EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5911
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Characterizations of (r, s)-Fuzzy b-Open Sets with
Applications in Double Fuzzy Topological Spaces

Islam M. Taha1,∗, Jawaher Al-Mufarrij2, Osama M. Taha1

1 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
2 Department of Mathematics, Women Section, King Saud University, Riyadh 12372,
Saudi Arabia

Abstract. In this paper, we displayed and characterized a novel class of fuzzy open sets (F-open
sets) in double fuzzy topological spaces (DFT Ss) based on Šostak,s sense, called (r, s)-fuzzy b-
open sets ((r, s)-F-b-open sets). This class is contained in the class of (r, s)-F-β-open sets and
contains all (r, s)-F-α-open sets, (r, s)-F-pre-open sets, and (r, s)-F-semi-open sets. Next, we
explored and studied the notion of DF-b-continuity between DFT Ss (G,ℑ,ℑ∗) and (Z,𭟋,𭟋∗).
We also defined and discussed the notions of DF-almost b-continuity and DF-weakly b-continuity,
which are weaker forms of DF-b-continuity. Thereafter, we presented and investigated novel DF-
mappings via (r, s)-F-b-open and (r, s)-F-b-closed sets. Finally, we introduced some novel types
of DF-separation axioms, called (r, s)-F-b-regular and (r, s)-F-b-normal spaces, and studied some
properties of them.

2020 Mathematics Subject Classifications: 54A05, 54A40, 54C05, 54C08, 54D15

Key Words and Phrases: DF-topology, (r, s)-F-b-open set, DF-b-closure operator, DF-b-
continuity, DF-b-irresoluteness, DF-b-openness, DF-b-closeness, (r, s)-F-b-normal space, (r, s)-F-
b-regular space

1. Introduction

The concept of a fuzzy set (F-set) of a nonempty set G is a mapping M : G → I
(where I = [0, 1]). This concept was first defined in 1965 by Zadeh [1]. The concept of
an F-topology was presented in 1968 by the author of [2]. Several authors have success-
fully generalized the theory of general topology to the fuzzy setting with crisp methods.
According to Šostak [3], the notion of an F-topology being a crisp subclass of the class
of F-sets and fuzziness in the notion of openness of an F-set have not been considered,
which seems to be a drawback in the process of fuzzification of a topological space. Thus,
the author of [3] introduced a novel definition of an F-topology as the concept of openness

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5911

Email addresses: imtaha2010@yahoo.com (I. M. Taha),
jmufarij@ksu.edu.sa (J. Al-Mufarrij), osama.taha2015@yahoo.com (O. M. Taha)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 2 of 27

of F-sets. It is an extension of an F-topology introduced by Chang [2]. Also, many re-
searchers (Ramadan [4], Chattopadhyay et. al. [5], El Gayyar et. al. [6], Höhle and Šostak
[7], Ramadan et. al. [8], Kim et. al. [9], Abbas [10, 11], Kim and Abbas [12], Aygun
and Abbas [13, 14], Li and Shi [15, 16], Shi and Li [17], Fang and Guo [18], El-Dardery
et. al. [19], Kalaivani and Roopkumar [20], Solovyov [21], Minana and Šostak [22]) have
redefined the same notion and studied FT Ss being unaware of Šostak,s work.

The notion of an intuitionistic F-set was defined by Atanassov [23, 24], which is a
generalization of an F-set [1]. Coker [25, 26] presented the notion of an intuitionistic
F-topology based on Chang,s sense [2]. After that, the notion of an intuitionistic F-
topology based on Šostak,s sense [3] was introduced by the authors of [27, 28]. The
name (intuitionistic) was replaced with the name (double) by Garcia and Rodabaugh
[29]. In addition, the notions of (r, s)-F-semi-open, (r, s)-F-pre-open, and (r, s)-F-α-open
sets were introduced by the authors of [30, 31] based on Šostak,s sense [3]. Also, lots
of creative studies about the theories of an intuitionistic F-set have been considered by
several researchers; see [32–39].

The layout of this study is as follows.

• In Section 3, we present and investigate a novel class of F-open sets in DFT Ss based
on Šostak,s sense [3], called (r, s)-F-b-open sets. Furthermore, we define and discuss the
notions of DF-b-closure operators and DF-b-interior operators.

• In Section 4, we introduce and discuss the concept of DF-b-continuity between
DFT Ss (G,ℑ,ℑ∗) and (Z,𭟋,𭟋∗). In addition, we display and characterize the concepts
of DF-weakly b-continuity and DF-almost b-continuity, which are weaker forms of DF-b-
continuity.

• In Section 5, we explore and characterize some novel DF-mappings using (r, s)-F-b-
open and (r, s)-F-b-closed sets. We also introduce novel types of DF-separation axioms,
called (r, s)-F-b-regular and (r, s)-F-b-normal spaces, and discuss some properties of them.

• In Section 6, we close this paper with conclusions and proposed future papers.

2. Preliminaries

In this study, nonempty sets will be denoted by G, Z, Q, etc. On G, IG is the class of
all F-sets. For any F-set M ∈ IG, Mc(g) = 1 −M(g), for each g ∈ G. Also, for θ ∈ I,
θ(g) = θ, for each g ∈ G.

An F-point gθ on G is an F-set, and is defined as follows: gθ(v) = θ if v = g, and
gθ(v) = 0 for any v ∈ G− {g}. Moreover, we say that gθ belongs to M ∈ IG (gθ ∈ M), if



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 3 of 27

θ ≤ M(g). On G, Pθ(G) is the class of all F-points.

On G, an F-set M ∈ IG is a quasi-coincident with N ∈ IG (M q N ), if there is g ∈ G,
with M(g) +N (g) > 1. Otherwise, M is not a quasi-coincident with N (M q N ).

Lemma 1. [40] Let M,N ∈ IG. Thus,

(i) M q N iff there is gθ ∈ M such that gθ q N ,

(ii) M∧N ≠ 0 if M q N ,

(iii) M q N iff M ≤ N c,

(iv) M ≤ N iff gθ ∈ M implies gθ ∈ N iff gθ q M implies gθ q N ,

(v) gθ q
∨

i∈ΓMi iff there is i◦ ∈ Γ such that gθ q Mi◦ .

Definition 1. [27, 35, 38] A double fuzzy topology (DFT ) on G is a pair (ℑ,ℑ∗) of the
mappings ℑ,ℑ∗ : IG −→ I, which satisfy the following conditions:

(i) ℑ(M) + ℑ∗(M) ≤ 1, ∀ M ∈ IG.

(ii) ℑ(M∧N ) ≥ ℑ(M) ∧ ℑ(N ) and ℑ∗(M∧N ) ≤ ℑ∗(M) ∨ ℑ∗(N ), ∀ M,N ∈ IG.

(iii) ℑ(
∨

i∈ΓMi) ≥
∧

i∈Γℑ(Mi) and ℑ∗(
∨

i∈ΓMi) ≤
∨

i∈Γℑ∗(Mi), ∀ {Mi}i∈Γ ⊂ IG.

Thus, (G,ℑ,ℑ∗) is said to be an DFT S based on Šostak,s sense [3].

Definition 2. [28, 30, 35] In an DFT S (G,ℑ,ℑ∗), for each M ∈ IG, r ∈ I◦, and s ∈ I1
(where I◦ = (0, 1] and I1 = [0, 1)), we define DF-operators Cℑ∗ and Iℑ∗ : IG×I◦×I1 → IG

as follows:

Cℑ∗(M, r, s) =
∧

{N ∈ IG : M ≤ N , ℑ(N c) ≥ r,ℑ∗(N c) ≤ s}.

Iℑ∗(M, r, s) =
∨

{N ∈ IG : N ≤ M, ℑ(N ) ≥ r,ℑ∗(N ) ≤ s}.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 4 of 27

Definition 3. [30, 31, 35] Let (G,ℑ,ℑ∗) be an DFT S, r ∈ I◦, and s ∈ I1. An F-
set M ∈ IG is said to be (r, s)-F-regularly-open (resp. (r, s)-F-pre-open, (r, s)-F-semi-
open, (r, s)-F-β-open, and (r, s)-F-α-open) if M = Iℑ∗(Cℑ∗(M, r, s), r, s) (resp. M ≤
Iℑ∗(Cℑ∗(M, r, s), r, s),M ≤ Cℑ∗(Iℑ∗(M, r, s), r, s),M ≤ Cℑ∗(Iℑ∗(Cℑ∗(M, r, s), r, s), r, s),
and M ≤ Iℑ∗(Cℑ∗(Iℑ∗(M, r, s), r, s), r, s)).

Definition 4. [28, 35, 38] An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is said to be

(i) DF-continuous if ℑ(P−1(N )) ≥ 𭟋(N ) and ℑ∗(P−1(N )) ≤ 𭟋∗(N ), ∀ N ∈ IZ ;

(ii) DF-open if 𭟋(P(M)) ≥ ℑ(M) and 𭟋∗(P(M)) ≤ ℑ∗(M), ∀ M ∈ IG;

(iii) DF-closed if 𭟋((P(M))c) ≥ ℑ(Mc) and 𭟋∗((P(M))c) ≤ ℑ∗(Mc), ∀ M ∈ IG.

Definition 5. [30, 31, 35] Let (G,ℑ,ℑ∗) and (Z,𭟋,𭟋∗) be DFT Ss, r ∈ I◦, and s ∈ I1.
An F-mapping P : IG −→ IZ is said to be DF-α-continuous (resp. DF-pre-continuous,
DF-semi-continuous, and DF-β-continuous) if P−1(N ) is an (r, s)-F-α-open set (resp.
(r, s)-F-pre-open set, (r, s)-F-semi-open set, and (r, s)-F-β-open set), for every N ∈ IZ

with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s.

Some basic notations and results that we need in the sequel are found in [27, 28, 30,
31, 35, 36].

3. On (r, s)-fuzzy b-open and b-closed sets

Here, we present and study a new class of F-open sets, called (r, s)-F-b-open sets
in DFT S (G,ℑ,ℑ∗) based on Šostak,s sense [3]. Also, we explore and investigate the
concepts of DF-b-interior operators and DF-b-closure operators.

Definition 6. Let (G,ℑ,ℑ∗) be an DFT S, r ∈ I◦, and s ∈ I1. An F-set M ∈ IG is said
to be an (r, s)-F-b-open set if M ≤ Cℑ∗(Iℑ∗(M, r, s), r, s) ∨ Iℑ∗(Cℑ∗(M, r, s), r, s).

Definition 7. Let (G,ℑ,ℑ∗) be an DFT S, r ∈ I◦, and s ∈ I1. An F-set M ∈ IG is said
to be an (r, s)-F-b-closed set if M ≥ Cℑ∗(Iℑ∗(M, r, s), r, s) ∧ Iℑ∗(Cℑ∗(M, r, s), r, s).

Remark 1. The complement of (r, s)-F-b-open sets (resp. (r, s)-F-b-closed sets) are
(r, s)-F-b-closed sets (resp. (r, s)-F-b-open sets).



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 5 of 27

Proposition 1. In an DFT S (G,ℑ,ℑ∗), for each M ∈ IG, r ∈ I◦, and s ∈ I1, then

(i) every (r, s)-F-pre-open set is (r, s)-F-b-open;

(ii) every (r, s)-F-b-open set is (r, s)-F-β-open;

(iii) every (r, s)-F-semi-open set is (r, s)-F-b-open.

Proof.
(i) If M is an (r, s)-F-pre-open set, then

M ≤ Iℑ∗(Cℑ∗(M, r, s), r, s)

≤ Iℑ∗(Cℑ∗(M, r, s), r, s) ∨ Iℑ∗(M, r, s)

≤ Iℑ∗(Cℑ∗(M, r, s), r, s) ∨ Cℑ∗(Iℑ∗(M, r, s), r, s).

Thus, M is (r, s)-F-b-open.

(ii) If M is an (r, s)-F-b-open set, then

M ≤ Cℑ∗(Iℑ∗(M, r, s), r, s) ∨ Iℑ∗(Cℑ∗(M, r, s), r, s)

≤ Cℑ∗(Iℑ∗(Cℑ∗(M, r, s), r, s), r, s) ∨ Iℑ∗(Cℑ∗(M, r, s), r, s)

≤ Cℑ∗(Iℑ∗(Cℑ∗(M, r, s), r, s), r, s).

Thus, M is (r, s)-F-β-open.

(iii) If M is an (r, s)-F-semi-open set, then

M ≤ Cℑ∗(Iℑ∗(M, r, s), r, s)

≤ Cℑ∗(Iℑ∗(M, r, s), r, s) ∨ Iℑ∗(M, r, s)

≤ Cℑ∗(Iℑ∗(M, r, s), r, s) ∨ Iℑ∗(Cℑ∗(M, r, s), r, s).

Thus, M is (r, s)-F-b-open.

Remark 2. From the previous discussions and definitions, we have the following diagram.

(r,s)-F-pre-open set

↗ ↓

(r,s)-F-α-open set −→ (r,s)-F-b-open set −→ (r,s)-F-β-open set

↘ ↑

(r,s)-F-semi-open set



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 6 of 27

Remark 3. The converse of the above diagram fails as Examples 1, 2, and 3 will show.

Example 1. Let G = {g1, g2} and define M,N ,U ∈ IG as follows: M = { g1
0.4 ,

g2
0.3},

N = { g1
0.2 ,

g2
0.6}, U = { g1

0.5 ,
g2
0.7}. Define ℑ,ℑ∗ : IG −→ I as follows:

ℑ(V) =



1, if V ∈ {1, 0},
1
4 , if V = N ,
1
2 , if V = M,
1
4 , if V = N ∧M,
1
2 , if V = N ∨M,

0, otherwise,

ℑ∗(V) =



0, if V ∈ {1, 0},
1
4 , if V = N ,
1
2 , if V = M,
1
2 , if V = N ∧M,
1
4 , if V = N ∨M,

1, otherwise.

Thus, U is an (14 ,
1
2)-F-b-open set, but it is neither (14 ,

1
2)-F-pre-open nor (14 ,

1
2)-F-α-

open.

Example 2. Let G = {g1, g2} and define M,N ,U ∈ IG as follows: M = { g1
0.3 ,

g2
0.2},

N = { g1
0.7 ,

g2
0.8}, U = { g1

0.5 ,
g2
0.4}. Define ℑ,ℑ∗ : IG −→ I as follows:

ℑ(V) =


1, if V ∈ {1, 0},
1
3 , if V = M,
1
2 , if V = N ,

0, otherwise,

ℑ∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = M,
1
3 , if V = N ,

1, otherwise.

Thus, U is an (13 ,
1
2)-F-b-open set, but it is not (13 ,

1
2)-F-semi-open.

Example 3. Let G = {g1, g2} and define M,U ∈ IG as follows: M = { g1
0.5 ,

g2
0.4}, U =

{ g1
0.4 ,

g2
0.5}. Define ℑ,ℑ∗ : IM −→ I as follows:

ℑ(V) =


1, if V ∈ {1, 0},
1
2 , if V = M,

0, otherwise,

ℑ∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = M,

1, otherwise.

Thus, U is an (13 ,
1
2)-F-β-open set, but it is not (13 ,

1
2)-F-b-open.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 7 of 27

Corollary 1. In an DFT S (G,ℑ,ℑ∗), r ∈ I◦, and s ∈ I1, we have the following properties:

(i) the union of (r, s)-F-b-open sets is (r, s)-F-b-open;

(ii) the intersection of (r, s)-F-b-closed sets is (r, s)-F-b-closed.

Proof. This is easily proved by Definitions 6 and 7.

Corollary 2. In an DFT S (G,ℑ,ℑ∗), for each (r, s)-F-b-closed set M ∈ IG:

(i) If M is (r, s)-F-regularly-open, then M is (r, s)-F-pre-closed.

(ii) If M is (r, s)-F-regularly-closed, then M is (r, s)-F-semi-closed.

(iii) If Iℑ∗(M, r, s) = 0, then M is (r, s)-F-semi-closed.

(iv) If Cℑ∗(M, r, s) = 0, then M is (r, s)-F-pre-closed.

Proof. The proof follows by Definitions 3 and 7.

Corollary 3. In an DFT S (G,ℑ,ℑ∗), for each (r, s)-F-b-open set N ∈ IG:

(i) If N is (r, s)-F-regularly-open, then N is (r, s)-F-semi-open.

(ii) If N is (r, s)-F-regularly-closed, then N is (r, s)-F-pre-open.

(iii) If Iℑ∗(N , r, s) = 0, then N is (r, s)-F-pre-open.

(iv) If Cℑ∗(N , r, s) = 0, then N is (r, s)-F-semi-open.

Proof. The proof follows by Definitions 3 and 6.

Definition 8. In an DFT S (G,ℑ,ℑ∗), for each M ∈ IG, r ∈ I◦, and s ∈ I1, we define
an DF-b-closure operator bCℑ∗ : IG × I◦ × I1 −→ IG as follows: bCℑ∗(M, r, s) =

∧
{N ∈

IG : M ≤ N , N is (r,s)-F-b-closed}.

Proposition 2. In an DFT S (G,ℑ,ℑ∗), for each M ∈ IG, r ∈ I◦, and s ∈ I1. An F-set
M is (r, s)-F-b-closed iff bCℑ∗(M, r, s) = M.

Proof. This is easily proved from Definition 8.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 8 of 27

Theorem 1. In an DFT S (G,ℑ,ℑ∗), for each M,N ∈ IG, r ∈ I◦, and s ∈ I1. An
DF-operator bCℑ∗ : IG × I◦ × I1 −→ IG satisfies the following properties.

(i) bCℑ∗(0, r, s) = 0.

(ii) M ≤ bCℑ∗(M, r, s) ≤ Cℑ∗(M, r, s).

(iii) bCℑ∗(M, r, s) ≤ bCℑ∗(N , r, s) if M ≤ N .

(iv) bCℑ∗(bCℑ∗(M, r, s), r, s) = bCℑ∗(M, r, s).

(v) bCℑ∗(M∨N , r, s) ≥ bCℑ∗(M, r, s) ∨ bCℑ∗(N , r, s).

(vi) bCℑ∗(Cℑ∗(M, r, s), r, s) = Cℑ∗(M, r, s).

Proof. (i), (ii), and (iii) are easily proved by Definition 8.

(iv) From (ii) and (iii), bCℑ∗(M, r, s) ≤ bCℑ∗(bCℑ∗(M, r, s), r, s). Now, we show
bCℑ∗(M, r, s) ≥ bCℑ∗(bCℑ∗(M, r, s), r, s). If bCℑ∗(M, r, s) does not contain bCℑ∗(bCℑ∗(M, r, s), r, s),
there is g ∈ G and θ ∈ (0, 1) with bCℑ∗(M, r, s)(g) < θ < bCℑ∗(bCℑ∗(M, r, s), r, s)(g). (G)

Since bCℑ∗(M, r, s)(g) < θ, by Definition 8, there is U ∈ IG as an (r, s)-F-b-closed set
and M ≤ U with bCℑ∗(M, r, s)(g) ≤ U(g) < θ. Since M ≤ U , then bCℑ∗(M, r, s) ≤ U .
Again, by the definition of bCℑ∗ , then bCℑ∗(bCℑ∗(M, r, s), r, s) ≤ U .

Hence, bCℑ∗(bCℑ∗(M, r, s), r, s)(g) ≤ U(g) < θ, which is a contradiction for (G).
Thus, bCℑ∗(M, r, s) ≥ bCℑ∗(bCℑ∗(M, r, s), r, s). Therefore, bCℑ∗(bCℑ∗(M, r, s), r, s) =
bCℑ∗(M, r, s).

(v) Since M ≤ M∨N and N ≤ M∨N , hence by (iii), bCℑ∗(M, r, s) ≤ bCℑ∗(M∨
N , r, s) and bCℑ∗(N , r, s) ≤ bCℑ∗(M∨N , r, s). Thus, bCℑ∗(M∨N , r, s) ≥ bCℑ∗(M, r, s)∨
bCℑ∗(N , r, s).

(vi) From Proposition 2 and the fact that Cℑ∗(M, r, s) is an (r, s)-F-b-closed set, then
bCℑ∗(Cℑ∗(M, r, s), r, s) = Cℑ∗(M, r, s).

Definition 9. In an DFT S (G,ℑ,ℑ∗), for each M ∈ IG, r ∈ I◦, and s ∈ I1, we define
an DF-b-interior operator bIℑ∗ : IG × I◦ × I1 −→ IG as follows: bIℑ∗(M, r, s) =

∨
{N ∈

IG : N ≤ M, N is (r, s)-F-b-open}.

Proposition 3. Let (G,ℑ,ℑ∗) be an DFT S, M ∈ IG, r ∈ I◦, and s ∈ I1. Then

(i) bCℑ∗(Mc, r, s) = (bIℑ∗(M, r, s))c;



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 9 of 27

(ii) bIℑ∗(Mc, r, s) = (bCℑ∗(M, r, s))c.

Proof. (i) For eachM ∈ IG, we have bCℑ∗(Mc, r, s) =
∧
{N ∈ IG : Mc ≤ N , N is (r, s)-F-b-closed} =

[
∨
{N c ∈ IG : N c ≤ M, N c is (r, s)-F-b-open}]c = (bIℑ∗(M, r, s))c.

(ii) This is similar to that of (i).

Proposition 4. In an DFT S (G,ℑ,ℑ∗), for each M ∈ IG, r ∈ I◦, and s ∈ I1. An F-set
M is (r, s)-F-b-open iff bIℑ∗(M, r, s) = M.

Proof. This is easily proved from Definition 9.

Theorem 2. In an DFT S (G,ℑ,ℑ∗), for each M,N ∈ IG, r ∈ I◦, and s ∈ I1. An
DF-operator bIℑ∗ : IG × I◦ × I1 −→ IG satisfies the following properties.

(i) bIℑ∗(1, r, s) = 1.

(ii) Iℑ∗(M, r, s) ≤ bIℑ∗(M, r, s) ≤ M.

(iii) bIℑ∗(M, r, s) ≤ bIℑ∗(N , r, s) if M ≤ N .

(iv) bIℑ∗(bIℑ∗(M, r, s), r, s) = bIℑ∗(M, r, s).

(v) bIℑ∗(M, r, s) ∧ bIℑ∗(N , r, s) ≥ bIℑ∗(M∧N , r, s).

Proof. The proof is similar to that of Theorem 1.

4. On double fuzzy b-continuity and b-irresoluteness

Here, we display and discuss the concept of DF-b-continuity between DFT Ss based on
Šostak,s sense [3]. Moreover, we present and study the notions of DF-almost b-continuity
and DF-weakly b-continuity.

Definition 10. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-b-continuous if
P−1(N ) is an (r, s)-F-b-open set, for each N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 10 of 27

Remark 4. From the previous definitions, we have the following diagram.

DF-pre-continuity

↗ ↓

DF-α-continuity −→ DF-b-continuity −→ DF-β-continuity

↘ ↑

DF-semi-continuity

Remark 5. The converse of the above diagram fails as Examples 4, 5, and 6 will show.

Example 4. Let G = {g1, g2} and define M,N ,U ∈ IG as follows: M = { g1
0.4 ,

g2
0.3},

N = { g1
0.2 ,

g2
0.6}, U = { g1

0.5 ,
g2
0.7}. Define ℑ,ℑ∗,𭟋,𭟋∗ : IG −→ I as follows:

ℑ(V) =



1, if V ∈ {1, 0},
1
4 , if V = N ,
1
2 , if V = M,
1
4 , if V = N ∧M,
1
2 , if V = N ∨M,

0, otherwise,

ℑ∗(V) =



0, if V ∈ {1, 0},
1
4 , if V = N ,
1
2 , if V = M,
1
2 , if V = N ∧M,
1
4 , if V = N ∨M,

1, otherwise,

𭟋(V) =


1, if V ∈ {1, 0},
1
4 , if V = U ,
0, otherwise,

𭟋∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = U ,
1, otherwise.

Thus, the identity F-mapping P : (G,ℑ,ℑ∗) −→ (G,𭟋,𭟋∗) is DF-b-continuous, but
it is neither DF-pre-continuous nor DF-α-continuous.

Example 5. Let G = {g1, g2} and define M,N ,U ∈ IG as follows: M = { g1
0.3 ,

g2
0.2},

N = { g1
0.7 ,

g2
0.8}, U = { g1

0.5 ,
g2
0.4}. Define ℑ,ℑ∗,𭟋,𭟋∗ : IG −→ I as follows:

ℑ(V) =


1, if V ∈ {1, 0},
1
3 , if V = M,
1
2 , if V = N ,

0, otherwise,

ℑ∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = M,
1
3 , if V = N ,

1, otherwise,



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 11 of 27

𭟋(V) =


1, if V ∈ {1, 0},
1
3 , if V = U ,
0, otherwise,

𭟋∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = U ,
1, otherwise.

Thus, the identity F-mapping P : (G,ℑ,ℑ∗) −→ (G,𭟋,𭟋∗) is DF-b-continuous, but
it is not DF-semi-continuous.

Example 6. Let G = {g1, g2} and define M,U ∈ IG as follows: M = { g1
0.5 ,

g2
0.4}, U =

{ g1
0.4 ,

g2
0.5}. Define ℑ,ℑ∗,𭟋,𭟋∗ : IG −→ I as follows:

ℑ(V) =


1, if V ∈ {1, 0},
1
2 , if V = M,

0, otherwise,

ℑ∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = M,

1, otherwise,

𭟋(V) =


1, if V ∈ {1, 0},
1
3 , if V = U ,
0, otherwise,

𭟋∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = U ,
1, otherwise.

Thus, the identity F-mapping P : (G,ℑ,ℑ∗) −→ (G,𭟋,𭟋∗) is DF-β-continuous, but
it is not DF-b-continuous.

Theorem 3. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is DF-b-continuous iff for any
gθ ∈ Pθ(G) and any N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s containing P(gθ), there is
M ∈ IG that is (r, s)-F-b-open containing gθ with P(M) ≤ N .

Proof. (⇒) Let gθ ∈ Pθ(G) and N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s containing
P(gθ), and then P−1(N ) ≤ bIℑ∗(P−1(N ), r, s). Since gθ ∈ P−1(N ), then we obtain gθ ∈
bIℑ∗(P−1(N ), r, s) = M (say). Hence, M ∈ IG is (r, s)-F-b-open containing gθ with
P(M) ≤ N .

(⇐) Let gθ ∈ Pθ(G) and N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s containing P(gθ).
According to the assumption there is M ∈ IG that is (r, s)-F-b-open containing gθ with
P(M) ≤ N . Hence, gθ ∈ M ≤ P−1(N ) and gθ ∈ bIℑ∗(P−1(N ), r, s). Thus, P−1(N ) ≤
bIℑ∗(P−1(N ), r, s), so P−1(N ) is an (r, s)-F-b-open set. Then, P is DF-b-continuous.

Theorem 4. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an F-mapping, r ∈ I◦, and s ∈ I1.
Then the following statements are equivalent for every M ∈ IG and N ∈ IZ :

(i) P is DF-b-continuous.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 12 of 27

(ii) P−1(N ) is (r, s)-F-b-closed, for every N ∈ IZ with 𭟋(N c) ≥ r and 𭟋∗(N c) ≤ s.

(iii) P(bCℑ∗(M, r, s)) ≤ C𭟋∗(P(M), r, s).

(iv) bCℑ∗(P−1(N ), r, s) ≤ P−1(C𭟋∗(N , r, s)).

(v) P−1(I𭟋∗(N , r, s)) ≤ bIℑ∗(P−1(N ), r, s).

Proof. (i) ⇔ (ii) The proof follows by P−1(N c) = (P−1(N ))c and Definition 10.

(ii) ⇒ (iii) Let M ∈ IG. By (ii), we have P−1(C𭟋∗(P(M), r, s)) is (r, s)-F-b-closed.
Thus,

bCℑ∗(M, r, s) ≤ bCℑ∗(P−1(P(M)), r, s) ≤ bCℑ∗(P−1(C𭟋∗(P(M), r, s)), r, s) = P−1(C𭟋∗(P(M), r, s)).

Therefore, P(bCℑ∗(M, r, s)) ≤ C𭟋∗(P(M), r, s).

(iii) ⇒ (iv) Let N ∈ IZ . By (iii), P(bCℑ∗(P−1(N ), r, s)) ≤ C𭟋∗(P(P−1(N )), r, s) ≤
C𭟋∗(N , r, s). Thus, bCℑ∗(P−1(N ), r, s) ≤ P−1(P(bCℑ∗(P−1(N ), r, s))) ≤ P−1(C𭟋∗(N , r, s)).

(iv) ⇔ (v) The proof follows by P−1(N c) = (P−1(N ))c and Proposition 3.

(v) ⇒ (i) Let N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s. By (v), we obtain P−1(N ) =
P−1(I𭟋∗(N , r, s)) ≤ bIℑ∗(P−1(N ), r, s) ≤ P−1(N ). Then, bIℑ∗(P−1(N ), r, s) = P−1(N ).
Thus, P−1(N ) is (r, s)-F-b-open, so P is DF-b-continuous.

Definition 11. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-b-irresolute if
P−1(N ) is an (r, s)-F-b-open set, for every (r, s)-F-b-open set N ∈ IZ .

Lemma 2. Every DF-b-irresolute mapping is DF-b-continuous.

Proof. The proof follows by Definitions 10 and 11.

Remark 6. The converse of Lemma 2 fails as Example 7 will show.

Example 7. Let G = {g1, g2} and define M,N ∈ IG as follows: M = { g1
0.5 ,

g2
0.5}, N =

{ g1
0.5 ,

g2
0.4}. Define ℑ,ℑ∗,𭟋,𭟋∗ : IG −→ I as follows:

ℑ(V) =


1, if V ∈ {1, 0},
1
2 , if V = N ,

0, otherwise,

ℑ∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = N ,

1, otherwise,



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 13 of 27

𭟋(V) =


1, if V ∈ {1, 0},
1
3 , if V = M,

0, otherwise,

𭟋∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = M,

1, otherwise.

Thus, the identity F-mapping P : (G,ℑ,ℑ∗) −→ (G,𭟋,𭟋∗) is DF-b-continuous, but
it is not DF-b-irresolute.

Theorem 5. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an F-mapping, r ∈ I◦, and s ∈ I1.
Then the following statements are equivalent for every M ∈ IG and N ∈ IZ :

(i) P is DF-b-irresolute.

(ii) P−1(N ) is (r, s)-F-b-closed, for every N is (r, s)-F-b-closed.

(iii) P(bCℑ∗(M, r, s)) ≤ bC𭟋∗(P(M), r, s).

(iv) bCℑ∗(P−1(N ), r, s) ≤ P−1(bC𭟋∗(N , r, s)).

(v) P−1(bI𭟋∗(N , r, s)) ≤ bIℑ∗(P−1(N ), r, s).

Proof. (i) ⇔ (ii) The proof follows by P−1(N c) = (P−1(N ))c and Definition 11.

(ii) ⇒ (iii) Let M ∈ IG. By (ii), we have P−1(bC𭟋∗(P(M), r, s)) is (r, s)-F-b-closed.
Thus,

bCℑ∗(M, r, s) ≤ bCℑ∗(P−1(P(M)), r, s) ≤ bCℑ∗(P−1(bC𭟋∗(P(M), r, s)), r, s) = P−1(bC𭟋∗(P(M), r, s)).

Therefore, P(bCℑ∗(M, r, s)) ≤ bC𭟋∗(P(M), r, s).

(iii) ⇒ (iv) Let N ∈ IZ . By (iii), P(bCℑ∗(P−1(N ), r, s)) ≤ bC𭟋∗(P(P−1(N )), r, s) ≤
bC𭟋∗(N , r, s). Thus, bCℑ∗(P−1(N ), r, s) ≤ P−1(P(bCℑ∗(P−1(N ), r, s))) ≤ P−1(bC𭟋∗(N , r, s)).

(iv) ⇔ (v) The proof follows by P−1(N c) = (P−1(N ))c and Proposition 3.

(v) ⇒ (i) Let N ∈ IZ be an (r, s)-F-b-open set. By (v),

P−1(N ) = P−1(bI𭟋∗(N , r, s)) ≤ bIℑ∗(P−1(N ), r, s) ≤ P−1(N ).

Thus, bIℑ∗(P−1(N ), r, s) = P−1(N ). Therefore, P−1(N ) is (r, s)-F-b-open, so P is DF-b-
irresolute.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 14 of 27

Proposition 5. Let (G,ℑ,ℑ∗), (Q, η, η∗) and (Z,𭟋,𭟋∗) beDFT Ss, and P : (G,ℑ,ℑ∗) −→
(Q, η, η∗), Y : (Q, η, η∗) −→ (Z,𭟋,𭟋∗) be two F-mappings. Then the composition Y◦P is
DF-b-irresolute (resp. DF-b-continuous) if P is DF-b-irresolute and Y is DF-b-irresolute
(resp. DF-b-continuous).

Proof. The proof follows from Definitions 10 and 11.

Definition 12. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-almost b-
continuous if P−1(N ) ≤ bIℑ∗(P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)), r, s), for every N ∈ IZ with
𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s.

Lemma 3. Every DF-b-continuous mapping is DF-almost b-continuous.

Proof. The proof follows by Definitions 10 and 12.

Remark 7. The converse of Lemma 3 fails as Example 8 will show.

Example 8. Let G = {g1, g2, g3} and define M,N ,U ∈ IG as follows: M = { g1
0.4 ,

g2
0.2 ,

g3
0.4},

N = { g1
0.5 ,

g2
0.5 ,

g3
0.4}, U = { g1

0.3 ,
g2
0.2 ,

g3
0.6}. Define ℑ,ℑ∗,𭟋,𭟋∗ : IG −→ I as follows:

ℑ(V) =


1, if V ∈ {0, 1},
2
3 , if V = M,
1
2 , if V = N ,

0, otherwise,

ℑ∗(V) =


0, if V ∈ {0, 1},
1
3 , if V = M,
1
3 , if V = N ,

1, otherwise,

𭟋(V) =


1, if V ∈ {0, 1},
1
2 , if V = U ,
0, otherwise.

𭟋∗(V) =


0, if V ∈ {0, 1},
1
3 , if V = U ,
1, otherwise.

Thus, the identity F-mapping P : (G,ℑ,ℑ∗) −→ (G,𭟋,𭟋∗) is DF-almost b-continuous,
but it is not DF-b-continuous.

Theorem 6. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is DF-almost b-continuous iff
for any gθ ∈ Pθ(G) and any N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s containing P(gθ),
there is M ∈ IG that is (r, s)-F-b-open containing gθ with P(M) ≤ I𭟋∗(C𭟋∗(N , r, s), r, s).



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 15 of 27

Proof. (⇒) Let gθ ∈ Pθ(G) and N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s containing
P(gθ), and then P−1(N ) ≤ bIℑ∗(P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)), r, s). Since gθ ∈ P−1(N ), then

gθ ∈ bIℑ∗(P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)), r, s) = M (say).

Therefore, M ∈ IG is (r, s)-F-b-open containing gθ with P(M) ≤ I𭟋∗(C𭟋∗(N , r, s), r, s).

(⇐) Let gθ ∈ Pθ(G) and N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s such that gθ ∈
P−1(N ). According to the assumption there is M ∈ IG that is (r, s)-F-b-open containing
gθ with P(M) ≤ I𭟋∗(C𭟋∗(N , r, s), r, s). Hence, gθ ∈ M ≤ P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)) and

gθ ∈ bIℑ(P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)), r, s).

Thus, P−1(N ) ≤ bIℑ∗(P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)), r, s). Therefore, P is DF-almost b-
continuous.

Theorem 7. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an F-mapping. Then the following
statements are equivalent:

(i) P is DF-almost b-continuous.

(ii) P−1(N ) is (r, s)-F-b-open, for every (r, s)-F-regularly open set N ∈ IZ .

(iii) P−1(N ) is (r, s)-F-b-closed, for every (r, s)-F-regularly closed set N ∈ IZ .

(iv) bCℑ∗(P−1(N ), r, s) ≤ P−1(C𭟋∗(N , r, s)), for every (r, s)-F-b-open set N ∈ IZ .

(v) bCℑ∗(P−1(N ), r, s) ≤ P−1(C𭟋∗(N , r, s)), for every (r, s)-F-semi-open set N ∈ IZ .

Proof. (i) ⇒ (ii) Let gθ ∈ Pθ(G) and N ∈ IZ be an (r, s)-F-regularly open set with
gθ ∈ P−1(N ). Hence, by (i), there is M ∈ IG that is (r, s)-F-b-open with gθ ∈ M and
P(M) ≤ I𭟋∗(C𭟋∗(N , r, s), r, s). Thus, M ≤ P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)) = P−1(N ) and
gθ ∈ bIℑ∗(P−1(N ), r, s). Therefore, P−1(N ) ≤ bIℑ∗(P−1(N ), r, s), so P−1(N ) is (r, s)-F-b-
open.

(ii) ⇒ (iii) If N ∈ IZ is (r, s)-F-regularly closed, then by (ii), P−1(N c) = (P−1(N ))c

is (r, s)-F-b-open. Thus, P−1(N ) is (r, s)-F-b-closed.

(iii) ⇒ (iv) If N ∈ IZ is (r, s)-F-b-open and since C𭟋∗(N , r, s) is (r, s)-F-regularly
closed, then by (iii), P−1(C𭟋∗(N , r, s)) is (r, s)-F-b-closed. Since P−1(N ) ≤ P−1(C𭟋∗(N , r, s)),
hence

bCℑ∗(P−1(N ), r, s) ≤ P−1(C𭟋∗(N , r, s)).



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 16 of 27

(iv) ⇒ (v) The proof follows from the fact that any (r, s)-F-semi-open set is (r, s)-F-
b-open.

(v) ⇒ (iii) If N ∈ IZ is (r, s)-F-regularly closed, then N is (r, s)-F-semi-open. By (v),

bCℑ∗(P−1(N ), r, s) ≤ P−1(C𭟋∗(N , r, s)) = P−1(N ).

Hence, P−1(N ) is (r, s)-F-b-closed.

(iii) ⇒ (i) If gθ ∈ Pθ(G) and N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s such that gθ ∈
P−1(N ), and then gθ ∈ P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)). Since [I𭟋∗(C𭟋∗(N , r, s), r, s)]c is (r, s)-
F-regularly closed, then by (iii), we have P−1([I𭟋∗(C𭟋∗(N , r, s), r, s)]c) is (r, s)-F-b-closed.
Hence, P−1(I𭟋∗(C𭟋∗(N ), r, s)) is (r, s)-F-b-open and gθ ∈ bIℑ∗(P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)), r, s).
Thus,

P−1(N ) ≤ bIℑ∗(P−1(I𭟋∗(C𭟋∗(N , r, s), r, s)), r, s).

Therefore, P is DF-almost b-continuous.

Definition 13. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-weakly b-
continuous if P−1(N ) ≤ bIℑ∗(P−1(C𭟋∗(N , r, s)), r, s), for every N ∈ IZ with 𭟋(N ) ≥ r
and 𭟋∗(N ) ≤ s.

Lemma 4. Every DF-b-continuous mapping is DF-weakly b-continuous.

Proof. The proof follows by Definitions 10 and 13.

Remark 8. The converse of Lemma 4 fails as Example 9 will show.

Example 9. Let G = {g1, g2, g3} and define M,N ,U ∈ IG as follows: M = { g1
0.4 ,

g2
0.2 ,

g3
0.4},

N = { g1
0.5 ,

g2
0.5 ,

g3
0.4}, U = { g1

0.3 ,
g2
0.2 ,

g3
0.6}. Define ℑ,ℑ∗,𭟋,𭟋∗ : IG −→ I as follows:

ℑ(V) =


1, if V ∈ {1, 0},
1
3 , if V = M,
1
2 , if V = N ,

0, otherwise,

ℑ∗(V) =


0, if V ∈ {1, 0},
1
3 , if V = M,
1
2 , if V = N ,

1, otherwise,

𭟋(V) =


1, if V ∈ {1, 0},
1
3 , if V = U ,
0, otherwise,

𭟋∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = U ,
1, otherwise.

Thus, the identity F-mapping P : (G,ℑ,ℑ∗) −→ (G,𭟋,𭟋∗) isDF-weakly b-continuous,
but it is not DF-b-continuous.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 17 of 27

Theorem 8. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is DF-weakly b-continuous iff
for any gθ ∈ Pθ(G) and any N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s containing P(gθ),
there is M ∈ IG that is (r, s)-F-b-open containing gθ with P(M) ≤ C𭟋∗(N , r, s).

Proof. (⇒) Let gθ ∈ Pθ(G) and N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s containing
P(gθ), and then P−1(N ) ≤ bIℑ∗(P−1(C𭟋∗(N , r, s)), r, s). Since gθ ∈ P−1(N ), then gθ ∈
bIℑ∗(P−1(C𭟋∗(N , r, s)), r, s) = M (say). Hence, M ∈ IG is (r, s)-F-b-open containing gθ
with P(M) ≤ C𭟋∗(N , r, s).

(⇐) Let gθ ∈ Pθ(G) and N ∈ IZ with 𭟋(N ) ≥ r and 𭟋∗(N ) ≤ s such that
gθ ∈ P−1(N ). According to the assumption there is M ∈ IG that is (r, s)-F-b-open
containing gθ with P(M) ≤ C𭟋∗(N , r, s). Hence, gθ ∈ M ≤ P−1(C𭟋∗(N , r, s)) and
gθ ∈ bIℑ∗(P−1(C𭟋∗(N , r, s)), r, s). Thus, P−1(N ) ≤ bIℑ∗(P−1(C𭟋∗(N , r, s)), r, s). There-
fore, P is DF-weakly b-continuous.

Theorem 9. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an F-mapping. Then the following
statements are equivalent:

(i) P is DF-weakly b-continuous.

(ii) P−1(N ) ≥ bCℑ∗(P−1(I𭟋∗(N , r, s)), r, s), if N ∈ IZ with 𭟋(N c) ≥ r and 𭟋∗(N c) ≤
s.

(iii) bIℑ∗(P−1(C𭟋∗(N , r, s)), r, s) ≥ P−1(I𭟋∗(N , r, s)).

(iv) bCℑ∗(P−1(I𭟋∗(N , r, s)), r, s) ≤ P−1(C𭟋∗(N , r, s)).

Proof. (i) ⇔ (ii) The proof follows by Proposition 3 and Definition 13.

(ii) ⇒ (iii) Let N ∈ IZ . Hence by (ii),

bCℑ∗(P−1(I𭟋∗(C𭟋∗(N c, r, s), r, s)), r, s) ≤ P−1(C𭟋∗(N c, r, s)).

Thus, P−1(I𭟋∗(N , r, s)) ≤ bIℑ∗(P−1(C𭟋∗(N , r, s)), r, s).

(iii) ⇔ (iv) The proof follows from Proposition 3.

(iv)⇒ (i) LetN ∈ IZ with𭟋(N ) ≥ r and𭟋∗(N ) ≤ s. Hence by (iv), bCℑ∗(P−1(I𭟋∗(N c, r, s)), r, s) ≤
P−1(C𭟋∗(N c, r, s)) = P−1(N c). Thus, P−1(N ) ≤ bIℑ∗(P−1(C𭟋∗(N , r, s)), r, s), so P is DF-
weakly b-continuous.

Lemma 5. Every DF-almost b-continuous mapping is DF-weakly b-continuous.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 18 of 27

Proof. The proof follows by Definitions 12 and 13.

Remark 9. The converse of Lemma 5 fails as Example 10 will show.

Example 10. LetG = {g1, g2, g3} and defineM,N ,U ∈ IG as follows: M = { g1
0.6 ,

g2
0.2 ,

g3
0.4},

N = { g1
0.3 ,

g2
0.2 ,

g3
0.5}, U = { g1

0.3 ,
g2
0.2 ,

g3
0.4}. Define ℑ,ℑ∗,𭟋,𭟋∗ : IG −→ I as follows:

ℑ(V) =


1, if V ∈ {1, 0},
1
4 , if V = M,
1
2 , if V = U ,
0, otherwise,

ℑ∗(V) =


0, if V ∈ {1, 0},
1
4 , if V = M,
1
2 , if V = U ,
1, otherwise,

𭟋(V) =


1, if V ∈ {1, 0},
1
4 , if V = N ,

0, otherwise,

𭟋∗(V) =


0, if V ∈ {1, 0},
1
2 , if V = N ,

1, otherwise.

Thus, the identity F-mapping P : (G,ℑ,ℑ∗) −→ (G,𭟋,𭟋∗) isDF-weakly b-continuous,
but it is not DF-almost b-continuous.

Remark 10. From the previous discussions and definitions, we have the following dia-
gram.

DF-b-continuity −→ DF-almost b-continuity −→ DF-weakly b-continuity

Proposition 6. Let (G,ℑ,ℑ∗), (Q, η, η∗) and (Z,𭟋,𭟋∗) beDFT Ss, and P : (G,ℑ,ℑ∗) −→
(Q, η, η∗), Y : (Q, η, η∗) −→ (Z,𭟋,𭟋∗) be two F-mappings. Then the composition Y ◦ P
is DF-almost b-continuous if P is DF-b-irresolute (resp. DF-b-continuous) and Y is DF-
almost b-continuous (resp. DF-continuous).

Proof. The proof follows by the previous definitions.

5. Some applications

Here, we present and study some new DF-mappings between DFT Ss (G,ℑ,ℑ∗) and
(Z,𭟋,𭟋∗) based on Šostak,s sense [3]. Next, we introduce and discuss new types of DF-
separation axioms via (r, s)-F-b-closed sets, called (r, s)-F-b-regular and (r, s)-F-b-normal
spaces.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 19 of 27

Definition 14. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-b-open if P(M)
is an (r, s)-F-b-open set, for each M ∈ IG with ℑ(M) ≥ r and ℑ∗(M) ≤ s.

Definition 15. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-b-irresolute
open if P(M) is an (r, s)-F-b-open set, for each (r, s)-F-b-open set M ∈ IG.

Lemma 6. Each DF-b-irresolute open mapping is DF-b-open.

Proof. The proof follows from Definitions 14 and 15.

Remark 11. The converse of Lemma 6 fails as Example 11 will show.

Example 11. Let G = {g1, g2} and define M,N ∈ IG as follows: M = { g1
0.5 ,

g2
0.5},

N = { g1
0.5 ,

g2
0.4}. Define ℑ,ℑ∗,𭟋,𭟋∗ : IG −→ I as follows:

ℑ(V) =


1, if V ∈ {1, 0},
1
5 , if V = M,

0, otherwise,

ℑ∗(V) =


0, if V ∈ {1, 0},
1
5 , if V = M,

1, otherwise,

𭟋(V) =


1, if V ∈ {1, 0},
1
5 , if V = N ,

0, otherwise,

𭟋∗(V) =


0, if V ∈ {1, 0},
1
5 , if V = N ,

1, otherwise.

Thus, the identity F-mapping P : (G,ℑ,ℑ∗) −→ (G,𭟋,𭟋∗) is DF-b-open, but it is not
DF-b-irresolute open.

Theorem 10. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an F-mapping. Then the following
statements are equivalent for every M ∈ IG and N ∈ IZ :

(i) P is DF-b-open.

(ii) P(Iℑ∗(M, r, s)) ≤ bI𭟋∗(P(M), r, s).

(iii) Iℑ∗(P−1(N ), r, s) ≤ P−1(bI𭟋∗(N , r, s)).

(iv) For every N and every M with ℑ(Mc) ≥ r, ℑ∗(Mc) ≤ s and P−1(N ) ≤ M, there
is U ∈ IZ is (r, s)-F-b-closed with N ≤ U and P−1(U) ≤ M.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 20 of 27

Proof. (i) ⇒ (ii) Since P(Iℑ∗(M, r, s)) ≤ P(M), hence by (i), P(Iℑ∗(M, r, s)) is (r, s)-
F-b-open. Thus,

P(Iℑ∗(M, r, s)) ≤ bI𭟋∗(P(M), r, s).

(ii)⇒ (iii) SetM = P−1(N ), hence by (ii), P(Iℑ∗(P−1(N ), r, s)) ≤ bI𭟋∗(P(P−1(N )), r, s) ≤
bI𭟋∗(N , r, s). Thus, Iℑ∗(P−1(N ), r, s) ≤ P−1(bI𭟋∗(N , r, s)).

(iii) ⇒ (iv) Let N ∈ IZ and M ∈ IG with ℑ(Mc) ≥ r and ℑ∗(Mc) ≤ s such that
P−1(N ) ≤ M. Since Mc ≤ P−1(N c), Mc = Iℑ∗(Mc, r, s) ≤ Iℑ∗(P−1(N c), r, s). Hence by
(iii), Mc ≤ Iℑ∗(P−1(N c), r, s) ≤ P−1(bI𭟋∗(N c, r, s)). Then, we have

M ≥ (P−1(bI𭟋∗(N c, r, s)))c = P−1(bC𭟋∗(N , r, s)).

Thus, bC𭟋∗(N , r, s) ∈ IZ is (r, s)-F-b-closed withN ≤ bC𭟋∗(N , r, s) and P−1(bC𭟋∗(N , r, s)) ≤
M.

(iv) ⇒ (i) Let V ∈ IG with ℑ(V) ≥ r and ℑ∗(V) ≤ s. Set N = (P(V))c and M = Vc,
then P−1(N ) = P−1((P(V))c) ≤ M. Hence by (iv), there is U ∈ IZ is (r, s)-F-b-closed
with N ≤ U and P−1(U) ≤ M = Vc. Thus, P(V) ≤ P(P−1(Uc)) ≤ Uc. On the other hand,
since N ≤ U , P(V) = N c ≥ Uc. Hence, P(V) = Uc, so P(V) is an (r, s)-F-b-open set.
Therefore, P is DF-b-open.

Theorem 11. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an F-mapping. Then the following
statements are equivalent for every M ∈ IG and N ∈ IZ :

(i) P is DF-b-irresolute open.

(ii) P(bIℑ∗(M, r, s)) ≤ bI𭟋∗(P(M), r, s).

(iii) bIℑ∗(P−1(N ), r, s) ≤ P−1(bI𭟋∗(N , r, s)).

(iv) For every N and every M is an (r, s)-F-b-closed set with P−1(N ) ≤ M, there is
U ∈ IZ is (r, s)-F-b-closed with N ≤ U and P−1(U) ≤ M.

Proof. The proof is similar to that of Theorem 10.

Definition 16. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-b-closed if
P(M) is an (r, s)-F-b-closed set, for each M ∈ IG with ℑ(Mc) ≥ r and ℑ∗(Mc) ≤ s.

Definition 17. An F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-b-irresolute
closed if P(M) is an (r, s)-F-b-closed set, for each (r, s)-F-b-closed set M ∈ IG.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 21 of 27

Lemma 7. Each DF-b-irresolute closed mapping is DF-b-closed.

Proof. The proof follows from Definitions 16 and 17.

Theorem 12. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an F-mapping. Then the following
statements are equivalent for every M ∈ IG and N ∈ IZ :

(i) P is DF-b-closed.

(ii) bC𭟋∗(P(M), r, s) ≤ P(Cℑ∗(M, r, s)).

(iii) P−1(bC𭟋∗(N , r, s)) ≤ Cℑ∗(P−1(N ), r, s).

(iv) For every N and every M with ℑ(M) ≥ r, ℑ∗(M) ≤ s and P−1(N ) ≤ M, there
is U ∈ IZ is (r, s)-F-b-open with N ≤ U and P−1(U) ≤ M.

Proof. The proof is similar to that of Theorem 10.

Theorem 13. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an F-mapping. Then the following
statements are equivalent for every M ∈ IG and N ∈ IZ :

(i) P is DF-b-irresolute closed.

(ii) bC𭟋∗(P(M), r, s) ≤ P(bCℑ∗(M, r, s)).

(iii) P−1(bC𭟋∗(N , r, s)) ≤ bCℑ∗(P−1(N ), r, s).

(iv) For every N and every M is an (r, s)-F-b-open set with P−1(N ) ≤ M, there is
U ∈ IZ is (r, s)-F-b-open with N ≤ U and P−1(U) ≤ M.

Proof. The proof is similar to that of Theorem 10.

Proposition 7. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be a bijective F-mapping, then P is
DF-b-irresolute open iff P is DF-b-irresolute closed.

Proof. The proof follows from:

P−1(bC𭟋∗(N , r, s)) ≤ bCℑ∗(P−1(N ), r, s) ⇐⇒ P−1(bI𭟋∗(N c, r, s)) ≤ bIℑ∗(P−1(N c), r, s).

Definition 18. A bijective F-mapping P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) is called DF-b-
irresolute homeomorphism if P and P−1 are DF-b-irresolute.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 22 of 27

The proof of the following corollary is easy and so is omitted.

Corollary 4. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be a bijective F-mapping. Then the
following statements are equivalent for every M ∈ IG and N ∈ IZ :

(i) P is DF-b-irresolute homeomorphism.

(ii) P is DF-b-irresolute closed and DF-b-irresolute.

(iii) P is DF-b-irresolute open and DF-b-irresolute.

(iv) P(bIℑ∗(M, r, s)) = bI𭟋∗(P(M), r, s).

(v) P(bCℑ∗(M, r, s)) = bC𭟋∗(P(M), r, s).

(vi) bIℑ∗(P−1(N ), r, s) = P−1(b𭟋∗(N , r, s)).

(vii) bCℑ∗(P−1(N ), r, s) = P−1(bC𭟋∗(N , r, s)).

Definition 19. Let gθ ∈ Pθ(G), M ∈ IG, r ∈ I◦, and s ∈ I1. An DFT S (G,ℑ,ℑ∗)
is called an (r, s)-F-b-regular space if gθ q M for each (r, s)-F-b-closed set M, there is
Ui ∈ IG with ℑ(Ui) ≥ r and ℑ∗(Ui) ≤ s for i = 1, 2, such that gθ ∈ U1, M ≤ U2, and
U1 q U2.

Definition 20. Let M,N ∈ IG, r ∈ I◦, and s ∈ I1. An DFT S (G,ℑ,ℑ∗) is called an
(r, s)-F-b-normal space if M q N for each (r, s)-F-b-closed sets M and N , there is Ui ∈ IG

with ℑ(Ui) ≥ r and ℑ∗(Ui) ≤ s for i = 1, 2, such that M ≤ U1, N ≤ U2, and U1 q U2.

Theorem 14. Let (G,ℑ,ℑ∗) be an DFT S, gθ ∈ Pθ(G), and M ∈ IG. Then the following
statements are equivalent:

(i) (G,ℑ,ℑ∗) is an (r, s)-F-b-regular space.

(ii) If gθ ∈ M for every (r, s)-F-b-open set M, there is N ∈ IG with ℑ(N ) ≥ r,
ℑ∗(N ) ≤ s, and gθ ∈ N ≤ Cℑ∗(N , r, s) ≤ M.

(iii) If gθ q M for each (r, s)-F-b-closed set M, there is Oi ∈ IG with ℑ(Oi) ≥ r and
ℑ∗(Oi) ≤ s for i = 1, 2, such that gθ ∈ O1, M ≤ O2, and Cℑ∗(O1, r, s) q Cℑ∗(O2, r, s).



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 23 of 27

Proof. (i) ⇒ (ii) Let gθ ∈ M for every (r, s)-F-b-open set M, then gθ q Mc. Since
(G,ℑ,ℑ∗) is (r, s)-F-b-regular, then there is N ,O ∈ IG with ℑ(N ) ≥ r, ℑ∗(N ) ≤ s,
ℑ(O) ≥ r, and ℑ∗(O) ≤ s, such that gθ ∈ N , Mc ≤ O, and N q O. Thus, gθ ∈ N ≤
Oc ≤ M, so gθ ∈ N ≤ Cℑ∗(N , r, s) ≤ M.

(ii) ⇒ (iii) Let gθ q M for each (r, s)-F-b-closed set M, then gθ ∈ Mc. By (ii), there is
O ∈ IG with ℑ(O) ≥ r, ℑ∗(O) ≤ s and gθ ∈ O ≤ Cℑ∗(O, r, s) ≤ Mc. Since ℑ(O) ≥ r and
ℑ∗(O) ≤ s, thenO is an (r, s)-F-b-open set and gθ ∈ O. Again, by (ii), there is V ∈ IG with
ℑ(V) ≥ r, ℑ∗(V) ≤ s, and gθ ∈ V ≤ Cℑ∗(V, r, s) ≤ O ≤ Cℑ∗(O, r, s) ≤ Mc. Hence, M ≤
(Cℑ∗(O, r, s))c = Iℑ∗(Oc, r, s) ≤ Oc. Set U = Iℑ∗(Oc, r, s), thus ℑ(U) ≥ r and ℑ∗(U) ≤ s.
Then, Cℑ∗(U , r, s) ≤ Oc ≤ (Cℑ∗(V, r, s))c. Therefore, Cℑ∗(U , r, s) q Cℑ∗(V, r, s).

(iii) ⇒ (i) This is easily proved by Definition 19.

Theorem 15. Let (G,ℑ,ℑ∗) be an DFT S, M,N ∈ IG. Then the following statements
are equivalent:

(i) (G,ℑ,ℑ∗) is an (r, s)-F-b-normal space.

(ii) If N ≤ M for every (r, s)-F-b-closed set N and (r, s)-F-b-open set M, there is
O ∈ IG with ℑ(O) ≥ r, ℑ∗(O) ≤ s, and N ≤ O ≤ Cℑ∗(O, r, s) ≤ M.

(iii) If M q N for each (r, s)-F-b-closed sets M and N , there is Oi ∈ IG with ℑ(Oi) ≥
r, ℑ∗(Oi) ≤ s for i = 1, 2, such that M ≤ O1, N ≤ O2, and Cℑ∗(O1, r, s) q Cℑ∗(O2, r, s).

Proof. The proof is similar to that of Theorem 14.

Theorem 16. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be a bijective DF-b-irresolute and DF-
open mapping. If (G,ℑ,ℑ∗) is an (r, s)-F-b-regular space (resp. (r, s)-F-b-normal space),
then (Z,𭟋,𭟋∗) is an (r, s)-F-b-regular space (resp. (r, s)-F-b-normal space).

Proof. If zθ q N for every (r, s)-F-b-closed set N ∈ IZ and P is DF-b-irresolute, then
P−1(N ) is an (r, s)-F-b-closed set. Set zθ = P(gθ), and then gθ q P−1(N ). Since (G,ℑ,ℑ∗)
is (r, s)-F-b-regular, there is O1, O2 ∈ IG with ℑ(O1) ≥ r, ℑ∗(O1) ≤ s, ℑ(O2) ≥ r, and
ℑ∗(O2) ≤ s such that gθ ∈ O1, P−1(N ) ≤ O2, and O1 q O2. Since P is a bijective DF-open
mapping, hence zθ ∈ P(O1), N = P(P−1(N )) ≤ P(O2), and P(O1) q P(O2). Therefore,
(Z,𭟋,𭟋∗) is an (r, s)-F-b-regular space. The other case also follows similar lines.

Theorem 17. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be an injective DF-continuous and DF-
b-irresolute closed mapping. If (Z,𭟋,𭟋∗) is an (r, s)-F-b-regular space (resp. (r, s)-F-
b-normal space), then (G,ℑ,ℑ∗) is an (r, s)-F-b-regular space (resp. (r, s)-F-b-normal
space).



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 24 of 27

Proof. If gθ q M for each (r, s)-F-b-closed set M ∈ IG and P is injective DF-
b-irresolute closed, hence P(M) is an (r, s)-F-b-closed set and P(gθ) q P(M). Since
(Z,𭟋,𭟋∗) is (r, s)-F-b-regular, there is O1,O2 ∈ IZ with 𭟋(O1) ≥ r, 𭟋∗(O1) ≤ s,
𭟋(O2) ≥ r, and 𭟋∗(O2) ≤ s such that P(gθ) ∈ O1, P(M) ≤ O2, and O1 q O2. Since P is
an DF-continuous mapping, then gθ ∈ P−1(O1) and M ≤ P−1(O2) with ℑ(P−1(O1)) ≥ r,
ℑ∗(P−1(O1)) ≤ s, ℑ(P−1(O2)) ≥ r, ℑ∗(P−1(O2)) ≤ s, and P−1(O1) q P−1(O2). Hence,
(G,ℑ,ℑ∗) is an (r, s)-F-b-regular space. The other case also follows similar lines.

Theorem 18. Let P : (G,ℑ,ℑ∗) −→ (Z,𭟋,𭟋∗) be a surjective DF-b-irresolute, DF-
open, and DF-closed mapping. If (G,ℑ,ℑ∗) is an (r, s)-F-b-regular space (resp. (r, s)-
F-b-normal space), then (Z,𭟋,𭟋∗) is an (r, s)-F-b-regular space (resp. (r, s)-F-b-normal
space).

Proof. The proof is similar to that of Theorem 16.

6. Conclusions

In the present paper, a novel class of generalized F-open sets, called (r, s)-F-b-open
sets, has been introduced in DFT S based on Šostak,s sense [3]. Furthermore, some
characterizations of (r, s)-F-b-open sets along with their mutual relationships have been
discussed. In addition, the notions of DF-b-closure operators and DF-b-interior operators
have been presented and investigated. Thereafter, the notion of DF-b-continuity between
DFT Ss (G,ℑ,ℑ∗) and (Z,𭟋,𭟋∗) has been defined and discussed. Moreover, the concepts
of DF-almost b-continuity and DF-weakly b-continuity, which are weaker forms of DF-
b-continuity, have been explored and characterized. After that, some new DF-mappings
using (r, s)-F-b-closed sets and (r, s)-F-b-open sets have been defined and studied. Lastly,
we introduced new types of DF-separation axioms using (r, s)-F-b-closed sets, and some
properties have been specified.

In upcoming works might look into the following topics: (i) defining upper (lower)
b-continuous DF-multifunctions and (r, s)-F-b-connected sets; (ii) introducing these novel
notions given here in the frame of fuzzy ideals as defined in [41–43]; and (iii) extending
these novel notions given here in the frame of fuzzy soft topological (r-minimal) spaces as
defined in [44–46].

Acknowledgements

We would like to thank the reviewers and editors whose constructive comments and
suggestions helped to improve this paper.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 25 of 27

References

[1] L. A. Zadeh. Fuzzy sets. Inform. Control, 8:338–353, 1965.
[2] C. L. Chang. Fuzzy topological spaces. J. Math. Anal. Appl., 24:182–190, 1968.
[3] A. P. Šostak. On a fuzzy topological structure. In In: Proceedings of the 13th winter

school on abstract analysis, Section of topology, Palermo: Circolo Matematico di
Palermo, pages 89–103, 1985.

[4] A. A. Ramadan. Smooth topological spaces. Fuzzy Set. Syst., 48:371–375, 1992.
[5] K. C. Chattopadhyay and S. K. Samanta. Fuzzy topology: fuzzy closure operator,

fuzzy compactness and fuzzy connectedness. Fuzzy Set. Syst., 54(2):207–212, 1993.
[6] M. K. El-Gayyar, E. E. Kerre, and A. A. Ramadan. Almost compactness and near

compactness in smooth topological spaces. Mathematics, 62(2):193–202, 1994.
[7] U. Höhle and A. P. Šostak. A general theory of fuzzy topological spaces. Fuzzy Set.

Syst., 73:131–149, 1995.
[8] A. A. Ramadan, S. E. Abbas, and Y. C. Kim. Fuzzy irresolute mappings in smooth

fuzzy topological spaces. J. Fuzzy Math., 9(4):865–877, 2001.
[9] Y. C. Kim, A. A. Ramadan, and S. E. Abbas. Weaker forms of continuity in Šostak’s

fuzzy topology. Indian J. Pure Appl. Math., 34(2):311–333, 2003.
[10] S. E. Abbas. Fuzzy super irresolute functions. Inter. J. Math. Mathematical Sci.,

42:2689–2700, 2003.
[11] S. E. Abbas. Fuzzy β-irresolute functions. Appl. Math. Comp., 157:369–380, 2004.
[12] Y. C. Kim and S. E. Abbas. On several types of r-fuzzy compactness. J. Fuzzy Math.,

12(4):827–844, 2004.
[13] H. Aygün and S. E. Abbas. On characterization of some covering properties in l-fuzzy

topological spaces in Šostak sense. Inform. Sciences, 165:221–233, 2004.
[14] H. Aygün and S. E. Abbas. Some good extensions of compactness in Šostak’s l-fuzzy

topology. Hacett. J. Math. Stat., 36(2):115–125, 2007.
[15] H. Y. Li and F. G Shi. Some separation axioms in i-fuzzy topological spaces. Fuzzy

Set. Syst., 159:573–587, 2008.
[16] H. Y. Li and F. G. Shi. Measures of fuzzy compactness in l-fuzzy topological spaces.

Comput. Math. Appl., 59:941–947, 2010.
[17] F. G. Shi and R. X. Li. Compactness in l-fuzzy topological spaces. Hacet. J. Math.

Stat., 40(6):767–774, 2011.
[18] J. Fang and Y. Guo. Quasi-coincident neighborhood structure of relative i-fuzzy

topology and its applications. Fuzzy Set. Syst., 190:105–117, 2012.
[19] M. El-Dardery, A. A. Ramadan, and Y. C. Kim. L-fuzzy topogenous orders and

l-fuzzy topologies. J. Intell. Fuzzy Syst., 24(4):685–691, 2013.
[20] C. Kalaivani and R. Roopkumar. Fuzzy perfect mappings and q-compactness in

smooth fuzzy topological spaces. Fuzzy Inform. Eng., 6(1):115–131, 2014.
[21] S. A. Solovyov. On fuzzification of topological categories. Fuzzy Set. Syst., 238:1–25,

2014.
[22] J. J. Minana and A. P. Šostak. Fuzzifying topology induced by a strong fuzzy metric.

Fuzzy Set. Syst., 300:24–39, 2016.



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 26 of 27

[23] K. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets Syst., 20:87–96, 1986.
[24] K. Atanassov. New operators defined over the intuitionistic fuzzy sets. Fuzzy Sets

Syst., 61:131–142, 1993.
[25] D. Coker. An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces.

J. Fuzzy Math., 4:749–764, 1996.
[26] D. Coker. An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets Syst.,

88:81–89, 1997.
[27] M. Demirci and D. Coker. An introduction to intuitionistic fuzzy topological spaces

in Šostak’s sense. Busefal, 67:67–76, 1996.
[28] S. K. Samanta and T. K. Mondal. Intuitionistic gradation of openness: intuitionistic

fuzzy topology. Busefal, 73:8–17, 1997.
[29] J. G. Garcia and S. E. Rodabaugh. Ordertheoretic, topological, categorical redun-

dancies of interval-valued sets, grey sets, vague sets, intervalvalued; intuitionistic sets,
intuitionistic fuzzy sets and topologies. Fuzzy Sets Syst., 156(3):445–484, 2005.

[30] E. P. Lee. Semiopen sets on intuitionistic fuzzy topological spaces in Šostak’s sense.
Int. J. Fuzzy Logic Intel. Sys., 14:234–238, 2004.

[31] E. P. Lee and J. I. Kim. Fuzzy strongly (r, s)-preopen and preclosed mappings.
Commun. Korean Math. Soc., 26(4):661–667, 2011.

[32] M. S. K. Samanta and T. K. Mondal. On intuitionistic gradation of openness. Fuzzy
Sets Syst., 131:323–336, 2002.

[33] S. E. Abbas. (r, s)-generalized intuitionistic fuzzy closed sets. J. Egyptian Math. Soc.,
14:331–351, 2006.

[34] S. E. Abbas and B. Krsteska. Some properties of intuitionistic (r, s)-t0 and (r, s)-t1
spaces. Int. J. Math. Math. Sci., 2008:1–11, 2008.

[35] A. M. Zahran, M. A. Abd-Allah, and A. Ghareeb. Several types of double fuzzy
irresolute functions. Int. J. Comput. Cognition, 8(2):19–23, 2010.

[36] F. M. Mohammed, M. S. M. Noorani, and A. Ghareeb. Several notions of general-
ized semi-compactness in double fuzzy topological spaces. Int. J. Pure Appl. Math.,
109(2):153–175, 2016.

[37] E. El-Sanousy and A. Atef. (r, s)-fuzzy g∗p-closed sets and its applications. Appl.
Math. Inf. Sci., 16(1):17–24, 2022.

[38] I. M. Taha. Some properties of (r, s)-generalized fuzzy semi-closed sets and some
applications. J. Math. Comput. Sci., 27(2):164–175, 2022.

[39] F. Alsharari, O. M. Taha, and I. M. Taha. Some new types of fuzzy closed sets,
separation axioms, and compactness via double fuzzy topologies. Eur. J. Pure Appl.
Math., 17(4):4093–4111, 2024.

[40] A. Kandil and M. E. El-Shafei. Regularity axioms in fuzzy topological spaces and
fri-proximities. Fuzzy Set. Syst., 27:217–231, 1988.

[41] I. M. Taha. On r-fuzzy ℓ-open sets and continuity of fuzzy multifunctions via fuzzy
ideals. J. Math. Comput. Sci., 10(6):2613–2633, 2020.

[42] I. M. Taha. On r-generalized fuzzy ℓ-closed sets: properties and applications. J.
Math., page 4483481, 2021.

[43] I. M. Taha. r-fuzzy δ-ℓ-open sets and fuzzy upper (lower) δ-ℓ-continuity via fuzzy



I. M. Taha, J. Al-Mufarrij, O. M. Taha / Eur. J. Pure Appl. Math, 18 (2) (2025), 5911 27 of 27

idealization. J. Math. Comput. Sci., 25(1):1–9, 2022.
[44] I. M. Taha. Some new separation axioms in fuzzy soft topological spaces. Filomat,

35:1775–1783, 2021.
[45] I. M. Taha. Compactness on fuzzy soft r-minimal spaces. Int. J. Fuzzy Logic Intell.

Syst., 21:251–258, 2021.
[46] I. M. Taha. Some new results on fuzzy soft r-minimal spaces. AIMS Mathematics,

7:12458–12470, 2022.