EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 760-772
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A view on Connectedness and Compactness in Fuzzy
Soft Bitopological Spaces

Abdelhamied Farrag Sayed

Mathematics Department, Al-Lith University College, Umm Al-Qura University
P.O. Box 112, Al-Lith 21961, Makkah Al Mukarramah, Kingdom of Saudi Arabia

Abstract. In the present paper, we introduce the notions of (1, 2)∗-fuzzy soft b-separated sets,
(1, 2)∗-fuzzy soft b-connectedness and (1, 2)∗-fuzzy soft b-compactness in fuzzy soft bitopological
spaces. Then, some basic topological properties of these notions are investigated. Also, some
illustrative examples are given to show the importance of the obtained theorems.

2020 Mathematics Subject Classifications: 54A05, 54A40, 54F99.

Key Words and Phrases: (1, 2)∗-fsb-separated, (1, 2)∗-fsb-connected, (1, 2)∗-fsb-compact

1. Introduction

In 1965, Zadeh [36], introduced the concept of fuzzy set theory and its applications can
be found in many branches of mathematical and engineering sciences including manage-
ment science, control engineering, computer science and artificial intelligence (see, [5, 7]).

In 1999, Russian researcher Molodtsov [16], initiated the concept of soft sets as a
new mathematical tool to deal with uncertainties while modeling problems in engineering
physics, computer science, economics, social sciences and medical sciences (see, [20, 28]).
In 2003, Maji, Biswas and Roy [22], studied the theory of soft sets initiated by Molodtsov.
They defined equality of two soft sets, subset and super set of a soft set, complement of a
soft set, null soft set and absolute soft set with examples. Soft binary operations like AND,
OR and also the operations of union and intersection were also defined. In 2005, D. Chen
[6], presented a new definition of soft set parametrization reduction and a comparison of
it with attribute reduction in rough set theory.

Recently, on soft sets, soft topological space has been studied increasingly Shabir
and Naz [32] defined the theory of soft topological space over an initial universe with a
fixed set of parameters. Çağman et al. [18] introduced a topology on a soft set called
”soft topology” and presented the foundations of the theory of soft topological spaces.
Moreover, many authors studied soft topology and its applications (e.g. [8, 9, 11]). Later
Tanay and Kandemir [35] introduced fuzzy soft topological space and established the basic

DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3980

Email address: dr.afsayed@hotmail.com, afssayed@uqu.edu.sa (A. F. Sayed)

http://www.ejpam.com 760 © 2021 EJPAM All rights reserved.



A. F. Sayed / Eur. J. Pure Appl. Math, 14 (3) (2021), 760-772 761

definitions of fuzzy soft topological space by incorporating the fuzzy topology and soft set.
Fuzzy soft topological space was applied in various ways say, game theory, analysis, etc.
Fuzzy soft set in topological space further studied by Roy [27]. The authors [13, 19, 33]
are successfully applied fuzzy soft topological space in real life.

In 1963, Kelly [14], first initiated the concept of bitopological spaces and other authors
have contributed to development and construction of some properties of such spaces (see,
[23, 24]) as generalizations of which are in general topology.

In 2014, Ittanagi [12], introduced and studied the concept of soft bitopological spaces
and other authors have contributed to development and construction of some properties
of such spaces (see, [2, 3, 25, 26]).
The notion of soft bitopological space was introduced using different soft topologies on an
initial universe set. On the other hand, the mixed type of soft set theory was given using
different soft topologies (see, [1, 4, 10, 21, 34]).

In 2015, Mukherjee and Park [17], first introduced the notion of fuzzy soft bitopological
space and they introduced the notions of τ1τ2-fuzzy soft open(closed) sets, τ1τ2-fuzzy soft
interior (resp. closure) and studied some of their basic properties. Also, Sayed ([29–31])
were extension and continuation of studying in this trend by characterizing new concepts
in fuzzy soft bitopological spaces. In the present paper, we introduce the notions of
(1, 2)∗-fuzzy soft b-separated sets, (1, 2)∗-fuzzy soft b-connectedness and (1, 2)∗-fuzzy soft
b-compactness in fuzzy soft bitopological spaces. Then, some basic topological properties
of these notions are investigated. Also, some illustrative examples are given to show the
importance of the obtained theorems.

2. Preliminaries

In this section we are going to present the basic definitions and results of fuzzy soft
set and fuzzy soft bitopological space which will be a central role in our paper.
Throughout our discussion, X refers to an initial universe, E the set of all parameters for
X and P (X) denotes the power set of X.

Definition 1. [36] A fuzzy set A in a non-empty set X is characterized by a membership
function µA : X → [0, 1] = I whose value µA(x) represents the ”degree of membership” of
x in A for every x in X. Let IX denotes the family of all fuzzy sets on X.

Definition 2. [36] The empty fuzzy set on X denoted by 0̃ is a function which maps each
x ∈ X to 0. That is, 0̃(x) = 0 for all x ∈ X.
A universal fuzzy set denoted by 1̃ is a function, which maps each x ∈ X to 1. That is,
1̃(x) = 1 for all x ∈ X.

Definition 3. [16] Let A ⊆ E. A pair (F,A) is called a soft set over X if F is a mapping
given by F : A→ P (X).

Definition 4. [15] Let A ⊆ E. A pair (f,A), denoted by fA, is called a fuzzy soft set over
X, where f is a mapping given by f : A→ IX defined by fA(e) = µefA where



A. F. Sayed / Eur. J. Pure Appl. Math, 14 (3) (2021), 760-772 762

µefA =

{
0̃, if e /∈ A;
otherwise, if e ∈ A.

(̃X,E) denotes the family of all fuzzy soft sets over (X,E).

Definition 5. [22] A fuzzy soft set fA∈̃(̃X,E) is said to be:

(a) NULL fuzzy soft set, denoted by φ̃, if for all e ∈ A, fA(e) = 0̃.

(b) absolute fuzzy soft set, denoted by Ẽ, if for all e ∈ E, fA(e) = 1̃.

Note that throughout our discussion in this paper, 0̃E and 1̃E will be denoted for φ̃
and Ẽ, respectively.

Definition 6. [27] The complement of a fuzzy soft set fA, denoted by f cA where f cA : E →
IX is a mapping given by µefcA

= 1̃−µefA, for all e ∈ E and where 1̃(x) = 1, for all x ∈ X.

Clearly (f cA)c = fA.

Definition 7. [27] Let fA, gB ∈ (̃X,E). fA is fuzzy soft subset of gB, denoted by fA⊆̃gB,
if A ⊆ B and µefA ≤ µ

e
gB

for all e ∈ A, that is, µefA(x) ≤ µegB (x) for all x ∈ X and for all
e ∈ A.

Definition 8. [27] Let fA, gB∈̃(̃X,E). The union of fA and gB is also a fuzzy soft set hC ,
where C = A∪B and for all e ∈ C, hC(e) = µehc = µefA ∨µ

e
gB

. Here we write hC = fA∪̃gB.

Definition 9. [27] Let fA, gB∈̃(̃X,E). The intersection of fA and gB is also a fuzzy soft
set dC , where C = A ∩ B and for all e ∈ C, dC(e) = µedc = µefA ∧ µ

e
gB

. Here we write
dC = fA∩̃gB.

Definition 10. [27] A fuzzy soft topology τ over (X,E) is a family of fuzzy soft sets over
(X,E) satisfying the following properties:

(i) 0̃E , 1̃E ∈ τ ,

(ii) if fA, gB ∈ τ, then fA∩̃gB ∈ τ ,

(iii) if fAα ∈τ for all α ∈ ∆ an index set, then
⋃̃
α∈∆fAα ∈ τ.

Definition 11. [17] If τ is a fuzzy soft topology on (X,E), then the triple (X,E, τ) is
said to be a fuzzy soft topological space. Also each member of τ is called a fuzzy soft open
set in (X,E, τ).

The complement of a fuzzy soft open set is a fuzzy soft closed set.

Definition 12. [17] Let (X,E, τ1) and (X,E, τ2) be two different fuzzy soft topologies on
(X,E). Then (X,E, τ1, τ2) is called a fuzzy soft bitopological space on which no seperation
axioms are assumed unless explicitly stated.
The members of τi(i = 1, 2) are called τi(i = 1, 2)-fuzzy soft open sets and the complement
of τi(i = 1, 2)-fuzzy soft open sets are called τi(i = 1, 2)-fuzzy soft closed sets.



A. F. Sayed / Eur. J. Pure Appl. Math, 14 (3) (2021), 760-772 763

Definition 13. [17] A fuzzy soft set fE∈̃(̃X,E) is called τ1τ2- fuzzy soft open set if fE =
gE∪̃hE such that gE∈̃τ1 and hE∈̃τ2.
The complement of τ1τ2- fuzzy soft open set is called τ1τ2- fuzzy soft closed set.

The family of all τ1τ2- fuzzy soft open (closed) sets in (X,E, τ1, τ2) is denoted by
τ1τ2FSO(X, τ1, τ2)E (τ1τ2FSC(X, τ1, τ2)E), respectively.

Definition 14. [17] Let (X,E, τ1, τ2) be a fuzzy soft bitopological space and fE∈̃(̃X,E).
Then the τ1τ2- fuzzy soft closure of fE, denoted by τ1τ2cl(fE), is the intersection of all
τ1τ2- fuzzy soft closed supersets of fE.
Clearly, τ1τ2cl(fE) is the smallest τ1τ2- fuzzy soft closed set over (X,E) which contains
fE.

Definition 15. Let (X,E, τ1, τ2) be a fuzzy soft bitopological space and fE∈̃(̃X,E). Then
fE is called (1, 2)∗-fuzzy soft b-open set (briefly, (1, 2)∗-fsb-open) if
fE⊆̃τ1τ2int (τ1τ2cl(fE)) ∪̃τ1τ2cl (τ1τ2int(fE)).

Definition 16. [30] Let (X,E, τ1, τ2) be a fuzzy soft bitopological space and fE∈̃(̃X,E).
(i) (1, 2)∗-fuzzy soft b-closure (briefly (1, 2)∗-fsbcl(fE)) of a set fE in (X,E, τ1, τ2) defined
by (1, 2)∗-fsbcl(fE) = ∩̃{gE⊇̃fE : gE is a (1, 2)∗-fuzzy soft b-closed set in (X,E, τ1, τ2)}.
(ii) (1, 2)∗-fuzzy soft b-interior (briefly (1, 2)∗-fsbint(fE)) of a set fE in (X,E, τ1, τ2) de-
fined by (1, 2)∗-fsbint(fE) = ∪̃{gE⊆̃fE : gE is a (1, 2)∗-fuzzy soft b-open set in (X,E, τ1, τ2)}.
(1, 2)∗-fsbcl(fE) is the smallest (1, 2)∗-fuzzy soft b-closed set in (X,E, τ1, τ2) which con-
tains fE and (1, 2)∗-fsbcl(fE) is the largest (1, 2)∗-fuzzy soft b-closed set in (X,E, τ1, τ2)
which is contained in fE.

Definition 17. [31] A fuzzy soft mapping (ϕ,ψ) : (X,E, τ1, τ2)→ (Y,K, σ1, σ2) is said to
be (1, 2)∗-fuzzy soft b-continuous (briefly (1, 2)∗-fsb-continuous) the inverse image of every
σ1σ2-fuzzy soft open set in (Y,K, σ1, σ2) is a (1, 2)∗-fuzzy soft b-open set in (X,E, τ1, τ2).

Definition 18. [31] A fuzzy soft mapping (ϕ,ψ) : (X,E, τ1, τ2)→ (Y,K, σ1, σ2) is said to
be (1, 2)∗-fuzzy soft b-irresolute mapping (briefly, (1, 2)∗-fsb-irresolute) if (ϕ,ψ)−1(gK) is
a (1, 2)∗-fuzzy soft b-closed set in (X,E, τ1, τ2) for every (1, 2)∗-fuzzy soft b-closed set gK
in (Y,K, σ1, σ2).

3. (1, 2)∗-Fuzzy Soft b-Connectedness

In this section we introduce the concepts of (1, 2)∗-fuzzy soft b-separated sets and
(1, 2)∗-fuzzy soft b-connectedness in fuzzy soft bitopological spaces. Also, some of the
main results and properties are studied and discussed.

Definition 19. Two non-empty fuzzy soft subsets fE , gE of (̃X,E) are said to be fuzzy
soft disjoint if fE∩̃gE = 0̃E.



A. F. Sayed / Eur. J. Pure Appl. Math, 14 (3) (2021), 760-772 764

Definition 20. Let (X,E, τ1, τ2) be a fuzzy soft bitopological space. Two non-empty fuzzy

soft disjoint fuzzy soft subsets fE , gE of (̃X,E) are called
(i) (1, 2)∗-fuzzy soft separated sets over X if τ1τ2cl(fE)∩̃gE = fE∩̃τ1τ2cl(gE) = 0̃E.
(ii) (1, 2)∗-fuzzy soft b-separated ((1, 2)∗-fsb-separated)sets over X if
((1, 2)∗-fsbcl(fE))∩̃gE = fE∩̃((1, 2)∗-fsbcl(gE)) = 0̃E.

Remark 1. From the fact that (1, 2)∗-fsbcl(fE)⊆̃τ1τ2cl(fE), for every fuzzy soft subset

fE of (̃X,E), every (1, 2)∗-fuzzy soft separated set is (1, 2)∗-fuzzy soft b-separated. But the
converse may not be true.

Definition 21. A (1, 2)∗-fuzzy soft b-separation ((1, 2)∗-fsb-separation) of a fuzzy soft
bitopological space (X,E, τ1, τ2) is a pair of (1, 2)∗-fuzzy soft b-separated sets fE and gE
whose fuzzy soft union is absolute fuzzy soft set 1̃E(that is fE∪̃gE = 1̃E).

Definition 22. Let (X,E, τ1, τ2) be a fuzzy soft bitopological space. Then (X,E, τ1, τ2) is
called (1, 2)∗-fuzzy soft b-connected space if 1̃E can not be expressed as the fuzzy soft union
of two (1, 2)∗-fuzzy soft b-separated sets.

Remark 2. In a fuzzy soft bitopological space (X,E, τ1, τ2):
(i) A fuzzy soft empty set is trivially (1, 2)∗-fuzzy soft b-connected set.
(ii) Every fuzzy soft singleton set is (1, 2)∗-fuzzy soft b-connected, since it can not be
expressd as a fuzzy soft union of two non-empty (1, 2)∗-fuzzy soft b-separated sets.

Theorem 1. Let (X,E, τ1, τ2) be a fuzzy soft bitopological space. Then the following
statements are equivalent:
(i) (X,E, τ1, τ2) is a (1, 2)∗-fuzzy soft b-connected space.
(ii) 1̃E and 0̃E are the only (1, 2)∗-fuzzy soft b-clopen (that is, closed and open) sets in
(X,E, τ1, τ2).
(iii) 1̃E can not be expressed as the fuzzy soft union of two fuzzy soft disjoint non-empty
(1, 2)∗-fuzzy soft b-open sets.
(iv) 1̃E can not be expressed as the fuzzy soft union of two fuzzy soft disjoint non-empty
(1, 2)∗-fuzzy soft b-closed sets.

Proof. (i) ⇒ (ii): Let (X,E, τ1, τ2) be a fuzzy soft bitopological space. Let fE be

non-empty proper fuzzy soft subset of (̃X,E) that is (1, 2)∗-fuzzy soft b-clopen. Then
1̃E \ fE is a non-empty (1, 2)∗-fuzzy soft b-clopen set and 1̃E = fE∪̃(1̃E \ fE). This is a
contradiction to (X,E, τ1, τ2) is a (1, 2)∗-fuzzy soft b-connected space. Therefore 1̃E and
0̃E are the only (1, 2)∗-fuzzy soft b-clopen sets in (X,E, τ1, τ2).
(ii) ⇒ (iii): Assume that 1̃E and 0̃E are the only (1, 2)∗-fuzzy soft b-clopen sets in
(X,E, τ1, τ2). suppose (iii) is false. Then 1̃E = fE∪̃gE where fE and gE are fuzzy soft
disjoint non-empty (1, 2)∗-fuzzy soft b-open sets. Then gE = 1̃E \ fE is (1, 2)∗-fuzzy soft
b-closed and non-empty. Thus gE is a non-empty proper (1, 2)∗-fuzzy soft b-clopen set in
(X,E, τ1, τ2), which contradicts (ii).
(iii) ⇒ (iv): Assume 1̃E cannot be expressed as the fuzzy soft union of two fuzzy soft
disjoint non-empty (1, 2)∗-fuzzy soft b-open sets. Suppose (iv) false. Then (1, 2)∗-fuzzy



A. F. Sayed / Eur. J. Pure Appl. Math, 14 (3) (2021), 760-772 765

soft b-closed sets. Then fE = 1̃E \ gE and gE = 1̃E \ fE are fuzzy soft disjoint non-empty
(1, 2)∗-fuzzy soft b-open sets in (X,E, τ1, τ2). Thus 1̃E is the fuzzy soft union of two fuzzy
soft disjoint non-empty (1, 2)∗-fuzzy soft b-open sets . This contradicts (iii).
(iv) ⇒ (i): Suppose (X,E, τ1, τ2) is not (1, 2)∗-fuzzy soft b-connected space. Then 1̃E =
fE∪̃gE where fE and gE are fuzzy soft disjoint non-empty (1, 2)∗-fuzzy soft b-open sets.
Then fE = 1̃E \ gE and gE = 1̃E \ fE are fuzzy soft disjoint non-empty (1, 2)∗-fuzzy soft
b-closed sets in (X,E, τ1, τ2). This is a contradiction to (iv).

Proposition 1. Every (1, 2)∗-fuzzy soft b-connected space is (1, 2)∗-fuzzy soft connected.

Proof. Let fE be a (1, 2)∗-fuzzy soft b-connected set in the fuzzy soft bitopological
space (X,E, τ1, τ2). Then there does not exist a (1, 2)∗-fuzzy soft b-separation of fE .
Since every τ1τ2-fuzzy soft open set is a (1, 2)∗-fuzzy soft b-open set, there does not exist
a (1, 2)∗-fuzzy soft separation of fE . Hence fE is a (1, 2)∗-fuzzy soft connected set in the
fuzzy soft bitopological space (X,E, τ1, τ2).

The converse is not true as shown in the following example.

Example 1. (1, 2)∗-fuzzy soft connectedness does not imply (1, 2)∗-fuzzy soft b-
connectedness.
Let (X,E, τ1, τ2) be a fuzzy soft bitopological space, where X = {x, y}, E = {e1, e2} and let
τ1 = {0̃E , 1̃E , f1E , f2E , f3E}, τ2 = {0̃E , 1̃E , g1E , g2E}, where
f1E = {f1(e1) = {x/0.2, y/0.0}, f1(e2) = {x/0.0, y/0.0} = 0̃},
f2E = {f2(e1) = {x/0.2, y/0.0}, f2(e2) = {x/0.7, y/0.0}},
f3E = {f3(e1) = {x/0.2, y/0.1}, f3(e2) = {x/0.7, y/0.0}},
g1E = {g1(e1) = {x/0.0, y/0.0} = 0̃, g1(e2) = {x/0.7, y/0.0}} and
g2E = {g2(e1) = {x/0.0, y/0.0} = 0̃, g2(e2) = {x/0.0, y/0.4}}.
Then τ1τ2-fuzzy soft open sets are {0̃E , 1̃E , f1E , f2E , f3E , g1E , g2E} and τ1τ2-fuzzy soft closed
sets are {0̃E , 1̃E , f c1E , f

c
2E
, f c3E , g

c
1E
, gc2E} where

f c1E = {f1(e1) = {x/0.0, y/0.1}, f1(e2) = {x/0.7, y/0.4}},
f c2E = {f2(e1) = {x/0.0, y/0.1}, f2(e2) = {x/0.0, y/0.4}},
f c3E = {f3(e1) = {x/0.0, y/0.0} = 0̃, f3(e2) = {x/0.0, y/0.4}}
gc1E = {g1(e1) = {x/0.2, y/0.1}, g1(e2) = {x/0.0, y/0.4}} and
gc2E = {g2(e1) = {x/0.2, y/0.1}, g2(e2) = {x/0.7, y/0.0}}.
It is clear that (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft connected since the only (1, 2)∗-fuzzy soft
clopen sets are 0̃E , 1̃E. Also (1, 2)∗-fuzzy soft b-open sets are {0̃E , 1̃E , f1E , f2E , f3E , f4E , f5E ,
g1E , g2E , g3E , g4E}, where f1E , f2E , f3E , g1E and g2E} are defined as above and
f4E = {f4(e1) = {x/0.0, y/0.1}, f4(e2) = {x/0.7, y/0.4}},
f5E = {f5(e1) = {x/0.0, y/0.1}, f5(e2) = {x/0.0, y/0.4}},
g3E = {g3(e1) = {x/0.2, y/0.1}, g3(e2) = {x/0.0, y/0.4}} and
g4E = {g4(e1) = {x/0.2, y/0.0}, g4(e2) = {x/0.0, y/0.4}}.
And (1, 2)∗-fuzzy soft b-closed sets are {0̃E , 1̃E , f c1E , f

c
2E
, f c3E , f

c
4E
, f c5E , g

c
1E
, gc2E , g

c
3E
, g

c

4E
},

where f c1E , f
c
2E
, f c3E , g

c
1E

and gc2E are obtained as above and

f c4E = {f4(e1) = {x/0.2, y/0.0}, f4(e2) = {x/0.0, y/0.0 = 0̃}},



A. F. Sayed / Eur. J. Pure Appl. Math, 14 (3) (2021), 760-772 766

f c5E = {f5(e1) = {x/0.2, y/0.0}, f5(e2) = {x/0.7, y/0.0}},
gc3E = {g3(e1) = {x/0.0, y/0.0} = 0̃, g3(e2) = {x/0.7, y/0.4}} and
gc4E = {g4(e1) = {x/0.0, y/0.1}, g4(e2) = {x/0.7, y/0.4}},
where 1̃E = f1E ∪̃f4E , then (1, 2)∗-fsbcl(f1E ) = f c4E , (1, 2)∗-fsbcl(f4E ) = gc4E and (1, 2)∗-

fsbcl(f1E )∩̃f4E = 0̃E , (1, 2)∗-fsbcl(f4E )∩̃g4E = 0̃E. Hence 1̃E can be expressed as a fuzzy
soft union of two (1, 2)∗-fuzzy soft b-separated sets f1E , f4E . There (X,E, τ1, τ2) is not
(1, 2)∗-fuzzy soft b-connected.

Example 2. (1, 2)∗-fuzzy soft b-connectivity is not hereditary property.
Consider the fuzzy soft bitopological space (X,E, τ1, τ2), where X = {x, y}, E = {e1, e2}
and let τ1 = {0̃E , 1̃E , f1E , f2E}, τ2 = {0̃E , 1̃E , g1E}, where
f1E = {f1(e1) = {x/0.2, y/0.0}, f1(e2) = {x/0.0, y/0.0} = 0̃},
f2E = {f2(e1) = {x/0.2, y/0.0}, f2(e2) = {x/0.7, y/0.0}} and
g1E = {g1(e1) = {x/0.2, y/0.1}, g1(e2) = {x/0.0, y/0.0} = 0̃}.
Then τ1τ2-fuzzy soft open sets are {0̃E , 1̃E , f1E , f2E , g1E , h1E}, where h1E = {h1(e1) =
{x/0.2, y/0.1}, h1(e2) = {x/0.7, y/0.0} = 0̃},
Also, (1, 2)∗-fuzzy soft b-open sets are {0̃E , 1̃E , f1E , f2E , g1E , h1E , h2E , h3E , h4E}, where
h2E = {h2(e1) = {x/0.2, y/0.0}, h2(e2) = {x/0.0, y/0.4}},
h3E = {h3(e1) = {x/0.2, y/0.0}, h3(e2) = {x/0.7, y/0.4}} and
h4E = {h3(e1) = {x/0.0, y/0.0}, h3(e2) = {x/0.7, y/0.0}} and (1, 2)∗-fuzzy soft b-closed
sets are {0̃E , 1̃E , f c1E , f

c
2E
, gc1E , h

c
1E
, hc2E , h

c
3E
, hc4E}, where

f c1E = {f1(e1) = {x/0.0, y/0.1}, f1(e2) = {x/0.7, y/0.4},
f c2E = {f2(e1) = {x/0.0, y/0.1}, f2(e2) = {x/0.0, y/0.4}},
gc1E = {g1(e1) = {x/0.0, y/0.0} = 0̃, g1(e2) = {x/0.7, y/0.4}},
hc1E = {h1(e1) = {x/0.0, y/0.0} = 0̃, h1(e2) = {x/0.0, y/0.4}},
hc2E = {h2(e1) = {x/0.0, y/0.1}, h2(e2) = {x/0.7, y/0.0}},
hc3E = {h3(e1) = {x/0.0, y/0.1}, h3(e2) = {x/0.0, y/0.0} = 0̃} and

hc4E = {h4(e1) = {x/0.2, y/0.1} = 0̃, h4(e2) = {x/0.0, y/0.4}}.
It is clear that (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft b-connected, since the only (1, 2)∗-fuzzy soft
clopen sets are 0̃E and 1̃E. Let Y = {x}⊆̃X and E = {e1, e2}. Let σ1 = {0̃E , 1̃E , f1E}, σ2 =
{0̃E , 1̃E , h4E}. Then σ1σ2-fuzzy soft open sets are {0̃E , 1̃E , f1E , h4E}. Also (1, 2)∗-fuzzy
soft b-clopen sets are {0̃E , 1̃E , f1E , h4E}. clearly (Y,E, σ1, σ2) is not (1, 2)∗-fuzzy soft b-
connected; since f1E and h4E are two (1, 2)∗-fuzzy soft b-clopen sets other than 0̃E and
1̃E.

Proposition 2. Let fE be a (1, 2)∗-fuzzy soft b-connected set, gE and hE are (1, 2)∗-fuzzy
soft b-separated sets. If fE⊆̃gE∪̃hE then either fE⊆̃gE or fE⊆̃hE.

Proof. Let fE be a (1, 2)∗-fuzzy soft b-connected set, gE and hE are (1, 2)∗-fuzzy soft b-

separated sets such that fE⊆̃gE∪̃hE . Let fE*̃gE and fE*̃hE . Suppose kE = gE∩̃fE 6= 0̃E
and lE = hE∩̃fE 6= 0̃E then fE = kE∪̃lE .
Since kE⊆̃gE , ((1, 2)∗-fsbcl(kE))⊆̃((1, 2)∗-fsbcl(gE)). Also ((1, 2)∗-fsbcl(gE))∩̃hE = 0̃E
then ((1, 2)∗-fsbcl(kE))∩̃lE = 0̃E . Since lE⊆̃hE , ((1, 2)∗-fsbcl(lE))⊆̃((1, 2)∗-fsbcl(hE)).
Also ((1, 2)∗-fsbcl(hE))∩̃gE = 0̃E then ((1, 2)∗-fsbcl(lE))∩̃kE = 0̃E . But fE = kE∪̃lE ,



A. F. Sayed / Eur. J. Pure Appl. Math, 14 (3) (2021), 760-772 767

therefore fE is not (1, 2)∗-fuzzy soft b-connected set which is not a contradiction. Then
either fE⊆̃gE or fE⊆̃hE .

Theorem 2. If fE is a (1, 2)∗-fuzzy soft b-connected set and fE⊆̃gE⊆̃((1, 2)∗-fsbcl(fE))
then gE is a (1, 2)∗-fuzzy soft b-connected.

Proof. Suppose gE is not (1, 2)∗-fuzzy soft b-connected then there exists two non-empty
fuzzy soft sets f1E and f2E such that ((1, 2)∗-fsbcl(f1E ))∩̃f2E = f1E ∩̃((1, 2)∗-fsbcl(f2E )) =
0̃E and fE = f1E ∪̃f2E . Since fE⊆̃gE then either fE⊆̃f1E or fE⊆̃f2E .
Suppose fE⊆̃f1E , then ((1, 2)∗-fsbcl(fE))⊆̃((1, 2)∗-fsbcl(f1E )), thus
((1, 2)∗-fsbcl(fE))∩̃f2E = fE∩̃((1, 2)∗-fsbcl(f2E )) = 0̃E . But f2E ⊆̃gE⊆̃((1, 2)∗-fsbcl(fE)),
thus ((1, 2)∗-fsbcl(fE))∩̃f2E = f2E . Therefore f2E = 0̃E , which is a contradiction.
If fE⊆̃f2E , then by the same way we can prove that f1E = 0̃E . This is a contradiction.
Thus gE be a (1, 2)∗-fuzzy soft b-connected.

Theorem 3. If fE is a (1, 2)∗-fuzzy soft b-connected set, then (1, 2)∗-fsbcl(fE) is (1, 2)∗-
fuzzy soft b-connected.

Proof. Suppose fE is (1, 2)∗-fuzzy soft b-connected and (1, 2)∗-fsbcl(fE) is not (1, 2)∗-
fuzzy soft b-connected. Then there exist two (1, 2)∗-fuzzy soft b-separated sets f1E and
f2E such that (1, 2)∗-fsbcl(fE) = f1E ∪̃f2E . But fE⊆̃(1, 2)∗-fsbcl(fE) then fE = f1E ∪̃f2E

and since fE is (1, 2)∗-fuzzy soft b-connected set, then either fE⊆̃f1E or fE⊆̃f2E .
If fE⊆̃f1E then (1, 2)∗-fsbcl(fE)⊆̃(1, 2)∗-fsbcl(f1E ). But (1, 2)∗-fsbcl(f1E )∩̃f2E = 0̃E ,
hence (1, 2)∗-fsbcl(fE)∩̃f2E = 0̃E . Since f2E ⊆̃(1, 2)∗-fsbcl(fE), then
(1, 2)∗-fsbcl(fE)∩̃f2E = f2E ; hence f2E = 0̃E which is a contradiction.
If fE⊆̃f1E , then by the same way we can prove that f1E = 0̃E , which is a contradiction.
Therefore (1, 2)∗-fsbcl(fE) is (1, 2)∗-fuzzy soft b-connected.

Theorem 4. The fuzzy soft union fE of any family {fiE : i ∈ I} of (1, 2)∗-fuzzy soft b-
connected sets having a non-empty fuzzy soft intersection is (1, 2)∗-fuzzy soft b-connected.

Proof. Let fE be fuzzy soft union of any family of (1, 2)∗-fuzzy soft b-connected sets
having a non-empty fuzzy soft intersection. Suppose that fE = f1E ∪̃f2E , where f1E and
f2E form a (1, 2)∗-fuzzy soft b-separation of fE . By hypothesis, we may choose a fuzzy

soft point fe∈̃
⋂̃
i∈IfiE . Then fe∈̃fiE for all i ∈ I. If fe∈̃fE , then either fe∈̃f1E or fe∈̃f2E

but not both. Since f1E and f2E are fuzzy soft disjoint, we must have fiE ⊆̃f1E , since fiE
is (1, 2)∗-fuzzy soft b-connected and it is true for all i ∈ I, and so fE⊆̃fiE . From this we
obtain that f2E = 0̃E ; which is a contradiction. Thus, there does not exist a (1, 2)∗-fuzzy
soft b-separation of fE . Therefore, fE is a (1, 2)∗-fuzzy soft b-connected set.

Theorem 5. (i) If ψ : (X,E, τ1, τ2) → (Y,E, σ1, σ2) is a (1, 2)∗-fuzzy soft b-continuous
surjection and (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft b-connected then (Y,E, σ1, σ2) is (1, 2)∗-
fuzzy soft connected.
(ii) If ψ : (X,E, τ1, τ2) → (Y,E, σ1, σ2) is a (1, 2)∗-fuzzy soft b-irresolute surjection
and (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft b-connected then (Y,E, σ1, σ2) is (1, 2)∗-fuzzy soft
b-connected.



A. F. Sayed / Eur. J. Pure Appl. Math, 14 (3) (2021), 760-772 768

Proof. (i) Suppose (Y,E, σ1, σ2) is not (1, 2)∗-fuzzy soft connected. Let Y = fE∪̃gE ,
where fE and gE are fuzzy soft disjoint non-empty σ1σ2)-fuzzy soft open sets in
(Y,E, σ1, σ2). Since ψ is (1, 2)∗-fuzzy soft b-continuous and onto; 1̃E = ψ−1(fE)∪̃ψ−1(gE),
where ψ−1(fE) and ψ−1(gE) are fuzzy soft disjoint non-empty (1, 2)∗-fuzzy soft b-open
sets in (X,E, τ1, τ2). This contradicts the fact that (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft b-
connected. Hence (Y,E, σ1, σ2) is (1, 2)∗-fuzzy soft connected.
(ii) Suppose (Y,E, σ1, σ2) is not (1, 2)∗-fuzzy soft b-connected. Let Y = fE∪̃gE , where
fE and gE are fuzzy soft disjoint non-empty (1, 2)∗-fuzzy soft b-open sets in (Y,E, σ1, σ2).
Since ψ is (1, 2)∗-fuzzy soft b-irresolute and onto; then 1̃E = ψ−1(fE)∪̃ψ−1(gE), where
ψ−1(fE) and ψ−1(gE) are fuzzy soft disjoint non-empty (1, 2)∗-fuzzy soft b-open sets in
(X,E, τ1, τ2). This contradicts the fact that (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft b-connected.
Hence (Y,E, σ1, σ2) is (1, 2)∗-fuzzy soft b-connected.

4. (1, 2)∗-Fuzzy Soft b-Compactness

In this section (1, 2)∗-fuzzy soft b-compactness is defined and some of the characteri-
zations are proved.

Definition 23. A collection {fiE : i ∈ Λ} of (1, 2)∗-fuzzy soft b-open sets in fuzzy soft

bitopological space (X,E, τ1, τ2) is called a (1, 2)∗-fuzzy soft b-cover of fE if fE⊆̃
⋃̃
{fiE :

i ∈ Λ} .

Definition 24. A fuzzy soft bitopological space (X,E, τ1, τ2) is called a (1, 2)∗-fuzzy soft
b-compact if every (1, 2)∗-fuzzy soft b-open cover of 1̃E has a finite subcover.

Definition 25. A fuzzy soft subset fE of fuzzy soft bitopological space (X,E, τ1, τ2) is
said to be (1, 2)∗-fuzzy soft b-compact relative to 1̃E, if for every collection {fiE : i ∈ Λ} of

(1, 2)∗-fuzzy soft b-open subsets of (X,E, τ1, τ2) such that fE if fE⊆̃
⋃̃
{fiE : i ∈ Λ} there

exists a finite subset Λ0 of Λ such that fE⊆̃
⋃̃
{fiE : i ∈ Λ0}.

Definition 26. A fuzzy soft subset fE of fuzzy soft bitopological space (X,E, τ1, τ2) is
said to be (1, 2)∗-fuzzy soft b-compact if fE is (1, 2)∗-fuzzy soft b-compact as a subspace of
(X,E, τ1, τ2).

Theorem 6. Every (1, 2)∗-fuzzy soft closed subset of fuzzy (1, 2)∗-fuzzy soft b-compact
space (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft b-compact relative to 1̃E.

Proof. Let fE be a (1, 2)∗-fuzzy soft closed subset of (X,E, τ1, τ2). Then f cE is a
(1, 2)∗-fuzzy soft open set in (X,E, τ1, τ2). Let S = {giE : i ∈ Λ} be a cover of fE by
(1, 2)∗-fuzzy soft open subsets in (X,E, τ1, τ2). Then S∪̃f cE is a (1, 2)∗-fuzzy soft b-open
cover for 1̃E . Since (X,E, τ1, τ2) is a (1, 2)∗-fuzzy soft b-compact; it has a finite subcover
say S = g1E ∪̃g1E ∪̃...∪̃gnE ∪̃f cE , giE ∈̃S, i = 1, 2, ..., n. But fE and f cE are fuzzy soft disjoint.
Hence fE⊆̃g1E ∪̃g1E ∪̃...∪̃gnE ∈̃S. Thus we have shown that any (1, 2)∗-fuzzy soft b-open
cover has a finite subcover. Therefore fE is (1, 2)∗-fuzzy soft b-compact relative to 1̃E .



REFERENCES 769

Theorem 7. A (1, 2)∗-fuzzy soft b-continuous image of a (1, 2)∗-fuzzy soft b-compact space
is (1, 2)∗-fuzzy soft compact.

Proof. Consider ψ : (X,E, τ1, τ2) → (Y,E, σ1, σ2) be a (1, 2)∗-fuzzy soft b-continuous
function. Let {fiE : i ∈ Λ} be a σ1σ2-fuzzy soft open cover of 1̃E in (Y,E, σ1, σ2).
Then {ψ−1(fiE ) : i ∈ Λ} is a (1, 2)∗-fuzzy soft b-open cover of 1̃E in (X,E, τ1, τ2). Since
(X,E, τ1, τ2) is (1, 2)∗-fuzzy soft b-compact; it has a finite subcover say,
{ψ−1(f1E ), ψ−1(f1E ), ..., ψ−1(fnE )}. Since ψ is onto, {f1E , f1E , ..., fnE} is a σ1σ2-fuzzy soft
open cover of 1̃E in (Y,E, σ1, σ2) and hence (Y,E, σ1, σ2) is (1, 2)∗-fuzzy soft compact.

Theorem 8. If a map ψ : (X,E, τ1, τ2)→ (Y,E, σ1, σ2) is a (1, 2)∗-fuzzy soft b-irresolute
and a fuzzy soft subset fE of (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft compact relative to 1̃E then
the image ψ(fE) is (1, 2)∗-fuzzy soft compact relative to 1̃E in (Y,E, σ1, σ2).

Proof. Let {fiE : i ∈ Λ} be a collection of (1, 2)∗-fuzzy soft b-open sets in (Y,E, σ1, σ2)

such that ψ(fE)⊆̃
⋃̃
{fiE : i ∈ Λ}. Then fE⊆̃

⋃̃
{ψ−1(fiE ) : i ∈ Λ}, where ψ−1(fiE ) is

(1, 2)∗-fuzzy soft b-open in (X,E, τ1, τ2) is (1, 2)∗-fuzzy soft compact relative to 1̃E in
(X,E, τ1, τ2), there exists a finite sub collection {f1E , f2E , ..., fnE} such that

fE⊆̃
⋃̃
{ψ−1(fiE ) : i = 1, 2, 3..., n} that is, ψ(fE)⊆̃

⋃̃
{fiE : i = 1, 2, 3..., n}. Hence ψ(fE) is

(1, 2)∗-fuzzy soft compact relative to 1̃E in (Y,E, σ1, σ2).

5. Conclusion

In this paper, we introduced the notions of (1, 2)∗-fuzzy soft b-separated sets, (1, 2)∗-
fuzzy soft b-connectedness and (1, 2)∗-fuzzy soft b-compactness in fuzzy soft bitopological
spaces. Then, some basic topological properties of these notions were investigated. Also,
some illustrative examples were given to show the importance of the obtained theorems.
We hope that this paper will be important for researchers to studying many other concepts
and also the generalization for some important results in topology.

Acknowledgements

The author is very grateful to the editor and the reviewers for their valuable sugges-
tions.

References

[1] N.A. Taş A. Açıkgöz and T.A. Noiri. A Decomposition of some types of mixed soft
continuity in soft topological spaces. Filomat, 30(2):379–385, 2016.

[2] S.A. El-Sheikh A. Kandil, O.A.E. Tantawy and S.A. Hazza. Pairwise open (closed)
soft sets in soft bitopological spaces. Annals of Fuzzy Mathematics and Informatics,
11(4):1–20, 2016.



REFERENCES 770

[3] S.A. El-Sheikh A. Kandil, O.A.E. Tantawy and S.A. Hazza. Pairwise soft separation
axioms in soft bitopological spaces. Annals of Fuzzy Mathematics and Informatics,
4(4):571–588, 2016.

[4] A. Açıkgöz and N.A. Taş. Some new mixed soft sets. Mathematical Sciences And
Applications E-Notes, 2(2):105–118, 2014.

[5] E. Yesil C. Eksin, M. Güzelkaya and I. Eksin. Fuzzy logic approach to mimic decision
making behaviour of humans in stock management game. Proceedings of the 2008
System Dynamics Conference, 2008.

[6] D. Chen. The parametrization reduction of soft sets and its applications. Computers
and Mathematics with Applications, 49(5-6):757–763, 2005.

[7] C.C. Chou, J.M. Yih, J.F. Ding, T.C. Han, Y.H.Lim, and et al. Application of a fuzzy
EOQ model to the stock management in the manufacture system. Key Engineering
Materials, 499:757–763, 2012.

[8] G. Şenel. A New Approach to Hausdorff Space Theory via the Soft Sets. Mathematical
Problems in Engineering, 2016:6 pages, 2016.

[9] G. Şenel. Soft Topology Generated by L-Soft Sets. Journal of New Theory, 4(24):88–
100, 2018.

[10] G. Şenel and N. Çağman. Soft Closed Sets on Soft Bitopological Space. Journal of
New Results in Science, 3(5):57–66, 2014.

[11] G. Şenel and N. Çağman. Soft topological subspaces. Annals of Fuzzy Mathematics
and Informatics, 10(4):525–535, 2015.

[12] B.M. Ittanagi. Soft Bitopological Spaces. International Journal of Computer Appli-
cations, 107(7):1–4, 2014.

[13] F. Karaca and N. Taş. Decision Making Problem for Life and Non Life Insurances.
Journal of Balikesir University Institute of Science and Technology, 20(1):572–588,
2018.

[14] J.C. Kelly. Bitopological spaces. Proceedings of the London Mathematical Society,
13:71–81, 1963.

[15] P.K. Maji, R. Biswas, and A.R. Roy. Fuzzy soft sets. Journal of Fuzzy Mathematics,
9(3):589–602, 2001.

[16] D. Molodtsov. Soft set theory-first results. Comput. Math. Appl., 37:19–31, 1999.

[17] P. Mukherjee and C. Park. On fuzzy soft bitopological spaces. Mathematics and
Computer Sciences Journal (MCSJ), 10(7):1–8, 2005.



REFERENCES 771

[18] S. Karataş N. Çağman and S. Enginoglu. Soft Topology. Comp. and Math. with App.,
62(1):351–358, 2011.

[19] N.Y. Özgür N. Taş and P. Demir. An Application of Soft Set and Fuzzy Soft Set
Theories to Stock Management. Süleyman Demirel University Journal of Natural
and Applied Sciences, 21(3):791–796, 2017.

[20] N.Y. Özgür and N. Taş. A note on ”Application of fuzzy soft sets to investment
decision making problem”. Journal of New Theory, 1(7):1–10, 2015.

[21] T.Y. Öztürk and M. Karademr. Soft pair-wise b-continuity on soft bitopological
spaces. Celal Bayar University Journal of Science, 13(2):413–422, 2017.

[22] R. Biswas P. Maji and A.R. Roy. Soft set theory. Computers and Mathematics with
Applications, 45:555–562, 2003.

[23] C.W. Patty. Bitopological spaces. Duke Mathematical Journal, 34:387–392, 1967.

[24] I.L. Reilly. On bitopological separation properties. Nanta Mathematica, 29:14–25,
1972.

[25] N. Revathi and K. Bageerathi. On Soft B-open sets in soft bitopological space.
International Journal of Applied Research, 1(11):615–623, 2015.

[26] N. Revathi and K. Bageerathi. (1, 2)∗-soft b-continuous and (1, 2)∗-soft b-closed map.
Asian Journal of Mathematics and Computer Research, 17(2):111–122, 2017.

[27] S. Roy and T.K. Samanta. A note on fuzzy soft topological spaces. Annals of Fuzzy
Mathematics and informatics, 3(2):305–311, 2012.

[28] G. Yildizdan S. Yüksel, T. Dizman and U. Sert. Application of soft sets to diagnose
the prostate cancer risk. Journal of Inequalities and Applications, 2013:2013–229,
2013.

[29] A.F. Sayed. Some separation axioms in fuzzy soft bitopological spaces. J. Math.
Comput. Sci., 8(1):28–45, 2018.

[30] A.F. Sayed. On fuzzy soft b-open sets in fuzzy soft bitopological space. Journal of
Mathematics and Computer Science, 21:31–44, 2020.

[31] A.F. Sayed. On (1, 2)∗-Fuzzy Soft b-Continuity in Fuzzy Soft Bitopological Spaces.
Soft computing, 2021.

[32] M. Shabir and M. Naz. On Soft Topological Spaces. Comput. Math. Appl., 61:1786–
1799, 2011.

[33] N. Taş. On the Pasting Lemma on a Fuzzy Soft Topological Space with Mixed
Structure. Mathematical Sciences and Applications E-Notes, 8(2):15–20, 2020.



REFERENCES 772

[34] N. Taş and A. Açıkgöz. Some mixed soft operations and extremally soft disconnect-
edness via two soft topologies. Applied Mathematical Sciences, 5:490–500, 2014.

[35] B. Tanay and M.B. Kandemir. Topological structure of fuzzy soft sets. Computers
and Mathematics with Applications, 61(10):2952–2957, 2011.

[36] L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965.