EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 4112-4134
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Fuzzy Soft α-open Sets, α-continuity, and
α-compactness: Some Novel Results

Wafa Alqurashi1, Islam M. Taha2,3,∗

1 Department of Mathematics, Faculty of Science, Umm Al-Qura University, Makkah,
Saudi Arabia
2 Department of Basic Sciences, Higher Institute of Engineering and Technology, Menoufia,
Egypt
3 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

Abstract. In this paper, we defined the notions of fuzzy soft α-interior (α-closure) operators via
fuzzy soft topologies based on the sense of Šostak and studied some topological properties of them.
Also, the notion of r-fuzzy soft α-connected sets was introduced and investigated. Thereafter,
we defined and characterized the notions of fuzzy soft weakly (almost) α-continuous mappings,
which are weaker forms of fuzzy soft α-continuous mappings. Moreover, we showed that fuzzy soft
α-continuity ⇒ fuzzy soft almost α-continuity ⇒ fuzzy soft weakly α-continuity, but the converse
may not be true. In addition, we investigated some properties of fuzzy soft α-continuity. Finally,
several types of fuzzy soft compactness via r-fuzzy soft α-open sets were given and the relationships
between them were studied with the help of some examples.

2020 Mathematics Subject Classifications: 54A05, 54A40, 54C05, 54C10, 54D30

Key Words and Phrases: Fuzzy soft topology, r-fuzzy soft α-open (α-closed) set, fuzzy soft α-
interior (α-closure) operator, connectedness, fuzzy soft weakly (almost) α-continuity, compactness

1. Introduction and preliminaries

The theory of soft sets was first introduced by Molodtsov [24], which is a completely
new approach for vagueness and modeling uncertainty. He demonstrated many appli-
cations of this theory in solving several practical problems in mathematics, engineering,
economics, social science, etc. In [28], the notion of soft sets was used to introduced soft
topologies. Moreover, the study in [28] was particularly important in the development of
the field of soft topology, see [10, 18, 33, 38]. Generalizations of soft open subsets play an
effective role in soft topologies through their use to improve on some known results or to
open the door to reintroduce and establish many of the soft topological notions such as
soft separation axioms [7, 20], soft continuity [25], soft connectedness [34, 36], etc. Akdag

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5330

Email addresses: wkqurashi@uqu.edu.sa (W. Alqurashi), imtaha2010@yahoo.com (I. M. Taha)

https://www.ejpam.com 4112 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4113

and Ozkan [3] defined the notion of soft α-open sets on soft topological spaces and some
properties are specified. The notion of soft β-open sets was defined and studied by the
authors of [2, 17]. Also, the concepts of soft semi-open, somewhere dense and Q-sets were
studied by the authors of [4, 6]. Moreover, Al-shami et al. [8] initiated the notion of
weakly soft β-open sets and examined weakly soft β-continuity. Kaur et al. [21] intro-
duced a new approach to studying soft continuous mappings using an induced mapping
based on soft sets. Al Ghour and Al-Mufarrij [5] defined two new notions of mappings
over soft topological spaces: soft somewhat-r-continuity and soft somewhat-r-openness.

The concept of fuzzy soft sets was defined by Maji et al. [22], which combines soft
sets [24] and fuzzy sets [37]. The concept of fuzzy soft topology was introduced and some
characterized such as fuzzy soft interior (closure) set, fuzzy soft continuity, and fuzzy soft
subspace were studied in [16, 19] based on fuzzy topologies in the sense of Šostak [35].
A new approach to studying separation and regularity axioms via fuzzy soft sets was
introduced by the author of [29, 31] based on the paper by Aygünoǧlu et al. [19]. The
concept of r-fuzzy soft regularly open sets was introduced by Çetkin and Aygün [15]. Also,
the concepts of r-fuzzy soft pre-open (resp. β-open) sets were defined by Taha [30]. In
2024, Alshammari and Taha [12] introduced and studied the notions of fuzzy soft almost
(weakly) β-continuous mappings, which are weaker forms of a fuzzy soft β-continuity in
fuzzy soft topological spaces. In addition, many authors have contributed to fuzzy soft set
theory in the different fields such as topology, see e.g. [9, 26, 27].

In our study, the layout is designed as follows.

• In Section 2, we introduce the concepts of fuzzy soft α-closure (α-interior) operators
in fuzzy soft topological space (W, τN ) based on the paper by Aygünoǧlu et al. [19] and
examine some of its properties. Also, the concept of r-fuzzy soft α-connected sets is
introduced and studied.

• In Section 3, we are going to investigate some properties of fuzzy soft α-continuous
mappings between two fuzzy soft topological spaces (W, τN ) and (V, ηF ). Moreover, we
define and study the concepts of fuzzy soft weakly (almost) α-continuous mappings, which
are weaker forms of fuzzy soft α-continuous mappings. Also, the relationships between
these classes of mappings are investigated with the help of some examples.

• In Section 4, several types of fuzzy soft compactness via r-fuzzy soft α-open sets are
defined, and the relationships between them are specified.

• Finally, we close this manuscript with some conclusions and proposed some future
works in Section 5.

In this work, nonempty sets will be denoted byW , V , etc. N is the set of all parameters
for W and C ⊆ N . The family of all fuzzy sets on W is denoted by IW (where I◦ =
(0, 1], I = [0, 1]), and for s ∈ I, s(w) = s, for all w ∈W.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4114

The following concepts and results will be used in the next sections.

Definition 1. [1, 14, 19] A fuzzy soft set hC on W is a mapping from N to IW , such
that hC(n) is a fuzzy set on W , for each n ∈ C and hC(n) = 0, if n ̸∈ C. The family of

all fuzzy soft sets on W is denoted by ˜(W,N).

Definition 2. [32] The difference between two fuzzy soft sets hC and gB is a fuzzy soft
set, defined as follows, for each n ∈ N :

(hC ⊓ gB)(n) =

{
0, if hC(n) ≤ gB(n),

hC(n) ∧ (gB(n))
c, otherwise.

Definition 3. [23] A fuzzy soft point nws on W is a fuzzy soft set, defined as follows:

nws(k) =

{
ws, if k = n,
0, if k ∈ N − {n},

where ws is a fuzzy point on W . A fuzzy soft point nws is called belong to a fuzzy soft
set fA, denoted by nws∈̃fA, if s ≤ fA(n)(w). The family of all fuzzy soft points on W is

denoted by P̃s(W ).

Definition 4. [13] A fuzzy soft point nws ∈ P̃s(W ) is called a soft quasi-coincident with

hC ∈ ˜(W,N) and denoted by nws q̃hC , if s+hC(n)(w) > 1. A fuzzy soft set hC ∈ ˜(W,N)

is called a soft quasi-coincident with gB ∈ ˜(W,N) and denoted by hC q̃gB, if there is n ∈ N
and w ∈W , such that hC(n)(w) + gB(n)(w) > 1, if hC is not soft quasi-coincident with
gB, hC ̸ q̃gB.

Definition 5. [19] A mapping τ : N −→ [0, 1]
˜(W,N) is called a fuzzy soft topology on W

if it satisfies the following, for each n ∈ N :

(1) τn(Φ) = τn(Ñ) = 1,

(2) τn(hC ⊓ gB) ≥ τn(hC) ∧ τn(gB), for each hC , gB ∈ ˜(W,N),

(3) τn(⊔δ∈∆(hC)δ) ≥ ∧δ∈∆τn((hC)δ), for each (hC)δ ∈ ˜(W,N), δ ∈ ∆.

Thus, (W, τN ) is called a fuzzy soft topological space (briefly, FSTS) in the sense of
Šostak [35].



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4115

Definition 6. [19] Let (W, τN ) and (V, ηF ) be an FSTSs. A fuzzy soft mapping φψ :

˜(W,N) −→ (̃V, F ) is called fuzzy soft continuous if τn(φ
−1
ψ (hC)) ≥ ηk(hC) for each hC ∈

(̃V, F ), n ∈ N , and (k = ψ(n)) ∈ F .

Definition 7. [15, 16] In an FSTS (W, τN ), for each hC ∈ ˜(W,N), n ∈ N , and r ∈ I0, we

define the fuzzy soft operators Cτ and Iτ : N × ˜(W,N)× I◦ → ˜(W,N) as follows:

Cτ (n, hC , r) = ⊓ {gB ∈ ˜(W,N) : hC ⊑ gB, τn(g
c
B) ≥ r},

Iτ (n, hC , r) = ⊔ {gB ∈ ˜(W,N) : gB ⊑ hC , τn(gB) ≥ r}.

Definition 8. Let (W, τN ) be an FSTS and r ∈ I0. A fuzzy soft set hC ∈ ˜(W,N) is called
r-fuzzy soft regularly open [15] (resp., β-open [30], pre-open [30], α-open [11], and semi-
open [11]) if hC = Iτ (n,Cτ (n, hC , r), r) (resp., hC ⊑ Cτ (n, Iτ (n,Cτ (n, hC , r), r), r), hC ⊑
Iτ (n,Cτ (n, hC , r), r), hC ⊑ Iτ (n,Cτ (n, Iτ (n, hC , r), r), r), and hC ⊑ Cτ (n, Iτ (n, hC , r), r))
for each n ∈ N .

Definition 9. [15] Let (W, τN ) be an FSTS and r ∈ I0. A fuzzy soft set hC ∈ ˜(W,N) is
called r-fuzzy soft regularly closed if hC = Cτ (n, Iτ (n, hC , r), r) for each n ∈ N .

Definition 10. [11] Let (W, τN ) and (V, ηF ) be an FSTSs and r ∈ I0. A fuzzy soft

mapping φψ : ˜(W,N) −→ (̃V, F ) is called fuzzy soft almost (resp., weakly) continuous if

for any nws ∈ P̃s(W ) and any fA ∈ (̃V, F ) with ηk(fA) ≥ r containing φψ(nws), there is

hC ∈ ˜(W,N) with τn(hC) ≥ r containing nws , such that φψ(hC) ⊑ Iη(k,Cη(k, fA, r), r)
(resp., φψ(hC) ⊑ Cη(k, fA, r)).

Remark 1. [11] From Definitions 6 and 10, we have: fuzzy soft continuity ⇒ fuzzy soft
almost continuity ⇒ fuzzy soft weakly continuity, but the converse may not be true.

Lemma 1. Let (W, τN ) and (V, ηF ) be an FSTSs and r ∈ I0. A fuzzy soft mapping

φψ : ˜(W,N) −→ (̃V, F ) is fuzzy soft almost continuous if τn(φ
−1
ψ (hC)) ≥ r for each

hC ∈ (̃V, F ) is r-fuzzy soft regularly open, n ∈ N , and (k = ψ(n)) ∈ F .

Proof. Easily proved from Definition 10.

Definition 11. Let (W, τN ) and (V, ηF ) be an FSTSs. A fuzzy soft mapping φψ :

˜(W,N) −→ (̃V, F ) is called fuzzy soft open if ηk(φψ(hC)) ≥ τn(hC) for each hC ∈ ˜(W,N),
n ∈ N , and (k = ψ(n)) ∈ F .

The basic concepts and results that we need in the next sections are found in [16, 19].



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4116

2. On r-fuzzy soft α-open sets

Here, we introduce and discuss the notions of fuzzy soft α-closure (α-interior) operators
in an FSTSs based on the paper by Aygünoǧlu et al. [19]. Also, the notion of r-fuzzy soft
α-connected sets has been defined and studied with help of fuzzy soft α-closure operators.

Definition 12. Let (W, τN ) be an FSTS and r ∈ I0. A fuzzy soft set hC is called r-fuzzy
soft α-closed (resp., semi-closed, β-closed, and pre-closed) if Cτ (n, Iτ (n,Cτ (n, hC , r), r), r)
⊑ hC (resp., Iτ (n,Cτ (n, hC , r), r) ⊑ hC , Iτ (n,Cτ (n, Iτ (n, hC , r), r), r) ⊑ hC , and
Cτ (n, Iτ (n, hC , r), r) ⊑ hC) for each n ∈ N .

Remark 2. The complement of r-fuzzy soft α-closed (resp., semi-closed, β-closed, and
pre-closed) set is r-fuzzy soft α-open [11] (resp., semi-open [11], β-open [30], and pre-open
[30]) set.

Lemma 2. Let (W, τN ) be an FSTS and r ∈ I0, then any intersection (resp., union) of
r-fuzzy soft α-closed (resp., α-open) sets is an r-fuzzy soft α-closed (resp., α-open) set.

Proof. Easily proved from Definitions 8 and 12.

Proposition 1. Let (W, τN ) be an FSTS, hC ∈ ˜(W,N), n ∈ N , and r ∈ I0, then the
following statements are equivalent.

(1) hC is r-fuzzy soft α-closed.

(2) hC is r-fuzzy soft semi-closed and r-fuzzy soft pre-closed.

Proof. (1) ⇒ (2) Let hC be an r-fuzzy soft α-closed,

hC ⊒ Cτ (n, Iτ (n,Cτ (n, hC , r), r), r) ⊒ Iτ (n,Cτ (n, hC , r), r).

This shows that hC is r-fuzzy soft semi-closed.

Since hC ⊒ Cτ (n, Iτ (n,Cτ (n, hC , r), r), r) and Cτ (n, hC , r) ⊒ hC , then

hC ⊒ Cτ (n, Iτ (n, hC , r), r).

Therefore, hC is r-fuzzy soft pre-closed

(2) ⇒ (1) Let hC be an r-fuzzy soft semi-closed and r-fuzzy soft pre-closed, then
hC ⊒ Cτ (n, Iτ (n, Iτ (n,Cτ (n, hC , r), r), r), r) = Cτ (n, Iτ (n,Cτ (n, hC , r), r), r). This shows
that hC is r-fuzzy soft α-closed.

Proposition 2. Let (W, τN ) be an FSTS, gB, hC ∈ ˜(W,N), n ∈ N , and r ∈ I0. If gB
is an r-fuzzy soft semi-closed set, such that gB ⊒ hC ⊒ Cτ (n, Iτ (n, gB, r), r), then hC is
r-fuzzy soft α-closed.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4117

Proof. Let gB be an r-fuzzy soft semi-closed and gB ⊒ hC , then

gB ⊒ Iτ (n,Cτ (n, gB, r), r) ⊒ Iτ (n,Cτ (n, hC , r), r).

Let hC ⊒ Cτ (n, Iτ (n, gB, r), r), then

hC ⊒ Cτ (n, Iτ (n, Iτ (n,Cτ (n, hC , r), r), r), r) = Cτ (n, Iτ (n,Cτ (n, hC , r), r), r).

Therefore, hC is r-fuzzy soft α-closed.

Lemma 3. Let (W, τN ) be an FSTS, gB, hC ∈ ˜(W,N), n ∈ N , and r ∈ I0. If gB is an
r-fuzzy soft α-closed set, such that gB ⊒ hC ⊒ Cτ (n, Iτ (n, gB, r), r), then hC is r-fuzzy
soft α-closed.

Proof. It is easily proved from every r-fuzzy soft α-closed set that is an r-fuzzy soft
semi-closed set.

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

fuzzy soft α− closed set

⇓ ⇓
fuzzy soft semi−closed set fuzzy soft pre−closed set

⇓ ⇓
fuzzy soft β − closed set

Remark 4. The converses of the above relationships may not be true, as shown by
Examples 1 and 2.

Example 1. Let W = {w1, w2}, N = {n1, n2}, and define fN , gN , hN ∈ ˜(W,N) as
follows: fN = {(n1, {w1

0.3 ,
w2
0.4}), (n2, {

w1
0.3 ,

w2
0.4})}, gN = {(n1, {w1

0.6 ,
w2
0.2}), (n2, {

w1
0.6 ,

w2
0.2})},

hN = {(n1, {w1
0.3 ,

w2
0.5}), (n2, {

w1
0.3 ,

w2
0.5})}. Define fuzzy soft topology τN : N −→ [0, 1]

˜(W,N)

as follows:

τn1(tN ) =



1, if tN ∈ {Φ, Ñ},
1
2 , if tN = fN ,
2
3 , if tN = gN ,
2
3 , if tN = fN ⊓ gN ,
1
2 , if tN = fN ⊔ gN ,
0, otherwise,

τn2(tN ) =



1, if tN ∈ {Φ, Ñ},
1
4 , if tN = fN ,
1
2 , if tN = gN ,
1
2 , if tN = fN ⊓ gN ,
1
4 , if tN = fN ⊔ gN ,
0, otherwise.

Thus, hN is 1
4 -fuzzy soft semi-closed and 1

4 -fuzzy soft β-closed, but it is neither 1
4 -fuzzy

soft α-closed nor 1
4 -fuzzy soft pre-closed.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4118

Example 2. Let W = {w1, w2}, N = {n1, n2}, and define fN , hN ∈ ˜(W,N) as fol-
lows: fN = {(n1, {w1

0.4 ,
w2
0.5}), (n2, {

w1
0.4 ,

w2
0.5})}, hN = {(n1, {w1

0.7 ,
w2
0.6}), (n2, {

w1
0.7 ,

w2
0.6})}. De-

fine fuzzy soft topology τN : N −→ [0, 1]
˜(W,N) as follows:

τn1(tN ) =


1, if tN ∈ {Φ, Ñ},
1
3 , if tN = fN ,

0, otherwise,

τn2(tN ) =


1, if tN ∈ {Φ, Ñ},
1
2 , if tN = fN ,

0, otherwise.

Thus, hN is 1
3 -fuzzy soft pre-closed and 1

3 -fuzzy soft β-closed, but it is neither 1
3 -fuzzy

soft semi-closed nor 1
3 -fuzzy soft α-closed.

Definition 13. In an FSTS (W, τN ), for each hC ∈ ˜(W,N), n ∈ N , and r ∈ I0, we define a

fuzzy soft α-closure operator αCτ : N × ˜(W,N)× I◦ → ˜(W,N) as follows: αCτ (n, hC , r) =

⊓ {fA ∈ ˜(W,N) : hC ⊑ fA, fA is r-fuzzy soft α-closed}.

Theorem 1. In an FSTS (W, τN ), for each gB, hC ∈ ˜(W,N), n ∈ N , and r ∈ I0, the

operator αCτ : N × ˜(W,N)× I◦ → ˜(W,N) satisfies the following properties.

(1) αCτ (n,Φ, r) = Φ.

(2) hC ⊑ αCτ (n, hC , r) ⊑ Cτ (n, hC , r).

(3) αCτ (n, hC , r) ⊑ αCτ (n, gB, r) if, hC ⊑ gB.

(4) αCτ (n, αCτ (n, hC , r), r) = αCτ (n, hC , r).

(5) αCτ (n, hC ⊔ gB, r) ⊒ αCτ (n, hC , r) ⊔ αCτ (n, gB, r).

(6) αCτ (n, hC , r) = hC iff hC is r-fuzzy soft α-closed.

(7) αCτ (n,Cτ (n, hC , r), r) = Cτ (n, hC , r).

Proof. (1), (2), (3), and (6) are easily proved from Definition 13.

(4) From (2) and (3), αCτ (n, hC , r) ⊑ αCτ (n, αCτ (n, hC , r), r). Now, we show that
αCτ (n, hC , r) ⊒ αCτ (n, αCτ (n, hC , r), r). Suppose that αCτ (n, hC , r) does not contain
αCτ (n, αCτ (n, hC , r), r), then there is w ∈W and s ∈ (0, 1), such that

αCτ (n, hC , r)(n)(w) < s < αCτ (n, αCτ (n, hC , r), r)(n)(w). (A)

Since αCτ (n, hC , r)(n)(w) < s, by the definition of αCτ , there is gB as an r-fuzzy
soft α-closed with hC ⊑ gB, such that αCτ (n, hC , r)(n)(w) ≤ gB(n)(w) < s. Since



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4119

hC ⊑ gB, we have αCτ (n, hC , r) ⊑ gB. Again, by the definition of αCτ , we have
αCτ (n, αCτ (n, hC , r), r) ⊑ gB. Hence, αCτ (n, αCτ (n, hC , r), r)(n)(w) ≤ gB(n)(w) < s,
which is a contradiction for (A). Thus, αCτ (n, hC , r) ⊒ αCτ (n, αCτ (n, hC , r), r), then
αCτ (n, αCτ (n, hC , r), r) = αCτ (n, hC , r).

(5) Since hC and gB ⊑ hC ⊔ gB, hence by (3), αCτ (n, hC , r) ⊑ αCτ (n, hC ⊔ gB, r)
and αCτ (n, gB, r) ⊑ αCτ (n, hC ⊔ gB, r). Thus, αCτ (n, hC ⊔ gB, r) ⊒ αCτ (n, hC , r) ⊔
αCτ (n, gB, r).

(7) From (6) and Cτ (n, hC , r) is r-fuzzy soft α-closed set, then αCτ (n,Cτ (n, hC , r), r) =
Cτ (n, hC , r).

Theorem 2. In an FSTS (W, τN ), for each hC ∈ ˜(W,N), n ∈ N , and r ∈ I0, we define a

fuzzy soft α-interior operator αIτ : N × ˜(W,N)× I◦ → ˜(W,N) as follows: αIτ (n, hC , r) =

⊔ {fA ∈ ˜(W,N) : fA ⊑ hC , fA is r-fuzzy soft α-open}. For each gB and hC ∈ ˜(W,N), the
operator αIτ satisfies the following properties.

(1) αIτ (n, Ñ , r) = Ñ .

(2) Iτ (n, hC , r) ⊑ αIτ (n, hC , r) ⊑ hC .

(3) αIτ (n, hC , r) ⊑ αIτ (n, gB, r) if, hC ⊑ gB.

(4) αIτ (n, αIτ (n, hC , r), r) = αIτ (n, hC , r).

(5) αIτ (n, hC , r) ⊓ αIτ (n, gB, r) ⊒ αIτ (n, hC ⊓ gB, r).

(6) αIτ (n, hC , r) = hC iff hC is r-fuzzy soft α-open.

(7) αIτ (n, h
c
C , r) = (αCτ (n, hC , r))

c.

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

(4) and (5) are easily proved by a similar way in Theorem 1.

(7) For each hC ∈ ˜(W,N), n ∈ N , and r ∈ I0, we have αIτ (n, h
c
C , r) = ⊔{fA ∈ ˜(W,N) :

fA ⊑ hcC , fA is r-fuzzy soft α-open}= [⊓{f cA ∈ ˜(W,N) : hC ⊑ f cA, f
c
A is r-fuzzy soft α-closed}]c

= (αCτ (n, hC , r))
c.

Definition 14. Let (W, τN ) be an FSTS, r ∈ I0, and gB, hC ∈ ˜(W,N), then we have



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4120

(1) Two fuzzy soft sets gB and hC are called r-fuzzy soft α-separated iff gB ̸ q̃ αCτ (n, hC , r)
and hC ̸ q̃ αCτ (n, gB, r) for each n ∈ N .

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

Theorem 3. In an FSTS (W, τN ), we have:

(1) If fA and gB ∈ ˜(W,N) are r-fuzzy soft α-separated and hC , tD ∈ ˜(W,N) such that
hC ⊑ fA and tD ⊑ gB, then hC and tD are r-fuzzy soft α-separated.

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

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

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

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

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

Theorem 4. In an FSTS (W, τN ), then fA, gB ∈ ˜(W,N) are r-fuzzy soft α-separated iff
there exist two r-fuzzy soft α-open sets hC and tD such that fA ⊑ hC , gB ⊑ tD, fA ̸ q̃ tD,
and gB ̸ q̃ hC .

Proof. (⇒) Let fA and gB ∈ ˜(W,N) be an r-fuzzy soft α-separated, fA ⊑ (αCτ (n, gB, r))
c =

hC and gB ⊑ (αCτ (n, fA, r))
c = tD, where tD and hC are r-fuzzy soft α-open, then

tD ̸ q̃αCτ (n, fA, r) and hC ̸ q̃αCτ (n, gB, r). Thus, gB ̸ q̃ hC and fA ̸ q̃ tD. Hence, we obtain
the required result.

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

Theorem 5. In an FSTS (W, τN ), if gB ∈ ˜(W,N) is r-fuzzy soft α-connected such that
gB ⊑ fA ⊑ αCτ (n, gB, r), then fA is r-fuzzy soft α-connected.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4121

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

α-separated sets h∗C and t∗D ∈ ˜(W,N) 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, by Theorem 3(1), hC
and tD are r-fuzzy soft α-separated, which is a contradiction. Thus, fA is r-fuzzy soft
α-connected, as required.

3. On fuzzy soft α-continuity

Here, we investigate some properties of fuzzy soft α-continuous mappings. Addition-
ally, we introduce and study the notions of fuzzy soft almost (weakly) α-continuous map-
pings, which are weaker forms of fuzzy soft α-continuous mappings. Also, we show that
fuzzy soft α-continuity ⇒ fuzzy soft almost α-continuity ⇒ fuzzy soft weakly α-continuity.

Definition 15. [11] Let (W, τN ) and (V, ηF ) be an FSTSs and r ∈ Io. A fuzzy soft

mapping φψ : ˜(W,N) −→ (̃V, F ) is called fuzzy soft α-continuous if φ−1
ψ (hC) is r-fuzzy

soft α-open set for each hC ∈ (̃V, F ) with ηk(hC) ≥ r, n ∈ N , and (k = ψ(n)) ∈ F .

Theorem 6. Let (W, τN ) and (V, ηF ) be an FSTSs, and φψ : ˜(W,N) −→ (̃V, F ) be a

fuzzy soft mapping. The following statements are equivalent for each fA ∈ (̃V, F ), n ∈ N ,
(k = ψ(n)) ∈ F , and r ∈ I◦:

(1) φψ is fuzzy soft α-continuous.

(2) For each fA with ηk(f
c
A) ≥ r, φ−1

ψ (fA) is r-fuzzy soft α-closed.

(3) αCτ (n, φ
−1
ψ (fA), r) ⊑ φ−1

ψ (Cη(k, fA, r)).

(4) φ−1
ψ (Iη(k, fA, r)) ⊑ αIτ (n, φ

−1
ψ (fA), r).

(5) Cτ (n, Iτ (n,Cτ (n, φ
−1
ψ (fA), r), r), r) ⊑ φ−1

ψ (Cη(k, fA, r)).

Proof.
(1) ⇔ (2) Follows from Remark 2 and φ−1

ψ (f cA) = (φ−1
ψ (fA))

c.

(2) ⇒ (3) Let fA ∈ (̃V, F ), hence by (2), φ−1
ψ (Cη(k, fA, r)) is r-fuzzy soft α-closed.

Then, we obtain αCτ (n, φ
−1
ψ (fA), r) ⊑ φ−1

ψ (Cη(k, fA, r)).

(3) ⇔ (4) Follows from Theorem 2(7).



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4122

(3)⇒ (5) Let fA ∈ (̃V, F ), hence by (3), we obtain Cτ (n, Iτ (n,Cτ (n, φ
−1
ψ (fA), r), r), r) ⊑

αCτ (n, φ
−1
ψ (fA), r) ⊑ φ−1

ψ (Cη(k, fA, r)).

(5) ⇒ (1) Let fA ∈ (̃V, F ) with ηk(fA) ≥ r, hence by (3), we obtain (φ−1
ψ (fA))

c =

φ−1
ψ (f cA) ⊒ Cτ (n, Iτ (n,Cτ (n, φ

−1
ψ (f cA), r), r), r) = (Iτ (n,Cτ (n, Iτ (n, φ

−1
ψ (fA), r), r), r))

c.

Then, φ−1
ψ (fA) ⊑ Iτ (n,Cτ (n, Iτ (n, φ

−1
ψ (fA), r), r), r), so φ

−1
ψ (fA) is r-fuzzy soft α-open.

Hence, φψ is fuzzy soft α-continuous.

Lemma 4. Every fuzzy soft continuous mapping [19] is fuzzy soft α-continuous.

Proof. Follows from Definitions 6 and 15.

Remark 5. The converse of Lemma 4 is not true, as shown by Example 3.

Example 3. Let W = {w1, w2, w3}, N = {n1, n2}, and define fN , gN , hN ∈ ˜(W,N) as:
fN = {(n1, {w1

0.4 ,
w2
0.5 ,

w3
0.5}), (n2, {

w1
0.4 ,

w2
0.5 ,

w3
0.5})}, gN = {(n1, {w1

0.3 ,
w2
0.3 ,

w3
0.4}), (n2, {

w1
0.3 ,

w2
0.3 ,

w3
0.4})},

hN = {(n1, {w1
0.3 ,

w2
0.4 ,

w3
0.4}), (n2, {

w1
0.3 ,

w2
0.4 ,

w3
0.4})}. Define fuzzy soft topologies τN , ηN : N −→

[0, 1]
˜(W,N) as follows: ∀n ∈ N ,

τn(tN ) =


1, if tN ∈ {Φ, Ñ},
1
2 , if tN = fN ,
2
3 , if tN = gN ,

0, otherwise,

ηn(tN ) =


1, if tN ∈ {Φ, Ñ},
1
2 , if tN = fN ,
1
3 , if tN = hN ,

0, otherwise.

Thus, the identity fuzzy soft mapping φψ : (W, τN ) −→ (W, ηN ) is fuzzy soft α-
continuous, but it is not fuzzy soft continuous.

Definition 16. Let (W, τN ) and (V, ηF ) be an FSTSs. A fuzzy soft mapping φψ :

˜(W,N) −→ (̃V, F ) is called fuzzy soft almost (resp., weakly) α-continuous if for each nws ∈
P̃s(W ) and each gB ∈ (̃V, F ) with ηk(gB) ≥ r containing φψ(nws), there is hC ∈ ˜(W,N)
that is an r-fuzzy soft α-open set containing nws , such that φψ(hC) ⊑ Iη(k,Cη(k, gB, r), r)
(resp., φψ(hC) ⊑ Cη(k, gB, r)), n ∈ N , (k = ψ(n)) ∈ F , and r ∈ I◦.

Lemma 5. (1) Every fuzzy soft α-continuous mapping is fuzzy soft almost α-continuous.

(2) Every fuzzy soft almost α-continuous mapping is fuzzy soft weakly α-continuous.

Proof. Follows from Definitions 15 and 16.

Remark 6. The converse of Lemma 5 is not true, as shown by Examples 4 and 5.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4123

Example 4. LetW = {w1, w2, w3}, N = {n1, n2}, and define gN , hN ∈ ˜(W,N) as follows:
gN = {(n1, {w1

0.5 ,
w2
0.5 ,

w3
0.4}), (n2, {

w1
0.5 ,

w2
0.5 ,

w3
0.4})}, hN = {(n1, {w1

0.3 ,
w2
0.3 ,

w3
0.4}), (n2, {

w1
0.3 ,

w2
0.3 ,

w3
0.4})}.

Define fuzzy soft topologies τN , ηN : N −→ [0, 1]
˜(W,N) as follows: ∀n ∈ N ,

τn(tN ) =


1, if tN ∈ {Φ, Ñ},
1
2 , if tN = gN ,

0, otherwise,

ηn(tN ) =


1, if tN ∈ {Φ, Ñ},
1
2 , if tN = gN ,
1
3 , if tN = hN ,

0, otherwise.

Thus, the identity fuzzy soft mapping φψ : (W, τN ) −→ (W, ηN ) is fuzzy soft almost
α-continuous, but it is not fuzzy soft α-continuous.

Example 5. LetW = {w1, w2, w3}, N = {n1, n2}, and define gN , hN ∈ ˜(W,N) as follows:
gN = {(n1, {w1

0.5 ,
w2
0.5 ,

w3
0.5}), (n2, {

w1
0.5 ,

w2
0.5 ,

w3
0.5})}, hN = {(n1, {w1

0.3 ,
w2
0 ,

w3
0.5}), (n2, {

w1
0.3 ,

w2
0 ,

w3
0.5})}.

Define fuzzy soft topologies τN , ηN : N −→ [0, 1]
˜(W,N) as follows: ∀n ∈ N ,

τn(tN ) =


1, if tN ∈ {Φ, Ñ},
2
3 , if tN = gN ,

0, otherwise,

ηn(tN ) =


1, if tN ∈ {Φ, Ñ},
1
3 , if tN = hN ,

0, otherwise.

Thus, the identity fuzzy soft mapping φψ : (W, τN ) −→ (W, ηN ) is fuzzy soft weakly
α-continuous, but it is not fuzzy soft almost α-continuous.

Theorem 7. Let (W, τN ) and (V, ηF ) be an FSTSs, and φψ : ˜(W,N) −→ (̃V, F ) be a

fuzzy soft mapping. The following statements are equivalent for each fA ∈ (̃V, F ), n ∈ N ,
(k = ψ(n)) ∈ F , and r ∈ I◦:

(1) φψ is fuzzy soft almost α-continuous.

(2) φ−1
ψ (fA) is r-fuzzy soft α-open, for each fA is r-fuzzy soft regularly open.

(3) φ−1
ψ (fA) is r-fuzzy soft α-closed, for each fA is r-fuzzy soft regularly closed.

(4) αCτ (n, φ
−1
ψ (fA), r) ⊑ φ−1

ψ (Cη(k, fA, r)), for each fA is r-fuzzy soft β-open.

(5) αCτ (n, φ
−1
ψ (fA), r) ⊑ φ−1

ψ (Cη(k, fA, r)), for each fA is r-fuzzy soft semi-open.

(6) αIτ (n, φ
−1
ψ (Iη(k,Cη(k, fA, r), r)), r) ⊒ φ−1

ψ (fA), for each fA with ηk(fA) ≥ r.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4124

Proof. (1) ⇒ (2) Let nws ∈ P̃s(W ) and fA ∈ (̃V, F ) be an r-fuzzy soft regularly open

set containing φψ(nws), hence by (1), there is hC ∈ ˜(W,N) is r-fuzzy soft α-open set
containing nws such that φψ(hC) ⊑ Iη(k,Cη(k, fA, r), r).

Thus, hC ⊑ φ−1
ψ (Iη(k,Cη(k, fA, r), r)) = φ−1

ψ (fA) and nws∈̃hC ⊑ φ−1
ψ (fA). Then,

nws∈̃Iτ (n,Cτ (n, Iτ (n, φ−1
ψ (fA), r), r), r) and φ

−1
ψ (fA) ⊑ Iτ (n,Cτ (n, Iτ (n, φ

−1
ψ (fA), r), r), r).

Therefore, φ−1
ψ (fA) is r-fuzzy soft α-open set.

(2) ⇒ (3) Let fA be an r-fuzzy soft regularly closed set, hence by (2), φ−1
ψ (f cA) =

(φ−1
ψ (fA))

c is r-fuzzy soft α-open set. Then, φ−1
ψ (fA) is r-fuzzy soft α-closed set.

(3) ⇒ (4) Let fA be an r-fuzzy soft β-open set. Since Cη(k, fA, r) is r-fuzzy soft
regularly closed set, hence by (3), φ−1

ψ (Cη(k, fA, r)) is r-fuzzy soft α-closed set. Since

φ−1
ψ (fA) ⊑ φ−1

ψ (Cη(k, fA, r)), then we have αCτ (n, φ
−1
ψ (fA), r) ⊑ φ−1

ψ (Cη(k, fA, r)).

(4) ⇒ (5) This is obvious from each r-fuzzy soft semi-open set that is an r-fuzzy soft
β-open.

(5) ⇒ (3) Let fA be an r-fuzzy soft regularly closed set, hence fA is r-fuzzy soft
semi-open. Then by (5), αCτ (n, φ

−1
ψ (fA), r) ⊑ φ−1

ψ (Cη(k, fA, r)) = φ−1
ψ (fA). Therefore,

φ−1
ψ (fA) is r-fuzzy soft α-closed set.

(3) ⇒ (6) Let fA ∈ (̃V, F ) with ηk(fA) ≥ r and nws∈̃φ−1
ψ (fA), then we have

nws∈̃φ−1
ψ (Iη(k,Cη(k, fA, r), r)).

Since [Iη(k,Cη(k, fA, r), r)]
c is r-fuzzy soft regularly closed set, φ−1

ψ ([Iη(k,Cη(k, fA, r), r)]
c)

is r-fuzzy soft α-closed set (from (3)). Thus, φ−1
ψ (Iη(k,Cη(k, fA, r), r)) is r-fuzzy soft α-

open set and nws∈̃αIτ (n, φ−1
ψ (Iη(k,Cη(k, fA, r), r)), r). Then,

φ−1
ψ (fA) ⊑ αIτ (n, φ

−1
ψ (Iη(k,Cη(k, fA, r), r)), r).

(6) ⇒ (1) Let nws ∈ P̃s(W ) and fA ∈ (̃V, F ) with ηk(fA) ≥ r containing φψ(nws),
hence by (6), φ−1

ψ (fA) ⊑ αIτ (n, φ
−1
ψ (Iη(k,Cη(k, fA, r), r)), r).

Since nws∈̃φ−1
ψ (fA), then we obtain nws∈̃αIτ (n, φ−1

ψ (Iη(k,Cη(k, fA, r), r)), r) = hC

(say). Hence, there is hC ∈ ˜(W,N) is r-fuzzy soft α-open set containing nws such that
φψ(hC) ⊑ Iη(k,Cη(k, fA, r), r). Therefore, φψ is fuzzy soft almost α-continuous.

In a similar way, we can prove the following theorem.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4125

Theorem 8. Let (W, τN ) and (V, ηF ) be an FSTSs, and φψ : ˜(W,N) −→ (̃V, F ) be a

fuzzy soft mapping. The following statements are equivalent for each fA ∈ (̃V, F ), n ∈ N ,
(k = ψ(n)) ∈ F , and r ∈ I◦:

(1) φψ is fuzzy soft weakly α-continuous.

(2) Iτ (n,Cτ (n, Iτ (n, φ
−1
ψ (Cη(k, fA, r)), r), r), r) ⊒ φ−1

ψ (fA), if ηk(fA) ≥ r.

(3) Cτ (n, Iτ (n,Cτ (n, φ
−1
ψ (Iη(k, fA, r)), r), r), r) ⊑ φ−1

ψ (fA), if ηk(f
c
A) ≥ r.

(4) αCτ (n, φ
−1
ψ (Iη(k, fA, r)), r) ⊑ φ−1

ψ (fA), if ηk(f
c
A) ≥ r.

(5) αCτ (n, φ
−1
ψ (Iη(k,Cη(k, fA, r), r)), r) ⊑ φ−1

ψ (Cη(k, fA, r)).

(6) αIτ (n, φ
−1
ψ (Cη(k, Iη(k, fA, r), r)), r) ⊒ φ−1

ψ (Iη(k, fA, r)).

(7) φ−1
ψ (fA) ⊑ αIτ (n, φ

−1
ψ (Cη(k, fA, r)), r), if ηk(fA) ≥ r.

Remark 7. 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 almost continuity ⇒ fuzzy soft almost α-continuity

⇓ ⇓

fuzzy soft weakly continuity ⇒ fuzzy soft weakly α-continuity

Proposition 3. Let (W, τN ), (V, ηF ) and (U, γE) be an FSTSs, and φψ : ˜(W,N) −→
(̃V, F ), φ∗

ψ∗ : (̃V, F ) −→ (̃U,E) be two fuzzy soft functions. Then, the composition φ∗
ψ∗◦φψ

is fuzzy soft almost α-continuous if φψ is fuzzy soft α-continuous and φ∗
ψ∗ is fuzzy soft

almost continuous (resp., continuous).

Proof. The proof is obvious.

Let H and I : N × ˜(W,N) × I◦ → ˜(W,N) be operators on ˜(W,N), and J and K
: F × (̃V, F )× I◦ → (̃V, F ) be operators on (̃V, F ).



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4126

Definition 17. [11] Let (W, τN ) and (V, ηF ) be an FSTSs. φψ : ˜(W,N) −→ (̃V, F ) is said
to be a fuzzy soft (H, I,J ,K)-continuous mapping if

H[n, φ−1
ψ (K(k, hC , r)), r] ⊓ I[n, φ−1

ψ (J (k, hC , r)), r] = Φ

for each hC ∈ (̃V, F ) with ηk(hC) ≥ r, n ∈ N , and (k = ψ(n)) ∈ F .

In (2023), Alshammari et al. [11] defined the notion of fuzzy soft α-continuous

mappings: φ−1
ψ (hC) ⊑ Iτ (n,Cτ (n, Iτ (n, φ

−1
ψ (hC), r), r), r), for each hC ∈ (̃V, F ) with

ηk(hC) ≥ r. We can see that Definition 17 generalizes the concept of fuzzy soft con-
tinuous functions when we choose H = identity operator, I = interior closure interior
operator, J = identity operator, and K = identity operator.

A historical justification of Definition 17:

(1) In Section 3, we obtained the notion of fuzzy soft almost α-continuous mappings:

φ−1
ψ (hC) ⊑ αIτ (n, φ

−1
ψ (Iη(k,Cη(k, hC , r), r)), r), for each hC ∈ (̃V, F ) with ηk(hC) ≥ r.

Here, H = identity operator, I = α-interior operator, J = interior closure operator, and
K = identity operator.

(2) In Section 3, we obtained the notion of fuzzy soft weakly α-continuous mappings:

φ−1
ψ (hC) ⊑ αIτ (n, φ

−1
ψ (Cη(k, hC , r)), r), for each hC ∈ (̃V, F ) with ηk(hC) ≥ r. Here, H

= identity operator, I = α-interior operator, J = closure operator, and K = identity
operator.

4. Fuzzy soft α-compactness

Here, some novel types of fuzzy soft compactness via r-fuzzy soft α-open sets were
introduced and the relationships between them were explored with the help of some ex-
amples.

Definition 18. Let (W, τN ) be an FSTS and r ∈ I◦, then hC ∈ ˜(W,N) is called an r-fuzzy

soft compact iff for every family {(gB)δ ∈ ˜(W,N) | τn((gB)δ) ≥ r for each n ∈ N}δ∈∆,
such that hC ⊑ ⊔δ∈∆(gB)δ, there is a finite subset ∆◦ of ∆, such that hC ⊑ ⊔δ∈∆◦(gB)δ.

Definition 19. Let (W, τN ) be an FSTS and r ∈ I◦, then hC ∈ ˜(W,N) is called an r-fuzzy

soft α-compact iff for every family {(gB)δ ∈ ˜(W,N) | (gB)δ is r-fuzzy soft α-open}δ∈∆,
such that hC ⊑ ⊔δ∈∆(gB)δ, there is a finite subset ∆◦ of ∆, such that hC ⊑ ⊔δ∈∆◦(gB)δ.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4127

Lemma 6. Let (W, τN ) be an FSTS and r ∈ I◦. If hC ∈ ˜(W,N) is r-fuzzy soft α-compact,
then hC is r-fuzzy soft compact.

Proof. Follows from Definitions 18 and 19.

Theorem 9. Let φψ : (W, τN ) −→ (V, ηF ) be a fuzzy soft α-continuous mapping. If

hC ∈ ˜(W,N) is r-fuzzy soft α-compact, then φψ(hC) is r-fuzzy soft compact.

Proof. Let {(gB)δ ∈ (̃V, F ) | ηk((gB)δ) ≥ r}δ∈∆ with φψ(hC) ⊑ ⊔δ∈∆(gB)δ for each

k ∈ F . Then, {φ−1
ψ ((gB)δ) ∈ ˜(W,N) | φ−1

ψ ((gB)δ) is r-fuzzy soft α-open}δ∈∆ (by φψ

is fuzzy soft α-continuous) such that hC ⊑ ⊔δ∈∆φ−1
ψ ((gB)δ). Since hC is r-fuzzy soft

α-compact, there is a finite subset ∆◦ of ∆ such that hC ⊑ ⊔δ∈∆◦φ
−1
ψ ((gB)δ). Then,

φψ(hC) ⊑ ⊔δ∈∆◦(gB)δ. Hence, the proof is completed.

Definition 20. Let (W, τN ) be an FSTS and r ∈ I◦, then hC ∈ ˜(W,N) is called an r-fuzzy

soft almost compact iff for every family {(gB)δ ∈ ˜(W,N) | τn((gB)δ) ≥ r}δ∈∆, such that
hC ⊑ ⊔δ∈∆(gB)δ, there is a finite subset ∆◦ of ∆, such that hC ⊑ ⊔δ∈∆◦Cτ (n, (gB)δ, r) for
each n ∈ N .

Definition 21. Let (W, τN ) be an FSTS and r ∈ I◦, then hC ∈ ˜(W,N) is called an r-fuzzy

soft almost α-compact iff for every family {(gB)δ ∈ ˜(W,N) | (gB)δ is r-fuzzy soft α-open}δ∈∆,
such that hC ⊑ ⊔δ∈∆(gB)δ, there is a finite subset ∆◦ of ∆, such that hC ⊑ ⊔δ∈∆◦Cτ (n, (gB)δ, r)
for each n ∈ N .

Lemma 7. Let (W, τN ) be an FSTS and r ∈ I◦. If hC ∈ ˜(W,N) is r-fuzzy soft almost
α-compact, then hC is r-fuzzy soft almost compact.

Proof. Follows from Definitions 20 and 21.

Lemma 8. Let (W, τN ) be an FSTS and r ∈ I◦. If hC ∈ ˜(W,N) is r-fuzzy soft compact
(resp., α-compact), then hC is r-fuzzy soft almost compact (resp., almost α-compact).

Proof. Follows from Definitions 18, 19, 20, and 21.

Remark 8. The converse of Lemma 8 may not be true, as shown by Example 6.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4128

Example 6. Let V = I, n ∈ N −{1}, and F = {k1, k2} be the parameter set of V. Define

gFn and fF1 ∈ (̃V, F ) as follows ∀ k ∈ F :

gFn(k)(v) =


0.8, if v = 0,

nv, if 0 < v ≤ 1
n ,

1 , if 1
n < v ≤ 1,

fF1(k)(v) =

{
1, if v = 0,
1
2 , otherwise.

Define fuzzy soft topology ηF : F −→ [0, 1](̃V,F ) as follows: ∀k ∈ F ,

ηk(tF ) =


4
5 , if tF ∈ {Φ, F̃},
2
3 , if tF ≤ fF1 ,
n
n+1 , if tF ≤ gFn ,

0, otherwise.

Thus, V is 1
2 -fuzzy soft almost compact, but it is not 1

2 -fuzzy soft compact.

Theorem 10. Let φψ : (W, τN ) −→ (V, ηF ) be a fuzzy soft continuous mapping. If

hC ∈ ˜(W,N) is r-fuzzy soft almost α-compact, then φψ(hC) is r-fuzzy soft almost compact.

Proof. Let {(gB)δ ∈ (̃V, F ) | ηk((gB)δ) ≥ r}δ∈∆ with φψ(hC) ⊑ ⊔δ∈∆(gB)δ for each

k ∈ F . Then, {φ−1
ψ ((gB)δ) ∈ ˜(W,N) | φ−1

ψ ((gB)δ) is r-fuzzy soft α-open}δ∈∆ (by φψ is

fuzzy soft α-continuous) such that hC ⊑ ⊔δ∈∆φ−1
ψ ((gB)δ). Since hC is r-fuzzy soft almost

α-compact, there is a finite subset ∆◦ of ∆ such that hC ⊑ ⊔δ∈∆◦Cτ (n, φ
−1
ψ ((gB)δ), r).

Since φψ is fuzzy soft continuous mapping, it follows

⊔δ∈∆◦Cτ (n, φ
−1
ψ ((gB)δ), r) ⊑

⊔δ∈∆◦φ
−1
ψ (Cη(k, (gB)δ, r)) =

φ−1
ψ (⊔δ∈∆◦Cη(k, (gB)δ, r)).

Then, φψ(hC) ⊑ ⊔δ∈∆◦Cη(k, (gB)δ, r). Hence, the proof is completed.

Definition 22. Let (W, τN ) be an FSTS and r ∈ I◦, then hC ∈ ˜(W,N) is called an r-fuzzy

soft nearly compact iff for every family {(gB)δ ∈ ˜(W,N) | τn((gB)δ) ≥ r}δ∈∆, such that
hC ⊑ ⊔δ∈∆(gB)δ, there is a finite subset ∆◦ of ∆, such that

hC ⊑ ⊔δ∈∆◦Iτ (n,Cτ (n, (gB)δ, r), r) for each n ∈ N.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4129

Definition 23. Let (W, τN ) be an FSTS and r ∈ I◦, then hC ∈ ˜(W,N) is called an r-fuzzy

soft nearly α-compact iff for every family {(gB)δ ∈ ˜(W,N) | (gB)δ is r-fuzzy soft α-open}δ∈∆,
such that hC ⊑ ⊔δ∈∆(gB)δ, there is a finite subset ∆◦ of ∆, such that

hC ⊑ ⊔δ∈∆◦Iτ (n,Cτ (n, (gB)δ, r), r) for each n ∈ N.

Lemma 9. Let (W, τN ) be an FSTS and r ∈ I◦. If hC ∈ ˜(W,N) is r-fuzzy soft nearly
α-compact, then hC is r-fuzzy soft nearly compact.

Proof. Follows from Definitions 22 and 23.

Lemma 10. Let (W, τN ) be an FSTS and r ∈ I◦. If hC ∈ ˜(W,N) is r-fuzzy soft compact
(resp., α-compact), then hC is r-fuzzy soft nearly compact (resp., nearly α-compact).

Proof. Follows from Definitions 18, 19, 22, and 23.

Remark 9. The converse of Lemma 10 may not be true, as shown by Example 7.

Example 7. Let V = I, 0 < n < 1, and F = {k1, k2} be the parameter set of V. Define

gFn , gF , and fF ∈ (̃V, F ) as follows ∀ k ∈ F :

gFn(k)(v) =

{
v
n , if 0 ≤ v ≤ n,
1−v
1−n , if n < v ≤ 1,

gF (k)(v) =

{
1, if v = 0,
1
2 , if 0 < v ≤ 1,

fF (k)(v) =

{
1
2 , if 0 ≤ v < 1,

1, if v = 1.

Define fuzzy soft topology ηF : F −→ [0, 1](̃V,F ) as follows: ∀k ∈ F ,

ηk(tF ) =


1, if tF ∈ {gF , fF ,Φ, F̃},
max({1− n, n}), if tF = gFn ,

0, otherwise.

Thus, V is 1
2 -fuzzy soft nearly compact, but it is not 1

2 -fuzzy soft compact.



W. Alqurashi, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4112-4134 4130

Theorem 11. Let φψ : (W, τN ) −→ (V, ηF ) be a fuzzy soft continuous and fuzzy soft

open mapping. If hC ∈ ˜(W,N) is r-fuzzy soft nearly α-compact, then φψ(hC) is r-fuzzy
soft nearly compact.

Proof. Let {(gB)δ ∈ (̃V, F ) | ηk((gB)δ) ≥ r}δ∈∆ with φψ(hC) ⊑ ⊔δ∈∆(gB)δ for each

k ∈ F . Then, {φ−1
ψ ((gB)δ) ∈ ˜(W,N) | φ−1

ψ ((gB)δ) is r-fuzzy soft α-open}δ∈∆ (by φψ is

fuzzy soft α-continuous) such that hC ⊑ ⊔δ∈∆φ−1
ψ ((gB)δ). Since hC is r-fuzzy soft nearly α-

compact, there is a finite subset ∆◦ of ∆ such that hC ⊑ ⊔δ∈∆◦Iτ (n,Cτ (n, φ
−1
ψ ((gB)δ), r), r).

Since φψ is fuzzy soft continuous and fuzzy soft open mapping, it follows

φψ(hC) ⊑ ⊔δ∈∆◦φψ(Iτ (n,Cτ (n, φ
−1
ψ ((gB)δ), r), r))

⊑ ⊔δ∈∆◦Iη(k, φψ(Cτ (n, φ
−1
ψ ((gB)δ), r)), r)

⊑ ⊔δ∈∆◦Iη(k, φψ(φ
−1
ψ (Cη(k, (gB)δ, r))), r)

⊑ ⊔δ∈∆◦Iη(k,Cη(k, (gB)δ, r), r).

Hence, the proof is completed.

Lemma 11. Let (W, τN ) be an FSTS and r ∈ I◦. If hC ∈ ˜(W,N) is r-fuzzy soft nearly α-
compact (resp., nearly compact), then hC is r-fuzzy soft almost α-compact (resp., almost
compact).

Proof. Follows from Definitions 20, 21, 22, and 23.

Remark 10. From the previous definitions and results, we can summarize the relation-
ships among different types of fuzzy soft compactness as in the next diagram.

fuzzy soft α-compactness ⇒ fuzzy soft compactness

⇓ ⇓

fuzzy soft nearly α-compactness ⇒ fuzzy soft nearly compactness

⇓ ⇓

fuzzy soft almost α-compactness ⇒ fuzzy soft almost compactness



REFERENCES 4131

5. Conclusion and future work

In this study, the concepts of fuzzy soft α-closure (α-interior) operators have been in-
troduced in an FSTSs based on the paper by Aygünoǧlu et al. [19] and some of their basic
properties have been investigated. Thereafter, the notion of r-fuzzy soft α-connected sets
has been defined and studied. Furthermore, some properties of fuzzy soft α-continuous
mappings have been obtained between two FSTSs (W, τN ) and (V, ηF ). Moreover, as a
weaker form of the notion of fuzzy soft α-continuous mappings, the notions of fuzzy soft
almost (weakly) α-continuous mappings have been introduced and some of their charac-
terizations have been investigated. Also, we have shown that fuzzy soft α-continuity ⇒
fuzzy soft almost α-continuity ⇒ fuzzy soft weakly α-continuity and we have the following:

• Fuzzy soft (idW , Iτ (Cτ (Iτ )), idV , idV )-continuous mapping is fuzzy soft α-continuous.

• Fuzzy soft (idW , αIτ , Iη(Cη), idV )-continuous mapping is fuzzy soft almost α-continuous.

• Fuzzy soft (idW , αIτ , Cη, idV )-continuous mapping is fuzzy soft weakly α-continuous.

In the end, new types of soft compactness via r-fuzzy soft α-open sets have been
explored and the relationships between them have been studied.

In upcoming papers, we will use the fuzzy soft α-closure operator to define some new
separation axioms in an FSTS based on the paper by Aygünoǧlu et al. [19]. Also, we shall
discuss some of the notions given here in the frames of fuzzy soft r-minimal structures
[30].

Acknowledgements

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

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