EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5733
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On r-Fuzzy Soft δ-Open Sets With Applications in
Fuzzy Soft Topological Spaces

Ibtesam Alshammari1, Osama Taha2, Mostafa K. El-Bably3,4, IslamM. Taha2,5,∗

1 Department of Mathematics, Faculty of Science, University of Hafr Al Batin, Saudi Arabia
2 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
3 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
4 Jadara University Research Center, Jadara University, Irbid, Jordan
5 Department of Basic Sciences, High Institute for Engineering and Technology, Sohag,
Egypt

Abstract. In this paper, we introduce the notion of r-fuzzy soft δ-open sets on fuzzy soft topo-
logical spaces in the sense of Šostak. Furthermore, we define and characterize the notions of fuzzy
soft δ-closure (δ-interior) operators using r-fuzzy soft δ-closed (δ-open) sets. After that, we explore
the notions of fuzzy soft δ-continuous (semi-continuous and pre-continuous) functions, which are
weaker forms of fuzzy soft continuity. Moreover, we study some properties of these functions along
with their mutual relationships with the help of some problems. We also present a decomposition
of fuzzy soft semi-continuity and a decomposition of fuzzy soft α-continuity. Additionally, as a
weaker form of fuzzy soft continuity, we define and study the notions of fuzzy soft almost (weakly)
continuous functions. Lastly, we explore the notion of continuity in a very general setting called
fuzzy soft (L,M,N ,O)-continuity and introduce a historical justification.

2020 Mathematics Subject Classifications: 54A05, 54A40, 54C05, 54C08, 54D05

Key Words and Phrases: Fuzzy soft topological space, fuzzy soft δ-closure operator, r-fuzzy
soft δ-connected set, weaker forms of fuzzy soft continuity, fuzzy soft (L,M,N ,O)-continuity

1. Introduction and preliminaries

In [25], the author proposed a novel notion of soft set theory, which is a completely new
approach for modeling uncertainty and vagueness. He studied many applications of this
theory in solving different problems in engineering, social science, medical science, etc. The
notion of soft sets was used to define soft topological spaces in [31]. The method in [31] was
particularly important in the development of the field of soft topology (see [6, 19, 39, 44]).
Generalizations of soft open sets play an effective role in soft topology through their use to

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5733

Email addresses: iealshamri@uhb.edu.sa (I. Alshammari), osama.taha2015@yahoo.com (O. M. Taha),
mkamel bably@yahoo.com (M. K. El-Bably), imtaha2010@yahoo.com (I. M. Taha)

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



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 2 of 21

improve on some known results or to open the door to explore some of the soft topological
notions such as soft separation axioms [21], soft connectedness [40, 42], soft continuity
[26], etc. Akdag and Ozkan [3] introduced and studied the notion of soft α-open sets in
soft topological spaces. Also, the notion of soft β-open sets was introduced and studied by
the authors of [2, 18]. Al-shami et al. [4] defined the notion of weakly soft β-open sets and
obtained weakly soft β-continuity. Kaur et al. [22] initiated a new approach to studying
soft continuous functions. Moreover, many authors have contributed to the theory of soft
sets in the different fields such as topology, algebra; see [7, 10, 11, 16, 17, 27, 30].

Maji et al. [23] defined the notion of fuzzy soft sets which combines soft sets [25] and
fuzzy sets [43]. The notion of fuzzy soft topology was defined and some of its properties
such as fuzzy soft continuity, interior fuzzy soft set, closure fuzzy soft set, and fuzzy soft
subspace topology were obtained in [15, 20] based on fuzzy topologies in Šostaks sense
[41]. A novel approach to studying separation axioms and regularity axioms via fuzzy soft
sets was defined by the author of [32, 36]. The notion of r-fuzzy soft regularly open sets
was defined by Çetkin and Aygün [14]. Also, the notions of r-fuzzy soft β-open (pre-open)
sets were also introduced by Taha [33]. In addition, several researchers have contributed
to the theory of fuzzy soft sets in many fields such as topology; see [5, 28, 29].

The organization of this paper is as follows:

• In Section 2, we define new types of fuzzy soft sets in fuzzy soft topological spaces
based on the paper by Aygünoǧlu et al. [20]. Also, the relations of these sets with each
other are established with the help of some examples. Moreover, the concept of r-fuzzy
soft δ-connected sets is introduced and characterized with the help of fuzzy soft δ-closure
operators.

• In Section 3, we define the concepts of fuzzy soft δ-continuous (semi-continuous and
pre-continuous) functions, which are weaker forms of fuzzy soft continuity [20]. Some
properties of these functions along with their mutual relationships are discussed. Also, a
decomposition of fuzzy soft semi-continuity is obtained.

• In Section 4, as a weaker form of a fuzzy soft continuity, the concepts of fuzzy soft
almost (weakly) continuous functions are introduced and some properties are specified.
Also, we show that fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft
weakly continuity, but the converse may not be true. In addition, we explore the notion
of continuity in a very general setting called fuzzy soft (L,M,N ,O)-continuous functions
and a historical justification is introduced.

• Finally, we close this paper with some conclusions and make a plan to suggest some
future works in Section 5.

Throughout this article, nonempty sets will be denoted by U , V , etc. E is the set of all
parameters for U and A ⊆ E. The family of all fuzzy sets on U is denoted by IU (where



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 3 of 21

I◦ = (0, 1], I = [0, 1]), and for t ∈ I, t(u) = t, for all u ∈ U. The following definitions will
be used in the next sections:

Definition 1. [1, 13, 20] A fuzzy soft set fA on U is a function from E to IU such that
fA(e) is a fuzzy set on U , for each e ∈ A and fA(e) = 0, if e ̸∈ A. The family of all fuzzy

soft sets on U is denoted by (̃U,E).

Definition 2. [24] A fuzzy soft point eut on U is a fuzzy soft set defined as follows:

eut(k) =

{
ut, if k = e,
0, if k ∈ E − {e},

where ut is a fuzzy point on U . eut is said to belong to a fuzzy soft set fA, denoted by

eut∈̃fA, if t ≤ fA(e)(u). The family of all fuzzy soft points on U is denoted by P̃t(U).

Definition 3. [12] A fuzzy soft point eut ∈ P̃t(U) is called a soft quasi-coincident with

fA ∈ (̃U,E) and denoted by eut q̃fA, if t+ fA(e)(u) > 1. A fuzzy soft set fA ∈ (̃U,E) is

called a soft quasi-coincident with gB ∈ (̃U,E) and denoted by fAq̃gB, if there is e ∈ E
and u ∈ U , such that fA(e)(u) + gB(e)(u) > 1. If fA is not soft quasi-coincident with
gB, fA ̸ q̃gB.

Definition 4. [20] A function τ : E −→ [0, 1](̃U,E) is called a fuzzy soft topology on U if
it satisfies the following conditions for every e ∈ E,

(i) τe(Φ) = τe(Ẽ) = 1,

(ii) τe(fA ⊓ gB) ≥ τe(fA) ∧ τe(gB), for every fA, gB ∈ (̃U,E),

(iii) τe(
⊔
δ∈∆(fA)δ) ≥

∧
δ∈∆ τe((fA)δ), for every (fA)δ ∈ (̃U,E), δ ∈ ∆.

Thus, (U, τE) is called a fuzzy soft topological space (FSTS) in Šostaks sense [41].

Definition 5. [20] Let (U, τE) and (V, τ∗F ) be an FSTSs. A fuzzy soft function φψ :

(̃U,E) −→ (̃V, F ) is said to be fuzzy soft continuous if τe(φ
−1
ψ (gB)) ≥ τ∗k (gB) for every

gB ∈ (̃V, F ), e ∈ E, and (k = ψ(e)) ∈ F .

Definition 6. [14, 15] In an FSTS (U, τE), for each fA ∈ (̃U,E), e ∈ E, and r ∈ I0, we

define the fuzzy soft operators Cτ and Iτ : E × (̃U,E)× I◦ → (̃U,E) as follows:

Cτ (e, fA, r) = ⊓ {gB ∈ (̃U,E) : fA ⊑ gB, τe(g
c
B) ≥ r}.

Iτ (e, fA, r) = ⊔ {gB ∈ (̃U,E) : gB ⊑ fA, τe(gB) ≥ r}.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 4 of 21

Definition 7. Let (U, τE) be an FSTS and r ∈ I0. A fuzzy soft set fA ∈ (̃U,E) is said to be
r-fuzzy soft regularly open [14] (pre-open [33] and β-open [33]) if fA = Iτ (e, Cτ (e, fA, r), r)
(fA ⊑ Iτ (e, Cτ (e, fA, r), r) and fA ⊑ Cτ (e, Iτ (e, Cτ (e, fA, r), r), r)) for every e ∈ E.

Lemma 1. [33] Every r-fuzzy soft regularly open set is r-fuzzy soft pre-open.

In general, the converse of Lemma 1 is not true, as shown by Example 1.

Example 1. [5] Let U = {u1, u2}, E = {e, k}, and define gE , fE ∈ (̃U,E) as follows:
gE = {(e, { u10.3 ,

u2
0.4}), (k, {

u1
0.3 ,

u2
0.4})}, fE = {(e, { u10.6 ,

u2
0.2}), (k, {

u1
0.6 ,

u2
0.2})}. Define fuzzy soft

topology τE : E −→ [0, 1](̃U,E) as follows:

τe(mE) =



1, if mE ∈ {Φ, Ẽ},
1
4 , if mE = gE ,
1
3 , if mE = fE ,
1
3 , if mE = gE ⊓ fE ,
1
4 , if mE = gE ⊔ fE ,
0, otherwise,

τk(mE) =



1, if mE ∈ {Φ, Ẽ},
1
4 , if mE = gE ,
1
2 , if mE = fE ,
1
2 , if mE = gE ⊓ fE ,
1
4 , if mE = gE ⊔ fE ,
0, otherwise.

Thus, fE is 1
4 -fuzzy soft pre-open set, but it is not 1

4 -fuzzy soft regularly open set.

The basic definitions and results that we need in the next sections are found in [15, 20].

2. On r-fuzzy soft δ-open sets

Here, we are going to give the concepts of r-fuzzy soft δ-open (semi-open) sets in an
FSTS. Some properties of these sets along with their mutual relationships are investigated
with the help of some examples. Also, the concept of an r-fuzzy soft δ-connected set is
defined and studied with the help of fuzzy soft δ-closure operators.

Definition 8. Let (U, τE) be an FSTS. A fuzzy soft set fA ∈ (̃U,E) is said to be an r-fuzzy
soft δ-open (resp., semi-open and α-open [8]) if Iτ (e, Cτ (e, fA, r), r) ⊑ Cτ (e, Iτ (e, fA, r), r)
(resp., fA ⊑ Cτ (e, Iτ (e, fA, r), r) and fA ⊑ Iτ (e, Cτ (e, Iτ (e, fA, r), r), r)) for every e ∈ E
and r ∈ I0.

Remark 1. The concepts of an r-fuzzy soft δ-open set and r-fuzzy soft β-open set [33]
are independent concepts, as shown by Examples 2 and 3.

Example 2. Let U = {u1, u2}, E = {e, k}, and define hE , gE , fE ∈ (̃U,E) as fol-
lows: hE = {(e, { u10.4 ,

u2
0.5}), (k, {

u1
0.4 ,

u2
0.5})}, gE = {(e, { u10.2 ,

u2
0.3}), (k, {

u1
0.2 ,

u2
0.3})}, fE =

{(e, { u10.8 ,
u2
0.7}), (k, {

u1
0.8 ,

u2
0.7})}. Define fuzzy soft topology τE : E −→ [0, 1](̃U,E) as follows:



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 5 of 21

τe(mE) =


1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = gE ,
2
3 , if mE = fE ,

0, otherwise,

τk(mE) =


1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = gE ,
1
2 , if mE = fE ,

0, otherwise.

Thus, hE is 1
3 -fuzzy soft β-open set, but it is neither 1

3 -fuzzy soft δ-open nor 1
3 -fuzzy

soft semi-open.

Example 3. Let U = {u1, u2, u3}, E = {e, k}, and define hE , gE , fE ∈ (̃U,E) as follows:
hE = {(e, {u10 ,

u2
1 ,

u3
1 }), (k, {u10 ,

u2
1 ,

u3
1 })}, gE = {(e, {u10 ,

u2
0 ,

u3
1 }), (k, {u10 ,

u2
0 ,

u3
1 })}, fE =

{(e, {u10 ,
u2
1 ,

u3
0 }), (k, {u10 ,

u2
1 ,

u3
0 })}. Define fuzzy soft topology τE : E −→ [0, 1](̃U,E) as

follows:

τe(mE) =



1, if mE ∈ {Φ, Ẽ},
1
4 , if mE = gE ,
1
2 , if mE = fE ,
1
3 , if mE = hE ,

0, otherwise,

τk(mE) =



1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = gE ,
1
2 , if mE = fE ,
1
3 , if mE = hE ,

0, otherwise.

Thus, hcE is 1
4 -fuzzy soft δ-open set, but it is neither 1

4 -fuzzy soft β-open nor 1
4 -fuzzy

soft semi-open.

Remark 2. The complement of an r-fuzzy soft δ-open (resp., semi-open, α-open and β-
open) set is said to be an r-fuzzy soft δ-closed (resp., semi-closed, α-closed and β-closed).

Proposition 1. Let (U, τE) be an FSTS, fA ∈ (̃U,E), e ∈ E, and r ∈ I0. The following
statements are equivalent:

(i) fA is an r-fuzzy soft semi-open.

(ii) fA is an r-fuzzy soft δ-open and r-fuzzy soft β-open.

Proof.
(i) ⇒ (ii) Let fA be an r-fuzzy soft semi-open, then fA ⊑ Cτ (e, Iτ (e, fA, r), r) ⊑

Cτ (e, Iτ (Cτ (e, fA, r), r), r). This shows that fA is r-fuzzy soft β-open. Moreover,

Iτ (e, Cτ (e, fA, r), r) ⊑ Cτ (e, fA, r) ⊑ Cτ (e, Cτ (e, Iτ (e, fA, r), r), r) = Cτ (e, Iτ (e, fA, r), r).



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 6 of 21

Therefore, fA is r-fuzzy soft δ-open.

(ii)⇒ (i) Let fA be an r-fuzzy soft δ-open and r-fuzzy soft β-open, then Iτ (e, Cτ (e, fA, r), r)
⊑ Cτ (e, Iτ (e, fA, r), r) and fA ⊑ Cτ (e, Iτ (e, Cτ (e, fA, r), r), r). Thus,

fA ⊑ Cτ (e, Iτ (e, Cτ (e, fA, r), r), r) ⊑ Cτ (e, Cτ (e, Iτ (e, fA, r), r), r) = Cτ (e, Iτ (e, fA, r), r).

This shows that fA is r-fuzzy soft semi-open.

Proposition 2. Let (U, τE) be an FSTS, fA ∈ (̃U,E), e ∈ E, and r ∈ I0. The following
statements are equivalent:

(i) fA is an r-fuzzy soft α-open.

(ii) fA is an r-fuzzy soft δ-open and r-fuzzy soft pre-open.

Proof. (i) ⇒ (ii) From Proposition 1 the proof is straightforward.

(ii) ⇒ (i) Let fA be an r-fuzzy soft pre-open and r-fuzzy soft δ-open. Then, fA ⊑
Iτ (e, Cτ (e, fA, r), r) ⊑ Iτ (e, Cτ (e, Iτ (e, fA, r), r), r). This shows that fA is r-fuzzy soft α-
open.

Remark 3. From the previous definitions and results, we can summarize the relationships
among different types of fuzzy soft open sets as in the next diagram.

fuzzy soft α-open set

↓ ↓

fuzzy soft pre-open set ↮ fuzzy soft semi-open set → fuzzy soft δ-open set

↓ ↓

fuzzy soft β-open set

Remark 4. In general, the converses of the above relationships are not true, as shown by
Examples 2, 3, 4, 5, and 6.

Example 4. Let U = {u1, u2}, E = {e, k}, and define gE , fE , hE ∈ (̃U,E) as fol-
lows: gE = {(e, { u10.3 ,

u2
0.4}), (k, {

u1
0.3 ,

u2
0.4})}, fE = {(e, { u10.6 ,

u2
0.2}), (k, {

u1
0.6 ,

u2
0.2})}, hE =

{(e, { u10.7 ,
u2
0.5}), (k, {

u1
0.7 ,

u2
0.5})}. Define fuzzy soft topology τE : E −→ [0, 1](̃U,E) as follows:



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 7 of 21

τe(mE) =



1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = gE ,
2
3 , if mE = fE ,
2
3 , if mE = gE ⊓ fE ,
1
2 , if mE = gE ⊔ fE ,
0, otherwise,

τk(mE) =



1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = gE ,
1
2 , if mE = fE ,
1
2 , if mE = gE ⊓ fE ,
1
3 , if mE = gE ⊔ fE ,
0, otherwise.

Thus, hE is 1
3 -fuzzy soft semi-open set, but it is neither 1

3 -fuzzy soft α-open nor 1
3 -fuzzy

soft pre-open.

Example 5. Let U = {u1, u2, u3}, E = {e, k}, and define gE , fE ∈ (̃U,E) as follows:
gE = {(e, { u10.2 ,

u2
0.3 ,

u3
0.2}), (k, {

u1
0.2 ,

u2
0.3 ,

u3
0.2})}, fE = {(e, { u10.3 ,

u2
0.4 ,

u3
0.8}), (k, {

u1
0.3 ,

u2
0.4 ,

u3
0.8})}.

Define fuzzy soft topology τE : E −→ [0, 1](̃U,E) as follows:

τe(mE) =


1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = gE ,

0, otherwise,

τk(mE) =


1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = gE ,

0, otherwise.

Thus, fE is 1
3 -fuzzy soft β-open set, but it is not 1

3 -fuzzy soft pre-open.

Example 6. Let U = {u1, u2}, E = {e, k}, and define gE , fE ∈ (̃U,E) as follows:
gE = {(e, { u10.4 ,

u2
0.5}), (k, {

u1
0.4 ,

u2
0.5})}, fE = {(e, { u10.3 ,

u2
0.4}), (k, {

u1
0.3 ,

u2
0.4})}. Define fuzzy soft

topology τE : E −→ [0, 1](̃U,E) as follows:

τe(mE) =


1, if mE ∈ {Φ, Ẽ},
1
4 , if mE = gE ,

0, otherwise,

τk(mE) =


1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = gE ,

0, otherwise.

Thus, fE is 1
4 -fuzzy soft pre-open set, but it is neither 1

4 -fuzzy soft α-open nor 1
4 -fuzzy

soft semi-open.

Theorem 1. Let (U, τE) be an FSTS, fA, gB ∈ (̃U,E), e ∈ E, and r ∈ I0. If fA is an
r-fuzzy soft δ-open set such that fA ⊑ gB ⊑ Cτ (e, fA, r), then gB is also r-fuzzy soft
δ-open.

Proof. Suppose that an fA is r-fuzzy soft δ-open and fA ⊑ gB ⊑ Cτ (e, fA, r). Then,
Iτ (e, Cτ (e, fA, r), r) ⊑ Cτ (e, Iτ (e, fA, r), r) ⊑ Cτ (e, Iτ (e, gB, r), r). Since gB ⊑ Cτ (e, fA, r),
Iτ (e, Cτ (e, gB, r), r) ⊑ Iτ (e, Cτ (e, fA, r), r) ⊑ Cτ (e, Iτ (e, gB, r), r). This shows that gB is
r-fuzzy soft δ-open.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 8 of 21

Definition 9. In an FSTS (U, τE), for each fA ∈ (̃U,E), e ∈ E, and r ∈ I0, we define a

fuzzy soft δ-closure operator δCτ : E × (̃U,E) × I◦ → (̃U,E) as follows: δCτ (e, fA, r) =

⊓ {gB ∈ (̃U,E) : fA ⊑ gB, gB is r-fuzzy soft δ-closed}.

Theorem 2. In an FSTS (U, τE), for each fA, gB ∈ (̃U,E), e ∈ E, and r ∈ I0, the operator

δCτ : E × (̃U,E)× I◦ → (̃U,E) satisfies the following properties.

(1) δCτ (e,Φ, r) = Φ.

(2) fA ⊑ δCτ (e, fA, r) ⊑ Cτ (e, fA, r).

(3) δCτ (e, fA, r) ⊑ δCτ (e, gB, r) if fA ⊑ gB.

(4) δCτ (e, δCτ (e, fA, r), r) = δCτ (e, fA, r).

(5) δCτ (e, fA ⊔ gB, r) ⊒ δCτ (e, fA, r) ⊔ δCτ (e, gB, r).

(6) δCτ (e, fA, r) = fA iff fA is r-fuzzy soft δ-closed.

(7) δCτ (e, Cτ (e, fA, r), r) = Cτ (e, fA, r).

Proof. (1), (2), (3), and (6) are easily proved from Definition 9.

(4) From (2) and (3), δCτ (e, fA, r) ⊑ δCτ (e, δCτ (e, fA, r), r).
Now, we show that δCτ (e, fA, r) ⊒ δCτ (e, δCτ (e, fA, r), r). Suppose that δCτ (e, fA, r)

does not contain δCτ (e, δCτ (e, fA, r), r), then there is u ∈ U and t ∈ (0, 1) such that

δCτ (e, fA, r)(e)(u) < t < δCτ (e, δCτ (e, fA, r), r)(e)(u). (A)

Since δCτ (e, fA, r)(e)(u) < t, by the definition of δCτ , there is gB as a r-fuzzy soft
δ-closed with fA ⊑ gB such that δCτ (e, fA, r)(e)(u) ≤ gB(e)(u) < t. Since fA ⊑ gB, then
δCτ (e, fA, r) ⊑ gB. Again, by the definition of δCτ , we have δCτ (e, δCτ (e, fA, r), r) ⊑ gB.
Hence, δCτ (e, δCτ (e, fA, r), r)(e)(u) ≤ gB(e)(u) < t, which is a contradiction for (A).
Thus, δCτ (e, fA, r) ⊒ δCτ (e, δCτ (e, fA, r), r), then δCτ (e, δCτ (e, fA, r), r) = δCτ (e, fA, r).

(5) Since fA and gB ⊑ fA ⊔ gB, hence by (3), δCτ (e, fA, r) ⊑ δCτ (e, fA ⊔ gB, r) and
δCτ (e, gB, r) ⊑ δCτ (e, fA ⊔ gB, r). Thus, δCτ (e, fA ⊔ gB, r) ⊒ δCτ (e, fA, r)⊔ δCτ (e, gB, r).

(7) From (6) and Cτ (e, fA, r) is r-fuzzy soft δ-closed set, hence δCτ (e, Cτ (e, fA, r), r) =
Cτ (e, fA, r).



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 9 of 21

Theorem 3. In an FSTS (U, τE), for each fA ∈ (̃U,E), e ∈ E, and r ∈ I0, we define a

fuzzy soft δ-interior operator δIτ : E × (̃U,E) × I◦ → (̃U,E) as follows: δIτ (e, fA, r) =

⊔ {gB ∈ (̃U,E) : gB ⊑ fA, gB is r-fuzzy soft δ-open}. Then, for each fA and gB ∈ (̃U,E),
the operator δIτ satisfies the following properties.

(1) δIτ (e, Ẽ, r) = Ẽ.

(2) Iτ (e, fA, r) ⊑ δIτ (e, fA, r) ⊑ fA.

(3) δIτ (e, fA, r) ⊑ δIτ (e, gB, r) if fA ⊑ gB.

(4) δIτ (e, δIτ (e, fA, r), r) = δIτ (e, fA, r).

(5) δIτ (e, fA, r) ⊓ δIτ (e, gB, r) ⊒ δIτ (e, fA ⊓ gB, r).

(6) δIτ (e, fA, r) = fA iff fA is r-fuzzy soft δ-open.

(7) δIτ (e, f
c
A, r) = (δCτ (e, fA, r))

c.

Proof. (1), (2), (3), and (6) are easily proved from the definition of δIτ .

(4) and (5) are easily proved by a similar way in Theorem 2.

(7) For each fA ∈ (̃U,E), e ∈ E, and r ∈ I0, we have δIτ (e, f
c
A, r) = ⊔{gB ∈ (̃U,E) :

gB ⊑ f cA, gB is r-fuzzy soft δ-open}= [⊓{gcB ∈ (̃U,E) : fA ⊑ gcB, g
c
B is r-fuzzy soft δ-closed}]c

= (δCτ (e, fA, r))
c.

Definition 10. Let (U, τE) be an FSTS, r ∈ I0, and fA, gB ∈ (̃U,E), then we have:

(1) Two fuzzy soft sets fA and gB are called r-fuzzy soft δ-separated iff gB ̸ q̃ δCτ (e, fA, r)
and fA ̸ q̃ δCτ (e, gB, r) for each e ∈ E.

(2) Any fuzzy soft set which cannot be expressed as the union of two r-fuzzy soft
δ-separated sets is called an r-fuzzy soft δ-connected.

Theorem 4. In an FSTS (U, τE), we have:

(1) If fA and gB ∈ (̃U,E) are r-fuzzy soft δ-separated and hC , tD ∈ (̃U,E) such that
hC ⊑ fA and tD ⊑ gB, then hC and tD are r-fuzzy soft δ-separated.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 10 of 21

(2) If fA ̸ q̃ gB and either both are r-fuzzy soft δ-open or both r-fuzzy soft δ-closed,
then fA and gB are r-fuzzy soft δ-separated.

(3) If fA and gB are either both r-fuzzy soft δ-open or both r-fuzzy soft δ-closed, then
fA ⊓ gcB and gB ⊓ f cA are r-fuzzy soft δ-separated.

Proof. (1) and (2) are obvious.

(3) Let fA and gB be an r-fuzzy soft δ-open. Since fA ⊓ gcB ⊑ gcB, δCτ (e, fA⊓gcB, r) ⊑
gcB and hence δCτ (e, fA ⊓ gcB, r)̸ q̃ gB. Then, δCτ (e, fA ⊓ gcB , r)̸ q̃ (gB ⊓ f cA).

Again, since gB ⊓ f cA ⊑ f cA, δCτ (e, gB ⊓ f cA, r) ⊑ f cA and hence δCτ (e, gB
⊓f cA, r)̸ q̃ fA. Then, δCτ (e, gB ⊓ f cA, r)̸ q̃ (fA ⊓ gcB). Thus, fA ⊓ gcB and gB ⊓ f cA are r-fuzzy
soft δ-separated. The other case follows similar lines.

Theorem 5. In an FSTS (U, τE), then fA, gB ∈ (̃U,E) are r-fuzzy soft δ-separated iff
there exist two r-fuzzy soft δ-open sets hC and tD such that fA ⊑ hC , gB ⊑ tD, fA ̸ q̃ tD
and gB ̸ q̃ hC .

Proof. (⇒) Let fA and gB ∈ (̃U,E) be an r-fuzzy soft δ-separated, fA ⊑ (δCτ (e, gB, r))
c =

hC and gB ⊑ (δCτ (e, fA, r))
c = tD, where tD and hC are r-fuzzy soft δ-open, then

tD ̸ q̃ δCτ (e, fA, r) and hC ̸ q̃ δCτ (e, gB, r). Thus, gB ̸ q̃ hC and fA ̸ q̃ tD. Hence, we ob-
tain the required result.

(⇐) Let hC and tD be an r-fuzzy soft δ-open such that gB ⊑ tD, fA ⊑ hC , gB ̸ q̃ hC and
fA ̸ q̃ tD. Then, gB ⊑ hcC and fA ⊑ tcD. Hence, δCτ (e, gB, r) ⊑ hcC and δCτ (e, fA, r) ⊑ tcD.
Then, δCτ (e, gB, r)̸ q̃ fA and δCτ (e, fA, r)̸ q̃ gB. Thus, fA and gB are r- fuzzy soft δ-
separated. Hence, we obtain the required result.

Theorem 6. In an FSTS (U, τE), if gB ∈ (̃U,E) is r-fuzzy soft δ-connected such that
gB ⊑ fA ⊑ δCτ (e, gB, r), then fA is r-fuzzy soft δ-connected.

Proof. Suppose that fA is not r-fuzzy soft δ-connected, then there is r-fuzzy soft

δ-separated sets h∗C and t∗D ∈ (̃U,E) such that fA = h∗C ⊔ t∗D. Let hC = gB ⊓ h∗C and
tD = gB ⊓ t∗D, then gB = tD ⊔ hC . Since hC ⊑ h∗C and tD ⊑ t∗D, hence by Theorem 4(1),
hC and tD are r-fuzzy soft δ-separated, it is a contradiction. Thus, fA is r-fuzzy soft
δ-connected, as required.

3. A decomposition of fuzzy soft semi-continuity

Here, we introduce the concepts of fuzzy soft δ-continuous (semi-continuous and pre-
continuous) functions, which are weaker forms of fuzzy soft continuity in an FSTSs in
Šostaks sense. Also, we study several relationships related to fuzzy soft δ-continuity with
the help of some problems. A decomposition of fuzzy soft semi-continuity is obtained.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 11 of 21

Definition 11. Let (U, τE) and (V, τ∗F ) be an FSTSs. A fuzzy soft function φψ : (̃U,E) −→
(̃V, F ) is said to be a fuzzy soft δ-continuous (resp., β-continuous [9], semi-continuous,
pre-continuous, and α-continuous [8]) if φ−1

ψ (gB) is r-fuzzy soft δ-open (resp., β-open,

semi-open, pre-open, and α-open) set for every gB ∈ (̃V, F ) with τ∗k (gB) ≥ r, e ∈ E,
(k = ψ(e)) ∈ F , and r ∈ Io.

Remark 5. Fuzzy soft δ-continuity and fuzzy soft β-continuity are independent concepts,
as shown by Examples 7 and 8.

Example 7. Let U = {u1, u2}, E = {e1, e2}, and define hE , gE , fE ∈ (̃U,E) as fol-
lows: hE = {(e1, { u10.4 ,

u2
0.5}), (e2, {

u1
0.4 ,

u2
0.5})}, gE = {(e1, { u10.2 ,

u2
0.3}), (e2, {

u1
0.2 ,

u2
0.3})}, fE =

{(e1, { u10.8 ,
u2
0.7}), (e2, {

u1
0.8 ,

u2
0.7})}. Define fuzzy soft topologies τE , τ

∗
E : E −→ [0, 1](̃U,E) as

follows: ∀e ∈ E,

τe(mE) =


1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = gE ,
2
3 , if mE = fE ,

0, otherwise,

τ∗e (mE) =


1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = hE ,

0, otherwise.

Thus, the identity fuzzy soft function φψ : (U, τE) −→ (U, τ∗E) is fuzzy soft β-continuous,
but it is neither fuzzy soft δ-continuous nor fuzzy soft semi-continuous.

Example 8. Let U = {u1, u2, u3}, E = {e1, e2}, and define hE , gE , fE ∈ (̃U,E) as follows:
hE = {(e1, {u10 ,

u2
1 ,

u3
1 }), (e2, {u10 ,

u2
1 ,

u3
1 })}, gE = {(e1, {u10 ,

u2
0 ,

u3
1 }), (e2, {u10 ,

u2
0 ,

u3
1 })},

fE = {(e1, {u10 ,
u2
1 ,

u3
0 }), (e2, {u10 ,

u2
1 ,

u3
0 })}. Define fuzzy soft topologies τE , τ

∗
E : E −→

[0, 1](̃U,E) as follows: ∀e ∈ E,

τe(mE) =



1, if mE ∈ {Φ, Ẽ},
1
4 , if mE = gE ,
1
2 , if mE = fE ,
1
3 , if mE = hE ,

0, otherwise,

τ∗e (mE) =


1, if mE ∈ {Φ, Ẽ},
1
4 , if mE = hcE ,

0, otherwise.

Thus, the identity fuzzy soft function φψ : (U, τE) −→ (U, τ∗E) is fuzzy soft δ-continuous,
but it is neither fuzzy soft β-continuous nor fuzzy soft semi-continuous.

Now, we have the following decomposition of fuzzy soft semi-continuity and decompo-
sition of fuzzy soft α-continuity, according to Propositions 1 and 2.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 12 of 21

Proposition 3. Let (U, τE) and (V, τ∗F ) be an FSTSs. φψ : (̃U,E) −→ (̃V, F ) is fuzzy soft
semi-continuous function iff it is both fuzzy soft δ-continuous and fuzzy soft β-continuous.

Proof. The proof is obvious by Proposition 1.

Proposition 4. Let (U, τE) and (V, τ∗F ) be an FSTSs. φψ : (̃U,E) −→ (̃V, F ) is fuzzy soft
α-continuous function iff it is both fuzzy soft δ-continuous and fuzzy soft pre-continuous.

Proof. The proof is obvious by Proposition 2.

Remark 6. From the previous definitions and results, we can summarize the relationships
among different types of fuzzy soft continuity as in the next diagram.

fuzzy soft continuity

↓

fuzzy soft α-continuity

↓ ↓

fuzzy soft pre-continuity ↮ fuzzy soft semi-continuity → fuzzy soft δ-continuity

↓ ↓

fuzzy soft β-continuity

Remark 7. In general, the converses of the above relationships are not true, as shown by
Examples 7, 8, 9, 10, and 11.

Example 9. Let U = {u1, u2}, E = {e1, e2}, and define gE , fE , hE ∈ (̃U,E) as fol-
lows: gE = {(e1, { u10.3 ,

u2
0.4}), (e2, {

u1
0.3 ,

u2
0.4})}, fE = {(e1, { u10.6 ,

u2
0.2}), (e2, {

u1
0.6 ,

u2
0.2})}, hE =

{(e1, { u10.7 ,
u2
0.5}), (e2, {

u1
0.7 ,

u2
0.5})}. Define fuzzy soft topologies τE , τ

∗
E : E −→ [0, 1](̃U,E) as

follows: ∀e ∈ E,

τe(mE) =



1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = gE ,
2
3 , if mE = fE ,
2
3 , if mE = gE ⊓ fE ,
1
2 , if mE = gE ⊔ fE ,
0, otherwise,

τ∗e (mE) =


1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = hE ,

0, otherwise.

Thus, the identity fuzzy soft function φψ : (U, τE) −→ (U, τ∗E) is fuzzy soft semi-
continuous, but it is neither fuzzy soft α-continuous nor fuzzy soft pre-continuous.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 13 of 21

Example 10. Let U = {u1, u2, u3}, E = {e1, e2}, and define gE , fE ∈ (̃U,E) as follows:
gE = {(e1, { u10.2 ,

u2
0.3 ,

u3
0.2}), (e2, {

u1
0.2 ,

u2
0.3 ,

u3
0.2})}, fE = {(e1, { u10.3 ,

u2
0.4 ,

u3
0.8}), (e2, {

u1
0.3 ,

u2
0.4 ,

u3
0.8})}.

Define fuzzy soft topologies τE , τ
∗
E : E −→ [0, 1](̃U,E) as follows: ∀e ∈ E,

τe(mE) =


1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = gE ,

0, otherwise,

τ∗e (mE) =


1, if mE ∈ {Φ, Ẽ},
1
3 , if mE = fE ,

0, otherwise.

Thus, the identity fuzzy soft function φψ : (U, τE) −→ (U, τ∗E) is fuzzy soft β-continuous,
but it is not fuzzy soft pre-continuous.

Example 11. Let U = {u1, u2}, E = {e1, e2}, and define gE , fE ∈ (̃U,E) as follows:
gE = {(e1, { u10.4 ,

u2
0.5}), (e2, {

u1
0.4 ,

u2
0.5})}, fE = {(e1, { u10.3 ,

u2
0.4}), (e2, {

u1
0.3 ,

u2
0.4})}. Define fuzzy

soft topologies τE , τ
∗
E : E −→ [0, 1](̃U,E) as follows: ∀e ∈ E,

τe(mE) =


1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = gE ,

0, otherwise,

τ∗e (mE) =


1, if mE ∈ {Φ, Ẽ},
1
4 , if mE = fE ,

0, otherwise.

Thus, the identity fuzzy soft function φψ : (U, τE) −→ (U, τ∗E) is fuzzy soft pre-
continuous, but it is neither fuzzy soft α-continuous nor fuzzy soft semi-continuous.

Theorem 7. Let (U, τE) and (V, τ∗F ) be an FSTSs and φψ : (̃U,E) −→ (̃V, F ) be a fuzzy

soft function. The following statements are equivalent for every gB ∈ (̃V, F ), e ∈ E,
(k = ψ(e)) ∈ F , and r ∈ I◦.

(i) φψ is fuzzy soft β-continuous.

(ii) Iτ (e, Cτ (e, Iτ (e, φ
−1
ψ (gB), r), r), r) ⊑ φ−1

ψ (gB), if τ
∗
k (g

c
B) ≥ r.

(iii) Iτ (e, Cτ (e, Iτ (e, φ
−1
ψ (gB), r), r), r) ⊑ φ−1

ψ (Cτ∗(k, gB, r)).

(iv) φ−1
ψ (Iτ∗(k, gB, r)) ⊑ Cτ (e, Iτ (e, Cτ (e, φ

−1
ψ (gB), r), r), r).

Proof. (i) ⇒ (ii) Let gB ∈ (̃V, F ) with τ∗k (g
c
B) ≥ r. Then by Definition 11,

(φ−1
ψ (gB))

c = φ−1
ψ (gcB) ⊑ Cτ (e, Iτ (e, Cτ (e, φ

−1
ψ (gcB), r), r), r) = (Iτ (e, Cτ (e, Iτ (e, φ

−1
ψ (gB), r), r), r))

c.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 14 of 21

Thus, Iτ (e, Cτ (e, Iτ (e, φ
−1
ψ (gB), r), r), r) ⊑ φ−1

ψ (gB).

(ii) ⇒ (iii) Obvious.

(iii)⇒ (iv) Since (Iτ (e, Cτ (e, Iτ (e, φ
−1
ψ (gB), r), r), r))

c = Cτ (e, Iτ (e, Cτ (e, φ
−1
ψ (gcB), r), r), r)

and (φ−1
ψ (Cτ∗(k, gB, r)))

c = φ−1
ψ (Iτ∗(k, g

c
B, r)). Then,

φ−1
ψ (Iτ∗(k, gB, r)) ⊑ Cτ (e, Iτ (e, Cτ (e, φ

−1
ψ (gB), r), r), r),

for each gB ∈ (̃V, F ).

(iv) ⇒ (i) Let gB ∈ (̃V, F ) with τ∗k (gB) ≥ r. Then by (iv) and gB = Iτ∗(k, gB, r),
φ−1
ψ (gB) ⊑ Cτ (e, Iτ (e, Cτ (e, φ

−1
ψ (gB), r), r), r). Thus, φψ is fuzzy soft β-continuous.

The following theorem is similarly proved as in Theorem 7.

Theorem 8. Let (U, τE) and (V, τ∗F ) be an FSTSs and φψ : (̃U,E) −→ (̃V, F ) be a fuzzy

soft function. The following statements are equivalent for every gB ∈ (̃V, F ), e ∈ E,
(k = ψ(e)) ∈ F , and r ∈ I◦.

(i) φψ is fuzzy soft δ-continuous.

(ii) Iτ (e, Cτ (e, φ
−1
ψ (gB), r), r) ⊑ Cτ (e, Iτ (e, φ

−1
ψ (gB), r), r), if τ

∗
k (g

c
B) ≥ r.

(iii) Iτ (e, Cτ (e, φ
−1
ψ (gB), r), r) ⊑ Cτ (e, Iτ (e, φ

−1
ψ (Cτ∗(k, gB, r)), r), r)

(iv) Iτ (e, Cτ (e, φ
−1
ψ (Iτ∗(k, gB, r)), r), r) ⊑ Cτ (e, Iτ (e, φ

−1
ψ (gB), r), r).

Proposition 5. Let (U, τE), (V, τ
∗
F ), and (W,γH) be an FSTSs and φψ : (̃U,E) −→ (̃V, F ),

φ∗
ψ∗ : (̃V, F ) −→ ˜(W,H) be two fuzzy soft functions. Then, the composition φ∗

ψ∗ ◦ φψ is
fuzzy soft δ-continuous (resp., β-continuous) if φψ is fuzzy soft δ-continuous (resp., β-
continuous) and φ∗

ψ∗ is fuzzy soft continuous.

Proof. Obvious.

4. Some weaker forms of fuzzy soft continuity

Here, as a weaker form of fuzzy soft continuity [20], the concepts of fuzzy soft almost
(weakly) continuous functions are introduced and some properties are obtained. Further-
more, we show that fuzzy soft continuity ⇒ fuzzy soft almost continuity ⇒ fuzzy soft
weakly continuity, but the converse may not be true. Finally, we introduce the notion of
continuity in a very general setting called fuzzy soft (L,M,N ,O)-continuous functions.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 15 of 21

Definition 12. Let (U, τE) and (V, τ∗F ) be an FSTSs. A fuzzy soft function φψ : (̃U,E) −→
(̃V, F ) is said to be fuzzy soft almost (resp., weakly) continuous if for each eut ∈ P̃t(U) and

each gB ∈ (̃V, F ) with τ∗k (gB) ≥ r containing φψ(eut), there is fA ∈ (̃U,E) with τe(fA) ≥ r
containing eut , such that φψ(fA) ⊑ Iτ∗(k,Cτ∗(k, gB, r), r) (resp., φψ(fA) ⊑ Cτ∗(k, gB, r)).

Theorem 9. Let (U, τE) and (V, τ∗F ) be an FSTSs and φψ : (̃U,E) −→ (̃V, F ) be a fuzzy

soft function. Suppose that one of the following holds for every gB ∈ (̃V, F ), e ∈ E,
(k = ψ(e)) ∈ F , and r ∈ I◦:

(i) If τ∗k (gB) ≥ r, φ−1
ψ (gB) ⊑ Iτ (e, φ

−1
ψ (Iτ∗(k,Cτ∗(k, gB, r), r)), r).

(ii) Cτ (e, φ
−1
ψ (Cτ∗(k, Iτ∗(k, gB, r), r)), r) ⊑ φ−1

ψ (gB) if τ
∗
k (g

c
B) ≥ r.

Then, φψ is fuzzy soft almost continuous.

Proof. (i) ⇒ (ii) Let gB ∈ (̃V, F ) with τ∗k (g
c
B) ≥ r. From (i), it follows

φ−1
ψ (gcB) ⊑ Iτ (e, φ

−1
ψ (Iτ∗(k,Cτ∗(k, g

c
B, r), r)), r) = Iτ (e, φ

−1
ψ ((Cτ∗(k, Iτ∗(k, gB, r), r))

c), r) =

Iτ (e, (φ
−1
ψ (Cτ∗(k, Iτ∗(k, gB, r), r)))

c, r) = (Cτ (e, φ
−1
ψ (Cτ∗(k, Iτ∗(k, gB, r), r)), r))

c.

Hence, Cτ (e, φ
−1
ψ (Cτ∗(k, Iτ∗(k, gB, r), r)), r) ⊑ φ−1

ψ (gB). Similarly, we get (ii) ⇒ (i).

Suppose that (i) holds. Let eut ∈ P̃t(U) and gB ∈ (̃V, F ) with τ∗k (gB) ≥ r containing

φψ(eut). Then, by (i), eut∈̃Iτ (e, φ−1
ψ (Iτ∗(k,Cτ∗(k, gB, r), r)), r), and so there is fA ∈ (̃U,E)

with τe(fA) ≥ r containing eut such that fA ⊑ φ−1
ψ (Iτ∗(k,Cτ∗(k, gB, r), r)). Hence, φψ(fA)

⊑ Iτ∗(k,Cτ∗(k, gB, r), r). Then, φψ is fuzzy soft almost continuous.

Lemma 2. Every fuzzy soft continuous function [20] is fuzzy soft almost continuous.

Proof. It follows from Definitions 5 and 12.

Remark 8. In general, the converse of Lemma 2 is not true, as shown by Example 12.

Example 12. Let U = {u1, u2}, E = {e1, e2}, and define gE , fE ∈ (̃U,E) as follows:
gE = {(e1, { u10.4 ,

u2
0.5}), (e2, {

u1
0.4 ,

u2
0.5})}, fE = {(e1, { u10.3 ,

u2
0.4}), (e2, {

u1
0.3 ,

u2
0.4})}. Define fuzzy

soft topologies τE , τ
∗
E : E −→ [0, 1](̃U,E) as follows: ∀e ∈ E,

τe(mE) =


1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = gE ,

0, otherwise,

τ∗e (mE) =


1, if mE ∈ {Φ, Ẽ},
1
4 , if mE ∈ {fE , gE},
0, otherwise.

Thus, the identity fuzzy soft function φψ : (U, τE) −→ (U, τ∗E) is fuzzy soft almost
continuous, but it is not fuzzy soft continuous.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 16 of 21

Theorem 10. Let (U, τE) and (V, τ∗F ) be an FSTSs and φψ : (̃U,E) −→ (̃V, F ) be a

fuzzy soft function. Suppose that one of the following holds for every gB ∈ (̃V, F ), e ∈ E,
(k = ψ(e)) ∈ F , and r ∈ I◦:

(i) φ−1
ψ (gB) ⊑ Iτ (e, φ

−1
ψ (Cτ∗(k, gB, r)), r) if τ∗k (gB) ≥ r.

(ii) Cτ (e, φ
−1
ψ (Iτ∗(k, gB, r)), r) ⊑ φ−1

ψ (gB) if τ∗k (g
c
B) ≥ r.

Then, φψ is fuzzy soft weakly continuous.

Proof. (i) ⇒ (ii) Let gB ∈ (̃V, F ) with τ∗k (g
c
B) ≥ r. From (i), it follows

φ−1
ψ (gcB) ⊑ Iτ (e, φ

−1
ψ (Cτ∗(k, g

c
B, r)), r)

= Iτ (e, φ
−1
ψ ((Iτ∗(k, gB, r))

c), r)

= Iτ (e, (φ
−1
ψ (Iτ∗(k, gB, r)))

c, r)

= (Cτ (e, φ
−1
ψ (Iτ∗(k, gB, r)), r))

c.

Hence, Cτ (e, φ
−1
ψ (Iτ∗(k, gB, r)), r) ⊑ φ−1

ψ (gB). Similarly, we get (ii) ⇒ (i).

Suppose that (i) holds. Let eut ∈ P̃t(U) and gB ∈ (̃V, F ) with τ∗k (gB) ≥ r contain-

ing φψ(eut). Then, by (i), eut∈̃Iτ (e, φ−1
ψ (Cτ∗(k, gB, r)), r), and so there is fA ∈ (̃U,E)

with τe(fA) ≥ r containing eut such that fA ⊑ φ−1
ψ (Cτ∗(k, gB, r)). Thus, φψ(fA) ⊑

Cτ∗(k, gB, r). Hence, φψ is fuzzy soft weakly continuous.

Lemma 3. Every fuzzy soft almost continuous function is fuzzy soft weakly continuous.

Proof. It follows from Definition 12.

Remark 9. In general, the converse of Lemma 3 is not true, as shown by Example 13.

Example 13. Let U = {u1, u2, u3}, E = {e1, e2}, and define gE , fE ∈ (̃U,E) as follows:
gE = {(e1, { u10.6 ,

u2
0.6 ,

u3
0.5}), (e2, {

u1
0.6 ,

u2
0.6 ,

u3
0.5})}, fE = {(e1, { u10.3 ,

u2
0 ,

u3
0.5}), (e2, {

u1
0.3 ,

u2
0 ,

u3
0.5})}.

Define fuzzy soft topologies τE , τ
∗
E : E −→ [0, 1](̃U,E) as follows: ∀e ∈ E,

τe(mE) =


1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = gE ,

0, otherwise,

τ∗e (mE) =


1, if mE ∈ {Φ, Ẽ},
1
2 , if mE = fE ,

0, otherwise.

Thus, the identity fuzzy soft function φψ : (U, τE) −→ (U, τ∗E) is fuzzy soft weakly
continuous, but it is not fuzzy soft almost continuous.



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 17 of 21

Remark 10. From the previous results, we have: fuzzy soft continuity ⇒ fuzzy soft
almost continuity ⇒ fuzzy soft weakly continuity.

In [38], the difference between fA and gB is a fuzzy soft set defined as follows:

(fA ⊓ gB)(e) =

{
0, if fA(e) ≤ gB(e),

fA(e) ∧ (gB(e))
c, otherwise,

∀e ∈ E.

Let L and M : E × (̃U,E) × I◦ → (̃U,E) be operators on (̃U,E), and N and O
: F × (̃V, F )× I◦ → (̃V, F ) be operators on (̃V, F ).

Definition 13. Let (U, τE) and (V, τ∗F ) be an FSTSs. φψ : (̃U,E) −→ (̃V, F ) is said to be
a fuzzy soft (L,M,N ,O)-continuous function if

L[e, φ−1
ψ (O(k, gB, r)), r] ⊓ M[e, φ−1

ψ (N (k, gB, r)), r] = Φ

for each gB ∈ (̃V, F ) with τ∗k (gB) ≥ r, e ∈ E, and (k = ψ(e)) ∈ F .

In (2014), Aygünoǧlu et al. [20] introduced the concept of fuzzy soft continuous func-

tions: τe(φ
−1
ψ (gB)) ≥ τ∗k (gB), for each gB ∈ (̃V, F ), e ∈ E, and (k = ψ(e)) ∈ F . We can

see that Definition 4.2 generalizes the concept of fuzzy soft continuous functions when we
choose L = identity operator, M = interior operator, N = identity operator, and O =
identity operator.

A historical justification of Definition 13:

(1) In Section 3, we introduced the concept of fuzzy soft δ-continuous functions:

Iτ (e, Cτ (e, φ
−1
ψ (gB), r), r) ⊑ Cτ (e, Iτ (e, φ

−1
ψ (gB), r), r), for each gB ∈ (̃V, F ) with τ∗k (gB) ≥

r. Here, L = interior closure operator, M = closure interior operator, N = identity
operator, and O = identity operator.

(2) In Section 3, we introduced the concept of fuzzy soft β-continuous functions:

φ−1
ψ (gB) ⊑ Cτ (e, Iτ (e, Cτ (e, φ

−1
ψ (gB), r), r), r), for each gB ∈ (̃V, F ) with τ∗k (gB) ≥ r.

Here, L = identity operator, M = closure interior closure operator, N = identity opera-
tor, and O = identity operator.

(3) In Section 3, we introduced the concept of fuzzy soft semi-continuous functions:

φ−1
ψ (gB) ⊑ Cτ (e, Iτ (e, φ

−1
ψ (gB), r), r), for each gB ∈ (̃V, F ) with τ∗k (gB) ≥ r. Here, L =



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 18 of 21

identity operator, M = closure interior operator, N = identity operator, and O = identity
operator.

(4) In Section 3, we introduced the concept of fuzzy soft pre-continuous functions:

φ−1
ψ (gB) ⊑ Iτ (e, Cτ (e, φ

−1
ψ (gB), r), r), for each gB ∈ (̃V, F ) with τ∗k (gB) ≥ r. Here, L =

identity operator, M = interior closure operator, N = identity operator, and O = identity
operator.

(5) In Section 3, we introduced the concept of fuzzy soft α-continuous functions:

φ−1
ψ (gB) ⊑ Iτ (e, Cτ (e, Iτ (e, φ

−1
ψ (gB), r), r), r), for each gB ∈ (̃V, F ) with τ∗k (gB) ≥ r. Here,

L = identity operator, M = interior closure interior operator, N = identity operator, and
O = identity operator.

(6) In Section 4, we introduced the concept of fuzzy soft almost continuous functions:

φ−1
ψ (gB) ⊑ Iτ (e, φ

−1
ψ (Iτ∗(k,Cτ∗(k, gB, r), r)), r), for each gB ∈ (̃V, F ) with τ∗k (gB) ≥ r.

Here, L = identity operator, M = interior operator, N = interior closure operator, and
O = identity operator.

(7) In Section 4, we introduced the concept of fuzzy soft weakly continuous functions:

φ−1
ψ (gB) ⊑ Iτ (e, φ

−1
ψ (Cτ∗(k, gB, r)), r), for each gB ∈ (̃V, F ) with τ∗k (gB) ≥ r. Here, L

= identity operator, M = interior operator, N = closure operator, and O = identity
operator.

5. Conclusion and future work

In this paper, some new types of a fuzzy soft open set called an r-fuzzy soft δ-open
(semi-open) set have been introduced in an FSTSs based on the paper by Aygünoǧlu
et al. [20]. In addition, the concepts of fuzzy soft δ-closure (δ-interior) operators have
been introduced and some properties of them have been investigated. Furthermore, the
concept of r-fuzzy soft δ-connected sets has been defined and studied with the help of
fuzzy soft δ-closure operators. Thereafter, the concepts of fuzzy soft δ-continuous (semi-
continuous and pre-continuous) functions have been introduced and the relations of these
functions with each other have been specified with the help of some illustrative examples.
Moreover, a decomposition of fuzzy soft semi-continuity and a decomposition of fuzzy soft
α-continuity have been obtained.

In the end, as a weaker form of fuzzy soft continuity [20], the concepts of fuzzy soft
almost (weakly) continuous functions have been introduced and some properties have been
obtained. Also, we have shown that fuzzy soft continuity ⇒ fuzzy soft almost continuity
⇒ fuzzy soft weakly continuity. Moreover, we have explored the notion of continuity in
a very general setting namely fuzzy soft (L,M,N ,O)-continuous functions. It is also we
have the following results:



I. Alshammari et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5733 19 of 21

• Fuzzy soft (idU , Iτ , idV , idV )-continuous function is a fuzzy soft continuous function
[20].

• Fuzzy soft (Iτ (Cτ ), Cτ (Iτ ), idV , idV )-continuous function is a fuzzy soft δ-continuous
function.

• Fuzzy soft (idU , Cτ (Iτ (Cτ )), idV , idV )-continuous function is a fuzzy soft β-continuous
function.

• Fuzzy soft (idU , Cτ (Iτ ), idV , idV )-continuous function is a fuzzy soft semi-continuous
function.

• Fuzzy soft (idU , Iτ (Cτ ), idV , idV )-continuous function is a fuzzy soft pre-continuous
function.

• Fuzzy soft (idU , Iτ (Cτ (Iτ )), idV , idV ))-continuous function is a fuzzy soft α-continuous
function.

• Fuzzy soft (idU , Iτ , Iτ∗(Cτ∗), idV )-continuous function is a fuzzy soft almost contin-
uous function.

• Fuzzy soft (idU , Iτ , Cτ∗ , idV )-continuous function is a fuzzy soft weakly continuous
function.

In upcoming papers, we will use r-fuzzy soft δ-open sets to define and study some new
higher separation axioms and to introduce the concept of δ-compact spaces in an FSTSs
based on the paper by Aygünoǧlu et al. [20] Also, introducing these novel notions given
here in the frame of fuzzy soft ideals as defined in [34, 35, 37].

Acknowledgements

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

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