ$ is called a fuzzy net in X and we denote $S (λ)$ , ∀λ ∈ Λ by $S_{λ}$ . Accordingly, the net $S$ is symbolized as { $S_{λ}$ , λ ∈ Λ}.

Definition 1.6 ([29])

Consider the fuzzy space X and let {A_n, n ∈ N} represent a net of fuzzy sets in X. Thus, we denote the fuzzy upper limit of the net {A_n, n ∈ N} in I^X by $F - \bar{lim_{N}} (A_{n})$ . Thus, the fuzzy set that is a union of all fuzzy points $p_{x}^{λ}$ in X such that for all n₀ ∈ N and each fuzzy open $Q$ -neighborhood $U$ of $p_{x}^{λ}$ in X, there exists an element n ∈ N for which n ≥ n₀ and $A_{n} q U$ . Otherwise, we write $F - \bar{lim_{N}} (A_{n}) = \bar{0}$ .

Definition 1.7 ([39–42])

Let X be a fuzzy space. Then, a fuzzy subset A of X is said to be

(i) fuzzy β-open (or fuzzy semi-preopen) if A ≤ Cl(Int(Cl(A))) and

(ii) fuzzy β-closed if Int(Cl(Int(A))) ≤ A.

The family of all fuzzy β-open (resp. fuzzy β-closed) sets of X is denoted by FβO(X) (resp. FβC(X)).

Definition 1.8 ([39–42])

Let X be a fuzzy space and A be a fuzzy subset of X. Then,

(i) the fuzzy β-closure of A is defined by the intersection of all fuzzy β-closed sets containing A and is symbolized by βCl(A), that is, βCl(A) = inf{K : A ≤ K, K ∈ FβC(X)}, and

(ii) the fuzzy β-interior of A, denoted by βInt(A), is defined as follows:

β I n t (A) = \sup {U : U \leq A, U \in F β O (X)} .

Definition 1.9 ([39–42])

2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space

In this section, we introduce and study the concept of fuzzy points and β-closed sets. Moreover, we give and discuss separation axioms in fuzzy topological space.

Definition 2.1

Let X be a fuzzy space and A ⊆ X. Then, A is said to be a fuzzy β-neighborhood of a fuzzy point $p_{x}^{λ}$ there exists a υ ∈ FβO(X) such that $p_{x}^{λ} \in v \leq A$ . Therefore, a fuzzy β-neighborhood is fuzzy β-open if A ∈ FβO(X).

Definition 2.2

Let X be a fuzzy space and A ⊆ X. Then, A is said to be a fuzzy $Q$ -β-neighborhood of $p_{x}^{λ}$ if there exists $B \in F β O (X)$ such that $p_{x}^{λ} q B$ and $B \leq A$ .

Remark 2.3

In general, we note that any point does not belong to its fuzzy $Q$ -β-neighborhood.

In the following, we denote a family of all fuzzy β-open $Q$ -β-neighborhoods of the fuzzy point $p_{x}^{λ}$ in X by $N_{Q - p - n} (p_{x}^{λ})$ . The set $N_{Q - p - n} (p_{x}^{λ})$ with relation ≤^* (that is, $U_{1} \leq^{*} U_{2}$ if and only if $U_{2} \leq U_{1}$ ) forms a directed set.

Proposition 2.4

Suppose that X is a fuzzy space and A ⊆ X. Then, a fuzzy point $p_{x}^{λ} \in β C l (A)$ if and only if for all $U \in F β O (X)$ for which $p_{x}^{λ} q U$ , we get $U q A$ .

Proof

First, a fuzzy point $p_{x}^{λ} \in β C l (A)$ if and only if $p_{x}^{λ} \in F$ , for each fuzzy β-closed set F of X for which A ≤ F. Thus, this is equivalent to $p_{x}^{λ} \in β C l (A)$ if and only if $λ \leq 1 - U (s)$ , for all fuzzy β-open sets $U$ for which $A \leq \bar{1} - U$ . Consequently, $p_{x}^{λ} \in β C l (A)$ if and only if $U (s) \leq 1 - λ$ , for each fuzzy β-open set $U$ for which $U \leq \bar{1} - A$ . Therefore, $p_{x}^{λ} \in β C l (A)$ if and only if for each fuzzy β-open set $U$ of X such that $U (s) > 1 - λ, U ≰ \bar{1} - A$ . Consequently, $p_{x}^{λ} \in β C l (A)$ if and only if for each fuzzy β-open set $U$ of X, $U (s) + λ > 1$ and $U q A$ . Thus, $p_{x}^{λ} \in β C l (A)$ if and only if for each fuzzy β-open set $U$ of X, $p_{x}^{λ} q U$ and $U q A$ .

Definition 2.5

Let X be a fuzzy space and A ⊆ X. Then, the fuzzy point $p_{x}^{λ}$ is called a β-boundary point of a fuzzy set A if and only if $p_{x}^{λ} \in β C l (A) \land (\bar{1} - β C l (A))$ . Moreover, the fuzzy set βCl(A) ∧ (1̄ − βCl(A)) is denoted by βBd(A).

Proposition 2.6

Suppose that X is a fuzzy space and A ⊆ X. Then, A ∨ βBd(A) ≤ βCl(A).

Proof

First, if $p_{x}^{λ} \in A \lor β B d (A)$ , then $p_{x}^{λ} \in A$ or $p_{x}^{λ} \in β B d (A)$ . Obviously, if $p_{x}^{λ} \in β B d (A)$ , then $p_{x}^{λ} \in β C l (A)$ . Now, let us suppose that $p_{x}^{λ} \in A$ . Then, we get βCl(A) = ∧{F : F ∈ I^X, F is β-closed and A ≤ F}. Accordingly, if $p_{x}^{λ} \in A$ , then $p_{x}^{λ} \in F$ , for each fuzzy β-closed set F of X for which A ≤ F, and hence, $p_{x}^{λ} \in β C l (A)$ .

Example 2.7

Consider a fuzzy topological space (X, $T$ ), where $X = {s, t}$ and $T = {\bar{0}, \bar{1}, p_{s}^{\frac{1}{2}}}$ . Then, the fuzzy sets A of X contained in the family of all fuzzy β-closed sets of X are

(i) A ∈ I^X such that $A (s) \in [0, \frac{1}{2})$ and $A (t) \in [0, 1]$ .

Consequently, Int(Cl(Int(A))) = Int(Cl(0̄)) = Cl(0̄) = 0̄ ≤ A.
(ii) A ∈ I^X such that $A (s) \in [\frac{1}{2}, 1]$ and $A (t) = 1$ .

Consequently, $I n t (C l (I n t (A))) = C l (p_{s}^{\frac{1}{2}}) \leq {(p_{s}^{\frac{1}{2}})}^{c} \leq A$ .

Similarly, the family of all fuzzy β-open sets of X is given by the following fuzzy sets $U$ of X:

(i) $U \in I^{X}$ such that $U (s) \in [0, \frac{1}{2}]$ and $U (t) = 0$ .

Consequently, $C l (I n t (C l (U))) = I n t ({(p_{s}^{\frac{1}{2}})}^{c}) \leq p_{s}^{\frac{1}{2}} \geq U$ .
(ii) $U \in I^{X}$ such that $U (s) \in [\frac{1}{2}, 1)$ and $U (t) \in [0, 1]$ .

Consequently, $C l (I n t (C l (U))) = I n t (\bar{1}) = \bar{1} \geq U$ .

Now, take a fuzzy set $B \in I^{X}$ , where $B = p_{s}^{\frac{2}{3}}$ . Accordingly, we get $β C l (B) = {(p_{s}^{\frac{1}{3}})}^{c}$ such that ${(p_{s}^{\frac{1}{3}})}^{c} (z) = \frac{2}{3}$ if $z = s$ , and ${(p_{s}^{\frac{1}{3}})}^{c} (z) = 1$ if $z = s$ .

Also, we get $\bar{1} - β C l (B) = p_{s}^{\frac{1}{3}}$ and $β B d (B) = β C l (B) \land (\bar{1} - β C l (B)) = p_{s}^{\frac{1}{3}}$ . Thus, B ∨ βBd(B) = B ≠ βCl(B).

Definition 2.8

If for every two fuzzy points $p_{s}^{λ}$ and $p_{t}^{μ}$ such that $p_{s}^{λ} \neq p_{t}^{μ}$ , either $p_{s}^{λ} \notin β C l (p_{t}^{μ})$ or $p_{t}^{μ} \notin β C l (p_{s}^{λ})$ , then a fuzzy space X is said to be β- $T_{0}$ .

Definition 2.9

If every fuzzy point is fuzzy β-closed, then the fuzzy space X is called β- $T_{1}$ .

Remark 2.10

Obviously, every β- $T_{1}$ fuzzy space is β- $T_{0}$ .

Proposition 2.11

A fuzzy space X is β- $T_{1}$ if and only if for each $s \in X$ and each λ ∈ [0, 1], there exists a fuzzy β-open set A such that $A (s) = 1 - λ$ and $A (t) = 1$ for all $t \neq s$ .

Proof

(Necessity) Suppose that λ = 0. We set A = 1̄. Then, A is a fuzzy β-open set such that $A (s) = 1 - 0$ and $A (t) = 1$ for all $t \neq s$ . Now, let λ ∈ (0, 1] and $s \in X$ . We set $A = {(p_{s}^{λ})}^{c}$ . The set A is fuzzy β-open such that $A (s) = 1 - λ$ and $A (t) = 1$ for all $t \neq s$ .

(Sufficiency) Let $p_{s}^{λ}$ be an arbitrary fuzzy point of X. We prove that the fuzzy point $p_{s}^{λ}$ is fuzzy β-closed. By assumption, there exists a fuzzy β-open set A such that $A (s) = 1 - λ$ and $A (t) = 1$ for all $t \neq s$ . Evidently, $A^{c} = p_{s}^{λ}$ . Accordingly, the fuzzy point $p_{s}^{λ}$ is fuzzy β-closed, and therefore, the fuzzy space X is β- $T_{1}$ .

Definition 2.12

A fuzzy space X is called a β-Hausdorff space if for any fuzzy points $p_{s}^{λ}$ and $p_{t}^{μ}$ for which $supp (p_{s}^{λ}) = s \neq supp (p_{t}^{μ}) = t$ , there exist two fuzzy β-open Q-β-neighborhoods $U$ and υ of $p_{s}^{λ}$ and $p_{t}^{μ}$ , respectively, such that $U \land v = \bar{0}$ .

Example 2.13

Consider the fuzzy space (X, $T$ ), where $X = {s, t}$ and $T = {\bar{0}, \bar{1}, p_{s}^{\frac{1}{2}}}$ .

The fuzzy point $p_{s}^{\frac{1}{2}}$ is not fuzzy β-closed. Certainly, we get $I n t (C l (I n t (p_{s}^{\frac{1}{2}}))) = C l {(p_{s}^{\frac{1}{2}})}^{c} ≰ p_{s}^{\frac{1}{2}}$ . Hence, the fuzzy space X is not β- $T_{1}$ . Meanwhile, it is clear that X is β- $T_{0}$ .

Example 2.14

Consider the fuzzy space (X, $T$ ), where $X = {s, t}$ and $T = {\bar{0}, \bar{1}}$ . Therefore, every fuzzy point $p_{s}^{λ}$ is fuzzy β-closed. Certainly, we get $I n t (C l (I n t (p_{s}^{λ}))) = \bar{0} \leq p_{s}^{λ}$ . Consequently, the fuzzy space X is β- $T_{1}$ and hence is β- $T_{0}$ . Meanwhile, it is clear that X is β-Hausdorff.

Note: According to the definitions of $T_{0}, T_{1}$ , and Hausdorff spaces in [21], it is clear that the fuzzy space X is not $T_{0}, T_{1}$ , and Hausdorff.

Definition 2.15

A fuzzy space X is called a β-regular space if for any fuzzy point $p_{x}^{λ}$ and fuzzy β-closed set F not containing $p_{x}^{λ}$ , there exist $U$ , υ ∈ FβO(X) such that $p_{x}^{λ} \in U$ , F ≤ υ and $U \land v = \bar{0}$ .

Example 2.16

Consider the fuzzy space (X, $T$ ), where $X = {s, t}$ and $T = {\bar{0}, \bar{1}}$ . Thus, X is β-Hausdorff but not β-regular, as illustrated as follows:

Suppose the fuzzy point $p_{s}^{\frac{1}{3}}$ and fuzzy set A of X such that $A (s) = \frac{1}{4}$ and $A (t) = 1$ .

For the fuzzy set A, we obtain Int(Cl(Int(A))) = 0̄ ≤ A. Thus, A is fuzzy β-closed. Also, we get $p_{s}^{\frac{1}{3}} \notin A$ . Now, let $U$ and υ be two arbitrary fuzzy β-open sets such that $p_{s}^{\frac{1}{3}} \in U$ and A ≤ υ. Then, $(U \land v) (s) \leq \frac{1}{4}$ , and hence, $U \land v \neq \bar{0}$ . Consequently, X is not β-regular.

Definition 2.17

The fuzzy space X is called quasi β- $T_{1}$ if for any fuzzy points $p_{s}^{λ}$ and $p_{y}^{μ}$ for which $supp (p_{s}^{λ}) = s \neq supp (p_{t}^{μ}) = t$ , there exists a fuzzy β-open set $U$ such that $p_{s}^{λ} \in U, p_{t}^{μ} \notin U$ and another υ such that $p_{x}^{λ} \notin v$ and $p_{t}^{μ} \in v$ .

Example 2.18

Consider the fuzzy space (X, $T$ ), where $X = {s, t}$ and $T = {\bar{0}, \bar{1}, p_{s}^{\frac{1}{2}}}$ . Thus, X is quasi β- $T_{1}$ but not β- $T_{1}$ .

Definition 2.19 ([43])

A fuzzy point $p_{s}^{λ}$ is called weak (resp. strong) if $λ \leq \frac{1}{2}$ (resp. $λ > \frac{1}{2}$ ).

Definition 2.20

A fuzzy set A of a fuzzy space X is called fuzzy β-generalized closed (or fβg-closed) if $β C l (A) \leq U$ whenever $A \leq U$ and $U$ is a fuzzy β-open set of X.

Proposition 2.21

Let X be a fuzzy space, and let $p_{x}^{λ}$ and $p_{t}^{μ}$ be weak and strong fuzzy points, respectively. If $p_{s}^{λ}$ is β-generalized closed, then

p_{t}^{μ} \in β C l (p_{s}^{λ}) \Rightarrow p_{s}^{λ} \in β C l (p_{t}^{μ}) .

Proof

Let $p_{t}^{μ} \in β C l (p_{s}^{λ})$ and $p_{s}^{λ} \notin β C l (p_{t}^{μ})$ . Then, $β C l (p_{t}^{μ}) (s) < λ$ . Also, $λ \leq \frac{1}{2}$ . Thus, $β C l (p_{t}^{μ}) (s) \leq 1 - λ$ , and then $λ \leq 1 - β C l (p_{t}^{μ}) (s)$ . Therefore, $p_{s}^{λ} \in {(β C l (p_{t}^{μ}))}^{c}$ , but $p_{s}^{λ}$ is β-generalized closed and ${(β C l (p_{t}^{μ}))}^{c}$ is fuzzy β-open. Thus, $β C l (p_{s}^{λ}) \leq {(β C l (p_{t}^{μ}))}^{c}$ . By assumption, we get $p_{t}^{μ} \in β C l (p_{s}^{λ})$ . Thus, $p_{t}^{μ} \in {(β C l (p_{t}^{μ}))}^{c}$ .

Now, we prove that this is a contradiction. Indeed, we have $μ \leq 1 - β C l (p_{t}^{μ}) (t)$ or $β C l (p_{t}^{μ}) (t) \leq 1 - μ$ . Also, $p_{t}^{μ} \in β C l (p_{t}^{μ})$ . Thus, μ ≤ 1 − μ.

However, $p_{t}^{μ}$ is a strong fuzzy point, that is, $μ > \frac{1}{2}$ . Thus, the above relation μ ≤ 1 − μ is a contradiction. Hence, $p_{s}^{λ} \in β C l (p_{t}^{μ})$ .

Proposition 2.22

If X is a quasi β-T₁ fuzzy space and $p_{s}^{λ}$ a weak fuzzy point in X, then ${(p_{x}^{λ})}^{c}$ is a fuzzy β-neighborhood of each fuzzy point $p_{t}^{μ}$ with $t \neq s$ .

Proof

Let $t \neq s$ and $p_{t}^{μ}$ be a fuzzy point of X. Because X is quasi β- $T_{1}$ , there exists a fuzzy β-open $U$ of X such that $p_{t}^{μ} \in U$ and $p_{x}^{λ} \notin U$ . This implies that $λ > U (s)$ . Also, $λ \leq \frac{1}{2}$ . Thus, $U (s) \leq 1 - λ$ . Hence, $U (t) \leq 1 = {(p_{s}^{λ})}^{c} (t)$ for every $t \in X \ {s}$ . Thus, $U \leq {(p_{s}^{λ})}^{c}$ . Therefore, the fuzzy point $p_{s}^{λ}$ is a β-neighborhood of ${(p_{t}^{μ})}^{c}$ .

Proposition 2.23

If X is a β-regular fuzzy space, then for any strong fuzzy point $p_{s}^{λ}$ and any fuzzy β-open set $U$ containing $p_{s}^{λ}$ , there exists a fuzzy β-open set W containing $p_{s}^{λ}$ such that $β C l (W) \leq U$ .

Proof

Suppose that $p_{s}^{λ}$ is any strong fuzzy point contained in $U \in F β O (X)$ . Then, $\frac{1}{2} < λ \leq U (s)$ . Thus, the complement of $U$ , that is, the fuzzy set $U^{c}$ , is a fuzzy β-closed set that does not contain the fuzzy point $p_{s}^{λ}$ . Thus, there exist W, V ∈ FβO(X) such that $p_{s}^{λ} \in W$

and $U^{c} < v$ with W ∧ υ = 0̄. Hence, we have W ≤ υ^c and βCl(W) ≤ βCl(υ^c) = υ^c. Now, $U^{c} \leq v$ implies $v^{c} \leq U$ . This means that $β C l (W) \leq U$ , which completes the proof.

Proposition 2.24

If X is a fuzzy β-regular space, then the strong fuzzy points in X are fβg-closed.

Proof

Let $p_{s}^{λ}$ be any strong fuzzy point in X and $U$ be a fuzzy open set such that $p_{s}^{λ} \in U$ . By Proposition 2.23, there exists a W ∈ FβO(X) such that $p_{s}^{λ} \in W$ and $β C l (W) \leq U$ . Now, we have $β C l (p_{s}^{λ}) \leq β C l (W) \leq U$ . Thus, the fuzzy point $p_{s}^{λ}$ is fβg-closed.

Definition 2.25

A fuzzy space X is called a weakly β-regular space if for any weak fuzzy point $p_{s}^{λ}$ and fuzzy β-closed set F not containing $p_{s}^{λ}$ , there exist $U$ , υ ∈ FβO(X) such that $p_{s}^{λ} \in U$ , F ≤ υ and U ∧ υ = 0̄. Observe that every β-regular fuzzy space is weakly β-regular.

Definition 2.26

Suppose that X is a fuzzy space. A fuzzy set $U$ in X is said to be fuzzy β-nearly crisp if $β C l (U) \land {(β C l (U))}^{c} = \bar{0}$ .

Proposition 2.27

Suppose that X is a fuzzy space. If for any weak fuzzy point $p_{s}^{λ}$ and any $U \in F β O (X)$ containing $p_{s}^{λ}$ , there exists a fuzzy β-open and β-nearly crisp fuzzy set W containing $p_{s}^{λ}$ such that $β C l (W) \leq U$ , then X is fuzzy weakly β-regular.

Proof

Assume that F is a fuzzy β-closed set not containing the weak fuzzy point $p_{s}^{λ}$ . Then, F^c is a fuzzy β-open set containing $p_{s}^{λ}$ . By hypothesis, there exists a fuzzy β-open and β-nearly crisp fuzzy set W such that $p_{s}^{λ} \in W$ and βCl(W) ≤ F^c. We set N = βInt(βCl(W)) and M = 1 − βCl(W). Then, N is fuzzy β-open, $p_{s}^{λ} \in N$ , and F ≤ M. We are going to prove that M ∧ N = 0̄. Now, assume that there exists $t \in X$ such that (N ∧ M)(y) = μ ≠ 0̄. Then, $p_{t}^{μ} \in N \land M$ . Hence, $p_{t}^{μ} \in β C l (W)$ and $p_{t}^{μ} \in {(β C l (W))}^{c}$ . This is a contradiction because W is β-nearly crisp. Thus, the fuzzy space X is weakly β-regular.

Definition 2.28

Let X be a fuzzy space. A fuzzy point $p_{s}^{λ}$ in X is said to be well-β-closed if there exists $p_{t}^{μ} \in β C l (p_{s}^{λ})$ such that $supp p_{s}^{λ} \neq supp (p_{t}^{μ})$ .

Proposition 2.29

If X is a fuzzy space and $p_{s}^{λ}$ is a fβg-closed and well-β-closed fuzzy point, then X is not a quasi β- $T_{1}$ space.

Proof

Let X be a fuzzy quasi β- $T_{1}$ space. By the fact that $p_{s}^{λ}$ is well-β-closed, there exists a fuzzy point $p_{t}^{μ}$ with $supp p_{s}^{λ} \neq supp (p_{t}^{μ})$ such that $p_{t}^{μ} \in β C l (p_{s}^{λ})$ . Then, there exists $U \in F β O (X)$ such that $p_{s}^{λ} \in U$ and $p_{t}^{μ} \notin U$ . Therefore, $β C l (p_{s}^{λ}) \leq U$ and $p_{t}^{μ} \in U$ . However, this is a contradiction, and hence, X cannot be a quasi β- $T_{1}$ space.

Definition 2.30

Let X be a fuzzy space. A fuzzy point $p_{s}^{λ}$ is said to be just-β-closed if the fuzzy set $β C l (p_{s}^{λ})$ is again a fuzzy point. Clearly, in a fuzzy β- $T_{1}$ space every fuzzy point is just-β-closed.

Proposition 2.31

Let X be a fuzzy space. If $p_{s}^{λ}$ and $p_{t}^{μ}$ are two fuzzy points such that λ < μ and $p_{t}^{μ}$ is fuzzy β-open, then $p_{s}^{λ}$ is just-β-closed if it is fβg-closed.

Proof

We prove that the fuzzy set $β C l (p_{s}^{λ})$ is again fuzzy point. We have $p_{s}^{λ} \in p_{s}^{μ}$ and the fuzzy set $p_{s}^{μ}$ is fuzzy β-open. Since $p_{s}^{λ}$ is fβg-closed we have $β C l (p_{s}^{λ}) \leq p_{s}^{μ}$ . Thus $β C l (p_{s}^{λ}) (s) \leq μ$ and $β C l (p_{s}^{λ}) (z) \leq 0$ , for every $z \in X \ {s}$ . So, the fuzzy set $β C l (p_{s}^{λ})$ is a fuzzy point.

3. Fuzzy β-Convergence and Fuzzy Points

In this section, we discuss the fuzzy β-convergence and fuzzy points, and some of their properties are investigated.

Definition 3.1

Let {A_n, n ∈ N} be a net of fuzzy sets in a fuzzy space X. Then, by $F - β - \bar{lim_{N}} (A_{n})$ , we denote the fuzzy β-upper limit of the net {A_n, n ∈ N} in $ℑ^{X}$ , that is, the fuzzy set which is the union of all fuzzy points $p_{s}^{λ}$ in X, such that for every n₀ ∈ N and every fuzzy β-open Q-β-neighborhood $U$ of $p_{x}^{λ}$ in X, there exists an element n ∈ N for which n ≥ n₀ and $A_{n} q U$ . In other cases, we set $F - β - \bar{lim_{N}} (A_{n}) = \bar{0}$ .

Example 3.2

Let (X, $T$ ) be a fuzzy space such that $X = {s, t}$ and $T = {0, \bar{1}, p_{s}^{\frac{1}{2}}}$ . Also, let {A_n, n ∈ N} be a net of fuzzy sets of X such that A_n(X) = {0.5} for every n ∈ N. The fuzzy point $p_{s}^{\frac{1}{2}} \in F - β - \bar{lim_{N}} (A_{n})$ . Indeed, for every n₀ ∈ N and for the only fuzzy open Q-neighborhood $U = \bar{1}$ of $p_{s}^{\frac{1}{2}}$ , there exists an element n ∈ N for which n ≥ n₀ and $A_{n} q U$ . The fuzzy point $p_{s}^{\frac{1}{2}} \notin F - β - \bar{lim_{N}} (A_{n})$ . Indeed, for every n₀ ∈ N and for the fuzzy β-open Q-β-neighborhood $U = p_{x}^{\frac{2}{3}}$ of $p_{s}^{\frac{1}{2}}$ , there is no element n ∈ N such that n ≥ n₀ and $A_{n} q U$ . However, from the above, we have $F - \bar{lim_{N}} (A_{n}) \neq F - β - \bar{lim_{N}} (A_{n})$ .

Definition 3.3

Let {A_n, n ∈ N} be a net of fuzzy sets in a fuzzy space X. Then, by $F - β - \frac{lim}{N} (A_{n})$ , we denote the fuzzy β-lower limit of the net {A_n, n ∈ N} in $J^{X}$ , that is, the fuzzy set which is the union of all fuzzy points $p_{s}^{λ}$ in X such that for every fuzzy β-open Q-neighborhood $U$ of $p_{s}^{λ}$ in X, there exists an element n₀ ∈ N such that $A_{n} q U$ , for every n ∈ N and n ≥ n₀. In other cases, we set $F - β - \frac{lim}{N} (A_{n}) = \bar{0}$ .

Definition 3.4

A net {A_n, n ∈ N} of fuzzy sets in a fuzzy topological space X is said to be fuzzy β-convergent to the fuzzy set A if $F - β - \frac{lim}{N} (A_{n}) = F - β - \bar{lim_{N}} (A_{n}) = A$ . We then write $F - β - lim_{N} (A_{n}) = A$ .

Proposition 3.5

Let {A_n, n ∈ N} and {B_n, n ∈ N} be two nets of fuzzy sets in X. Then, the following statements are true:

(i) The fuzzy β-upper limit is β-closed.
(ii) $F - β - \bar{lim_{N}} (A_{n}) = F - \bar{lim_{N}} (β C l (A))$ .
(iii) If A_n = A for every n ∈ N, then $F - β - \bar{lim_{N}} (A_{n}) = β C l (A)$ .
(iv) The fuzzy upper limit is not affected by changing a finite number of A_n.
(v) $F - β - \bar{lim_{N}} (A_{n}) \leq β C l (\lor {A_{n} : n \in N})$ .
(vi) If A_n ≤ B_n for every n ∈ N, then $F - β - \bar{lim_{N}} (A_{n}) \leq F - β - \bar{lim_{N}} (B_{n})$ .
(vii) $F - β - \bar{lim_{N}} (A_{n} \lor B_{n}) = F - β - \bar{lim_{N}} (A_{n}) \lor F - β - \bar{lim_{N}} (B_{n})$ .
(viii) $F - β - \bar{lim_{N}} (A_{n} \land B_{n}) \leq F - β - \bar{lim_{N}} (A_{n}) \land F - β - \bar{lim_{N}} (B_{n})$ .

Proof

We prove only Statements (i) to (v).

(i) It is sufficient to prove that $β C l (F - β - \bar{lim_{N}} (A_{n})) \leq F - β - \bar{lim_{N}} (A_{n})$ .

Let $p_{s}^{λ} \in β C l (F - β - \bar{lim_{N}} (A_{n}))$ , and let $U$ be an arbitrary fuzzy β-open Q-β-neighborhood of $p_{t}^{r}$ . Then, we have $U q F - β - \bar{lim_{N}} (A_{n})$ .

Hence, there exists an element $s^{'} \in X$ such that $U (s^{'}) + F - β - \bar{lim_{N}} (A_{n}) (s^{'}) > 1$ .

Let $F - β - \bar{lim_{N}} (A_{n}) (t^{'}) = k$ . Then, for the fuzzy point $p_{s^{'}}^{k}$ in X, we have $p_{s^{'}}^{k} q U$ and $p_{s^{'}}^{k} \in F - β - \bar{lim_{N}} (A_{n})$ .

Thus, for every element n₀ ∈ N, there exists n ≥ n₀, n ∈ N such that $A_{n} q U$ . This means that $p_{s}^{r} \in F - β - \bar{lim_{N}} (A_{n})$ .
(ii) Clearly, it is sufficient to prove that for every fuzzy β-open set $U$ , the condition $U q A_{n}$ is equivalent to $U q β C l (A_{n})$ .

Let $U q A_{n}$ . Then, there exists an element $s \in X$ such that $U (t) + A_{n} (s) > 1$ . As we have $U (s) + β C l (A_{n}) (s) > 1$ , then $U q β C l (A_{n})$ . Conversely, let $U q β C l (A_{n})$ . Then, A_n ≤ βCl(A_n). Then, there exists an element $s \in X$ such that $U (s) + β C l (A_{n}) (s) > 1$ . Let $β C l (A_{n}) (s) = r$ . Then, $p_{s}^{r} \in β C l (A_{n})$ , and the fuzzy β-open set $U$ is a fuzzy β-open Q-β-neighborhood of $p_{s}^{r}$ . Thus, $U q A_{n}$ .
(iii) This follows by Proposition 2.4 and the definition of the fuzzy β-upper limit.
(iv) This follows by definition of the fuzzy β-upper limit.
(v) Let $p_{s}^{r} \in F - β - \bar{lim_{N}} (A_{n})$ and $U$ be a fuzzy β-open Q-β-neighborhood of $p_{s}^{r}$ in X. Then, for every n₀ ∈ N, there exists n ∈ N, n ≥ n₀ such that $A_{n} q U$ , and therefore, $\lor {A_{n}, n \in N} q U$ . Thus, $p_{s}^{r} \in β C l (\lor {A_{n}, n \in N})$ .

Proposition 3.6

Let {A_n, n ∈ N} and {B_n, n ∈ N} be two nets of fuzzy sets in Y. Then, the following statements are true:

(1) The fuzzy β-lower limit is β-closed.
(2) $F - β - \frac{lim}{N} (A_{n}) = F - β - \frac{lim}{N} (β C l (A_{n})$ .
(3) If A_n = A for every n ∈ N, then $F - β - \frac{lim}{N} (A_{n}) = β C l (A)$ .
(4) The fuzzy upper limit is not affected by changing a finite number of A_n.
(5) $\land {A_{n} : n \in N} \leq F - β - \frac{lim}{N} (A_{n})$ .
(6) $F - β - \frac{lim}{N} (A_{n}) \leq β C l (\lor {A_{n} : n \in N})$ .
(7) If A_n ≤ B_n for every n ∈ N, then $F - β - \frac{lim}{N} (A_{n}) \leq F - β - \frac{lim}{N} (B_{n})$ .
(8) $F - β - \frac{lim}{N} (A_{n} \lor B_{n}) \geq F - β - \frac{lim}{N} (A_{n}) \lor F - β - \frac{lim}{N} (B_{n})$ .
(9) $F - β - \frac{lim}{N} (A_{n} \land B_{n}) \leq F - β - \frac{lim}{N} (A_{n}) \land F - β - \frac{lim}{N} (B_{n})$ .

Proof

The proof is similar to that of Proposition 3.5.

Proposition 3.7

For the fuzzy upper and lower limit, we have the relation $F - β - \frac{lim}{N} (A_{n}) \leq F - β - \bar{lim_{N}} (A_{n})$ .

Proof

This is a consequence of the definitions of fuzzy β-upper and fuzzy β-lower limits.

Proposition 3.8

(1) $β C l (F - β - lim_{N} (A_{n})) = F - β - lim_{N} (A_{n}) = F - β - lim_{N} (β C l (A_{n}))$ .
(2) If A_n = A for every n ∈ N, then $F - β - lim_{N} (A_{n}) = β C l (A)$ .
(3) If A_n ≤ B_n for every n ∈ N, then $F - β - lim_{N} (A_{n}) \leq F - β - lim_{N} (B_{n})$ .
(4) $F - β - lim_{N} (A_{n} \lor B_{n}) = F - β - lim_{N} (A_{n}) \lor F - β - lim_{N} (B_{n})$ .

Proof

The proof of this proposition follows by Propositions 3.5 and 3.6.

4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points

The main goals of this part is to study and discuss the notions of fuzzy β-continuous functions, fuzzy β-continuous convergence, and fuzzy points.

Definition 4.1

A function f from a fuzzy space Y into a fuzzy space Z is called fuzzy β-continuous if for every fuzzy point $p_{s}^{λ}$ in Y and every fuzzy β-open Q-β-neighborhood V of $f (p_{s}^{λ})$ , there exists a fuzzy β-open Q-β-neighborhood $U$ of $p_{s}^{λ}$ such that $f (U) \leq v$ .

Let Y and Z be two fuzzy spaces. Then, by FβC(Y, Z), we denote the set of all fuzzy β-continuous maps of Y into Z.

Example 4.2

Let (Y, $T_{1}$ ) and (Y, $T_{2}$ ) be two fuzzy spaces such that $Y = {s, t}, T_{1} = {\bar{0}, \bar{1}}$ and $T_{2} = {\bar{0}, \bar{1}, p_{s}^{\frac{1}{2}}}$ .

We consider the map $i : (Y, T_{1}) \to (Y, T_{2})$ for which i(z) = z for every z ∈ Y. We prove that the map i is not fuzzy continuous at the fuzzy point $p_{s}^{0.8}$ but is fuzzy β-continuous at the fuzzy point $p_{s}^{0.8}$ . Indeed, for the fuzzy open Q-neighborhood $v = p_{s}^{\frac{1}{2}}$ of $i (p_{s}^{0.8}) = p_{s}^{0.8}$ , a fuzzy open Q-neighborhood $U$ of $p_{s}^{0.8}$ such that $i (U) \leq v$ does not exist. The only fuzzy open Q-neighborhood $U$ of $p_{s}^{0.8}$ in (Y, $T_{1}$ ) is the fuzzy set 1̄, and i(1̄) ≰ υ.

Now, we prove that the map i is fuzzy β-continuous at the fuzzy point $p_{s}^{0.8}$ . Let υ be an arbitrary fuzzy β-open Q-β-neighborhood υ of $i (p_{s}^{0.8}) = p_{s}^{0.8}$ .

The family of all fuzzy β-open sets in (Y, $T_{2}$ ) is given by the following fuzzy sets υ of Y:

(i) υ ∈ I^Y such that $v (s) \in [0, \frac{1}{2}]$ and $v (t) = 0$ and
(ii) υ ∈ I^Y such that $v (s) \in (0, \frac{1}{2}]$ and $v (t) = [0, 1]$ .

The above fuzzy sets V (cases (i) and (ii)) are also fuzzy β-open sets of (Y, $T_{1}$ ). Thus, for every fuzzy β-open Q-β-neighborhood V of $i (p_{s}^{0.8})$ in (Y, $T_{2}$ ), there exists the fuzzy β-open Q-β-neighborhood U = V of $p_{s}^{0.8}$ in (Y, $T_{1}$ ) such that i(U) ≤ V.

Definition 4.3

A fuzzy net S = {s_λ, λ ∈ Λ} in a fuzzy space (X, $T$ ) is said to be β-convergent to a fuzzy point e in X relative to $T$ and write β lim s_λ = e if for every fuzzy β-open Q-β-neighborhood U of e and for every λ ∈ Λ, there exists m ∈ Λ such that Uqs_m and m ≥ λ.

Proposition 4.4

Let f : Y → Z be a fuzzy β-continuous map. Then, $p$ is a fuzzy point in $U$ , and $U$ , υ are fuzzy β-open Q-neighborhoods of p and $f (p)$ , respectively, such that $f (U) ≰ v$ . Then, there exists a fuzzy point $p_{1}$ in Y such that $p_{1} q U$ and $f (p_{1}) q̸ v$ .

Proof

Because $f (U) ≰ v, U ≰ f^{- 1} (v)$ , and hence, $s \in Y$ such that $U (s) > f^{- 1} (v) (s)$ or $U (s) - f^{- 1} (v) (s) > 0$ , and therefore, $U (s) + 1 - f^{- 1} (v) (s) > 1$ or $U (s) + {(f^{- 1} (v))}^{c} (s) > 1$ . Let ${(f^{- 1} (v))}^{c} (s) = r$ . Clearly, for the fuzzy point $p_{s}^{r}$ , we have $p_{s}^{r} q U$ and $p_{s}^{r} \in {(f^{- 1} (v))}^{c}$ . Hence, for the fuzzy point $p_{1} \equiv p_{s}^{r}$ , then $p_{1} q U$ and $f (p_{1}) q̸ v$ .

Definition 4.5

A net {f_μ, μ ∈ M} in FβC(Y, Z) fuzzy β-continuously converges to f ∈ FβC(Y, Z) if for every fuzzy net { $p_{λ}$ , λ ∈ Λ} in Y, which β-converges to a fuzzy point $p$ in Y. Thus, we have that the fuzzy net { $f_{μ} (p_{λ})$ , (λ, μ) ∈ Λ × M} β-converges to the fuzzy point $f (p)$ in Z.

Proposition 4.6

A net {f_μ, μ ∈ M}in FβC(Y, Z) fuzzy β-continuously converges to f ∈ FβC(Y, Z) if and only if for every fuzzy point $p$ in Y and for every fuzzy β-open Q-β-neighborhood υ of $f (p)$ in Z, there exist an element μ₀ ∈ M and fuzzy β-open Q-β-neighborhood $U$ of $p$ in Y such that $f_{μ} (U) \leq v$ for every μ ≥ μ₀, μ ∈ M.

Proof

Let $p$ be a fuzzy point in Y and υ be a fuzzy β-open Q-β-neighborhood of $f (p)$ in Z such that for every μ ∈ M and every fuzzy β-open Q-β-neighborhood $U$ of $p$ in Y, there exists μ′ ≥ μ such that $f_{μ^{'}} (U) ≰ v$ . Then, for every fuzzy β-open Q-neighborhood $U$ of $p$ in Y, we can choose a fuzzy point $p U$ in Y (Proposition 4.5) such that $p U q U$ and $f_{μ^{'}} (p U) q̸ v$ .

Clearly, the fuzzy net $p U, U \in N_{Q - p - n} (P)}$ β-converges to $p$ , but the fuzzy net { $f_{μ} (p U), (U, μ) \in N_{Q - p - n} (P) \times M$ } does not β-converge to $f (p)$ in Z.

Conversely, let { $p_{λ}$ , λ ∈ Λ} be a fuzzy net in FβC(Y, Z), which β-converges to the fuzzy point $p$ in Y, and let υ be an arbitrary fuzzy β-open Q-β-neighborhood of f(p) in Z. By assumption, there exist a fuzzy β-open Q-β-neighborhood $U$ of $p$ in Y and an element μ₀ ∈ M such that $f_{μ} (U) \leq v$ for every μ ≥ μ₀, μ ∈ M. Because the fuzzy net {p_λ, λ ∈ Λ} β-converges to $p$ in Y, there exists λ₀ ∈ Λ such that $p_{λ} q U$ for every λ ∈ Λ, λ ≥ λ₀. Let (λ₀, μ₀) ∈ Λ × M. Then, for every (λ, μ) ∈ Λ × M, (λ, μ) ≥ (λ₀, μ₀), we have $f_{μ} (p_{λ}) q f_{μ} (U) \leq v$ , that is, $f_{μ} (p_{λ}) q v$ . Thus, the net { $f_{μ} (p_{λ})$ , (λ, μ) ∈ Λ × M} β-converges to $f (p)$ , and the net {f_μ, μ ∈ M} fuzzy β-continuously converges to f.

Proposition 4.7

A net {f_λ, λ ∈ Λ} in FβC(Y, Z) fuzzy β-continuously converges to f ∈ FβC(Y, Z) if and only if

(1)

F - β - \bar{lim_{Λ}} (f_{λ}^{- 1} (K)) \leq f^{- 1} (K),

for every fuzzy β-closed subset K of Z.

Proof

Let {f_λ, λ ∈ Λ} be a net in FβC(Y, Z), which fuzzy β-continuously converges to f, and let K be an arbitrary fuzzy β-closed subset of Z. Let $p \in F - β - \bar{lim_{Λ}} (f_{λ}^{- 1} (K))$

and W be an arbitrary fuzzy β-open Q-β-neighborhood of $f (p)$ in Z. Because the net {f_λ, λ ∈ Λ} fuzzy β-continuously converges to f, there exist a fuzzy β-open Q-β-neighborhood V of $p$ in Y and an element λ₀ ∈ Λ such that f_λ(υ) ≤ W for every λ ∈ Λ, λ ≥ λ₀. Meanwhile, there exists an element λ ≥ λ₀ such that $v q f_{λ}^{- 1} (K)$ . Hence, f_λ(υ)qK, and therefore, WqK. This means that $f (p) \in β C l (K) = K$ . Thus, $p \in f^{- 1} (K)$ .

Conversely, let {f_λ, λ ∈ Λ} be a net in FβC(Y, Z) and f ∈ FβC(Y, Z) such that the relation (1) holds for every fuzzy β-closed subset K of Z. We prove that the net {f_λ, λ ∈ Λ} fuzzy continuously converges to f. Let $p$ be a fuzzy point of Y and W be a fuzzy β-open Q-β-neighborhood of $f (p)$ in Z. Because $p \notin f^{- 1} (K)$ , where K = W^c, we have $p \notin F - β - \bar{lim_{Λ}} (f_{λ}^{- 1} (K))$ . This means that there exist an element λ₀ ∈ Λ and fuzzy β-open Q-β-neighborhood υ of $p$ in Y such that $f_{λ}^{- 1} (K) q̸ v$ for every λ ∈ Λ, λ ≥ λ₀. Then, we have $v \leq {(f_{λ}^{- 1} (K))}^{c} = f_{λ}^{- 1} (K^{c}) = f_{λ}^{- 1} (W)$ . Therefore, f_λ(υ) ≤ W for every λ ∈ Λ, λ ≥ λ₀, that is, the net {f_λ, λ ∈ Λ} fuzzy β-continuously converges to f.

Proposition 4.8

The following statements are true:

(1) If {f_λ, λ ∈ Λ} is a net in FβC(Y, Z) such that f_λ = f, for every λ ∈ Λ, then {f_λ, λ ∈ Λ} fuzzy β-continuously converges to f ∈ FβC(Y, Z).
(2) If {f_λ, λ ∈ Λ} is a net in FβC(Y, Z), which fuzzy β-continuously converges to f ∈ FβC(Y, Z) and {g_μ, μ ∈ M} is a subnet of {f_λ, λ ∈ Λ}, then the net {g_μ, μ ∈ M} fuzzy β-continuously converges to f.

5. Conclusion

The notions of the sets, separation axioms, and functions in fuzzy topological spaces are highly developed and are used extensively in many practical and engineering problems, computational topology for geometric design, computer-aided geometric design, engineering design research, and mathematical science. In this paper, we discussed and studied some topological structures of fuzzy point applications in fuzzy topological spaces. Namely, we defined the fuzzy β-neighborhood, fuzzy Q-β-neighborhood, β-boundary point, fuzzy β-generalized closed, β- $T_{0}$ , β- $T_{1}$ , β-Hausdorff, β-regular-quasi β- $T_{1}$ , and other related topics. Moreover, many results and examples were investigated to illustrate the importance of our methods. In future works, we will study and apply these concepts in many fields, including bipolar hypersoft topological spaces [24], topological soft rough sets [44], covering rough sets [45], and neighborhood systems of rough sets [45].

Abstract
1. Introduction
2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space
3. Fuzzy β-Convergence and Fuzzy Points
4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points
5. Conclusion

References

Abo-Tabl, ESA, and El-Bably, MK (2022). Rough topological structure based on reflexivity with some applications. AIMS Mathematics. 7, 9911-9925. http://dx.doi.org/10.3934/math.2022553
Abu-Gdairi, R, El-Gayar, MA, El-Bably, MK, and Fleifel, KK (2021). Two different views for generalized rough sets with applications. Mathematics. 9. article no. 2275
El-Bably, MK, and El Atik, AA (2021). Soft β-rough sets and its application to determine COVID-19. Turkish Journal of Mathematics. 45. article no. 4
Nawar, AS, El-Gayar, MA, El-Bably, MK, and Hosny, RA (2022). θβ-ideal approximation spaces and their applications. AIMS Mathematics. 7, 2479-2497. https://doi.org/10.3934/math.2022139
Abd El-Monsef, ME, El-Gayar, MA, and Aqeel, RM (2014). On relationships between revised rough fuzzy approximation operators and fuzzy topological spaces. International Journal of Granular Computing, Rough Sets and Intelligent Systems. 3, 257-271. https://doi.org/10.1504/IJGCRSIS.2014.068022
Abd El-Monsef, ME, El-Gayar, MA, and Aqeel, RM (2017). A comparison of three types of rough fuzzy sets based on two universal sets. International Journal of Machine Learning and Cybernetics. 8, 343-353. https://doi.org/10.1007/s13042-015-0327-8
Abu-Gdairi, R, El-Gayar, MA, Al-shami, TM, Nawar, AS, and El-Bably, MK (2022). Some topological approaches for generalized rough sets and their decision-making applications. Symmetry. 14. article no. 95
El-Gayar, MA, and El Atik, AA (2022). Topological models of rough sets and decision making of COVID-19. Complexity. 2022. article no. 2989236
Lu, H, Khalil, AM, Alharbi, W, and El-Gayar, MA (2021). A new type of generalized picture fuzzy soft set and its application in decision making. Journal of Intelligent & Fuzzy Systems. 40, 12459-12475. https://doi.org/10.3233/JIFS-201706
Azzam, AA, Khalil, AM, and Li, SG (2020). Medical applications via minimal topological structure. Journal of Intelligent & Fuzzy Systems. 39, 4723-4730. https://doi.org/10.3233/JIFS-200651
Nasef, AA, and El-Sayed, MK (2022). Properties of topological concepts associated with weak systems. Mathematical Statistician and Engineering Applications. 71, 6631-6641.
El Atik, AEFA, and Nasef, AA (2022). Topological kernel of sets and application on fractals. Journal of Contemporary Technology and Applied Engineering. 1, 1-8. https://doi.org/10.21608/jctae.2022.143392.1000
Nasef, AA, El Atik, AA, and Revathi, GK (2022). A model of hepatitis C and blood pressure via nano topological spaces. Thai Journal of Mathematics. 20, 669-677.
El-Bably, MK, Abu-Gdairi, R, and El-Gayar, MA (2023). Medical diagnosis for the problem of Chikungunya disease using soft rough sets. AIMS Mathematics. 8, 9082-9105. http://dx.doi.org/10.3934/math.2023455
El-Bably, MK, and Abo-Tabl, EA (2021). A topological reduction for predicting of a lung cancer disease based on generalized rough sets. Journal of Intelligent & Fuzzy Systems. 41, 3045-3060. https://doi.org/10.3233/JIFS-210167
El-Bably, MK, Ali, MI, and Abo-Tabl, EA (2021). New topological approaches to generalized soft rough approximations with medical applications. Journal of Mathematics. 2021. article no. 2559495
Stadler, BMR, and Stadler, PF (2002). Generalized topological spaces in evolutionary theory and combinatorial chemistry. Journal of Chemical Information and Modeling. 42, 577-585. https://doi.org/10.1021/ci0100898
Stadler, BM, and Stadler, PF (2004). The topology of evolutionary Biology. Modeling in Molecular Biology. Heidelberg, Germany: Springer, pp. 267-286 https://doi.org/10.1007/978-3-642-18734-6
El-Sharkasy, MM (2018). Topological model for recombination of DNA and RNA. International Journal of Biomathematics. 11. article no. 1850097
El-Sayed, MK, and Nasef, AA (2021). Aapplications of rough sets in plant sciences and disease diagnosis. Journal of Computational and Theoretical Nanoscience. 18, 1759-1762. https://doi.org/10.1166/jctn.2021.9720
Hosny, RA, Abu-Gdairi, R, and El-Bably, MK (2023). Approximations by ideal minimal structure with chemical application. Intelligent Automation and Soft Computing. 36, 3073-3085. http://dx.doi.org/10.32604/iasc.2023.034234
Alghazzwi, D, Ali, A, Almutlg, A, Abo-Tabl, EA, and Azzam, AA (2023). A novel structure of q-rung orthopair fuzzy sets in ring theory. AIMS Mathematics. 8, 8365-8385. http://dx.doi.org/10.3934/math.2023422
Ameen, ZA, Al-Shami, TM, Azzam, AA, and Mhemdi, A (2022). A novel fuzzy structure: infra-fuzzy topological spaces. Journal of Function Spaces. 2022. article no. 97778069
Musa, SY, and Asaad, BA (2023). Separation axioms on bipolar hypersoft topological spaces. International Journal of Neutrosophic Science. 20, 150-164. http://dx.doi.org/10.54216/IJNS.200112
Asaad, BA, and Musa, SY (2023). A novel class of bipolar soft separation axioms concerning crisp points. Demonstratio Mathematica. 56. article no. 20220189
Saleh, HY, Asaad, BA, and Mohammed, RA (2022). Bipolar soft limit points in bipolar soft generalized topological spaces. Mathematics and Statistics. 10, 1264-1274. https://doi.org/10.13189/ms.2022.100612
Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Wong, CK (1974). Fuzzy points and local properties of fuzzy topology. Journal of Mathematical Analysis and Applications. 46, 316-328. https://doi.org/10.1016/0022-247X(74)90242-X
Pu, PM, and Liu, YM (1980). Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence. Journal of Mathematical Analysis and Applications. 76, 571-599. https://doi.org/10.1016/0022-247X(80)90048-7
Pu, PM, and Liu, YM (1980). Fuzzy topology. II. Product and quotient spaces. Journal of Mathematical Analysis and Applications. 77, 20-37. https://doi.org/10.1016/0022-247X(80)90258-9
Azad, KK (1981). On fuzzy semi continuity, fuzzy almost continuity, and fuzzy weakly continuity. Journal of Mathematical Analysis and Applications. 82, 14-32. https://doi.org/10.1016/0022-247X(81)90222-5
Ajmal, N, and Tyagi, BK (1991). On fuzzy almost continuous functions. Fuzzy Sets and Systems. 41, 221-232. https://doi.org/10.1016/0165-0114(91)90226-G
Chang, CL (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 24, 182-190. https://doi.org/10.1016/0022-247X(68)90057-7
Hu, CM (1985). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 110, 141-178. https://doi.org/10.1016/0022-247X(85)90340-3
Saha, S (1987). Fuzzy δ-continuous mappings. Journal of Mathematical Analysis and Applications. 126, 130-142. https://doi.org/10.1016/0022-247X(87)90081-3
Wong, CK (1975). Fuzzy topology. Fuzzy Sets and their Applications to Cognitive and Decision Processes. Amsterdam, Netherlands: Elsevier, pp. 171-190
Yalvac, TH (1987). Fuzzy sets and functions on fuzzy spaces. Journal of Mathematical Analysis and Applications. 126, 409-423. https://doi.org/10.1016/0022-247X(87)90050-3
Georgiou, DN, and Papadopoulos, BK (1999). Convergences in fuzzy topological spaces. Fuzzy Sets and Systems. 101, 495-504. https://doi.org/10.1016/S0165-0114(97)00112-7
Fathalla, MA 1984. On fuzzy topological spaces. Ph.D. dissertation. Assuit University. Sohag, Egypt.
Hanafy, IM (2004). Fuzzy β-compactness and fuzzy β-closed spaces. Turkish Journal of Mathematics. 28, 281-293.
Thakur, SS, and Singh, S (1998). On fuzzy semi-preopen sets and fuzzy semi-precontinuity. Fuzzy Sets and Systems. 98, 383-391. https://doi.org/10.1016/s0165-0114(96)00363-6
Mukherjee, MN, and Sinha, SP (1990). On some near-fuzzy continuous functions between fuzzy topological spaces. Fuzzy Sets and Systems. 34, 245-254. https://doi.org/10.1016/0165-0114(90)90163-Z
Johnson, TP, and Mathew, SC (2000). On fuzzy point in topological spaces. Far East Journal of Mathematical Sciences, Part I (Geometry and Topology), 75-86.
Ali, MI, El-Bably, MK, and Abo-Tabl, EA (2022). Topological approach to generalized soft rough sets via near concepts. Soft Computing. 26, 499-509. https://doi.org/10.1007/s00500-021-06456-z
Nawar, AS, El-Bably, MK, and El-Atik, AA (2020). Certain types of coverings based rough sets with application. Journal of Intelligent & Fuzzy Systems. 39, 3085-3098. https://doi.org/10.3233/JIFS-191542
El-Bably, MK, Al-shami, TA, Nawar, AS, and Mhemdi, A (2021). Corrigendum to “Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space”. Journal of Intelligent & Fuzzy Systems. 41, 7353-7361. https://doi.org/10.3233/JIFS-211198

Article Tools

Metrics

View942
Download356
SCOPUS 7

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

On Fuzzy Point Applications of Fuzzy Topological Spaces

Abstract

1. Introduction

Definition 1.1 ([27])

Definition 1.2 ([27])

Definition 1.3 ([29, 30])

Definition 1.4 ([29, 30])

Definition 1.5 ([28])

Definition 1.6 ([29])

Definition 1.7 ([39–42])

Definition 1.8 ([39–42])

Definition 1.9 ([39–42])

2. Study of Fuzzy Points, β-Closed Sets, and Separation Axioms in Fuzzy Topological Space

Definition 2.1

Definition 2.2

Remark 2.3

Proposition 2.4

Definition 2.5

Proposition 2.6

Example 2.7

Definition 2.8

Definition 2.9

Remark 2.10

Proposition 2.11

Definition 2.12

Example 2.13

Example 2.14

Definition 2.15

Example 2.16

Definition 2.17

Example 2.18

Definition 2.19 ([43])

Definition 2.20

Proposition 2.21

Proposition 2.22

Proposition 2.23

Proposition 2.24

Definition 2.25

Definition 2.26

Proposition 2.27

Definition 2.28

Proposition 2.29

Definition 2.30

Proposition 2.31

3. Fuzzy β-Convergence and Fuzzy Points

Definition 3.1

Example 3.2

Definition 3.3

Definition 3.4

Proposition 3.5

Proposition 3.6

Proposition 3.7

Proposition 3.8

4. Fuzzy-Continuous Functions, Fuzzy-Continuous Convergence, and Fuzzy Points

Definition 4.1

Example 4.2

Definition 4.3

Proposition 4.4

Definition 4.5

Proposition 4.6

Proposition 4.7

Proposition 4.8

5. Conclusion

Conflict of Interest

References

Biographies

Article

Original Article

On Fuzzy Point Applications of Fuzzy Topological Spaces

Abstract

1. Introduction

Definition 1.1 ([27])

Definition 1.2 ([27])

Definition 1.3 ([29, 30])

Definition 1.4 ([29, 30])

Definition 1.5 ([28])

Definition 1.6 ([29])

Definition 1.7 ([39–42])

International Journal
of Fuzzy Logic and
Intelligent Systems

Related articles in IJFIS

Prime Ideal Theorem for Fuzzy Lattices

Aparna Jain
International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(4): 436-447 https://doi.org/10.5391/IJFIS.2023.23.4.436