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Original Article

Fuzzy soft connected sets in fuzzy soft topological spaces II
A. Kandil a, O.A. El-Tantawy b, S.A. El-Sheikh c, Sawsan S.S. El-Sayed c,∗
a

Mathematics Department, Faculty of Science, Helwan University, Helwan, Egypt
Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
c
Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt
b

a r t i c l e

i n f o

Article history:
Received 18 October 2016

Revised 27 December 2016
Accepted 8 January 2017
Available online xxx
Keywords:
Fuzzy soft
Fuzzy soft
Fuzzy soft
Fuzzy soft
Fuzzy soft

a b s t r a c t
In this paper, we introduce some different types of fuzzy soft connected components related to the different types of fuzzy soft connectedness and based on an equivalence relation defined on the set of fuzzy
soft points of X. We have investigated some very interesting properties for fuzzy soft connected components. We show that the fuzzy soft C5 -connected component may be not exists and if it exists, it may not
be fuzzy soft closed set. Also, we introduced some very interesting properties for fuzzy soft connected
components in discrete fuzzy soft topological spaces which is a departure from the general topology.

sets
topological space
separated sets
connected sets
connected components

1. Introduction
The concept of a fuzzy set was introduced by Zadeh [15] in his
classical paper of 1965. In 1968, Chang [2] gave the definition of
fuzzy topology. Since Chang applied fuzzy set theory into topology
many topological notions were investigated in a fuzzy setting.
In 1999, the Russian researcher Molodtsov [9] introduced the
soft set theory which is a completely new approach for modeling uncertainty. He established the fundamental results of this new
theory and successfully applied the soft set theory into several directions. Maji et al. [8] defined and studied several basic notions of

soft set theory in 2003. Shabir and Naz [12] introduced the concept
of soft topological space.
Maji et al. [7] initiated the study involving both fuzzy sets and
soft sets. In this paper, Maji et al. combined fuzzy sets and soft
sets and introduced the concept of fuzzy soft sets. In 2011, Tanay
Kandemir [14] gave the topological structure of fuzzy soft sets.
The notions of fuzzy soft connected sets and fuzzy soft connected components are very important in fuzzy soft topological
spaces which in turn reflect the intrinsic nature of it that is in
fact its peculiarity. In fuzzy soft setting, connectedness has been
introduced by Mahanta and Das [6] and Karatas¸ et al. [5]. Recently, Kandil et al. [4] introduced some types of separated sets
and some types of connected sets. They studied the relationship
between these types.

∗

Corresponding author.
E-mail addresses: , (S.S.S. El-Sayed).

© 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license.
( />
In this paper, we extend the notion of connected components of
fuzzy topological space to fuzzy soft topological space. In Section 3,
we introduce and investigate some very interesting properties for
fuzzy soft connected components. We define an equivalence relation on the set of fuzzy soft points. The union of equivalence
classes turns out to be a maximal fuzzy soft connected set which is
called a fuzzy soft connected component. There are many types of
connected components deduced from the many types of connected
sets due to Kandil et al. [4]. Furthermore, we show that some of
these connected components may be not exists and the some if

exists, it may not be fuzzy soft closed set. Moreover, we introduced
some very interesting properties for fuzzy soft connected components in discrete fuzzy soft topological spaces which is a departure
from the general topology.

2. Preliminaries
Throughout this paper X denotes initial universe, E denotes the
set of all possible parameters which are attributes, characteristic
or properties of the objects in X. In this section, we present the
basic definitions and results of fuzzy soft set theory which will be
needed in the sequel.
Definition 2.1. [2] A fuzzy set A of 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 x ∈ X.
Let IX denotes the family of all fuzzy sets on X.

/>1110-256X/© 2017 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license.
( />
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A. Kandil et al. / Journal of the Egyptian Mathematical Society 000 (2017) 1–7

Definition 2.2. [9] Let A be a non-empty subset of E. A pair (F,
A) denoted by FA is called a soft set over X , where F is a mapping given by F: A → P(X). In other words, a soft set over X is a
parametrized family of subsets of the universe X . For a particular

e ∈ A , F(e) may be considered the set of e-approximate elements
of the soft set (F, A) and if e ∈ A, then F (e ) = φ i.e. F = {F (e ) : e ∈
A ⊆ E, F : A → P (X )}.

Definition 2.13. [10,11] Let (X, τ , E) be a fuzzy soft topological
space and fA ∈ FSS(X)E . The fuzzy soft closure of fA , denoted by
Fcl(fA ) is the intersection of all fuzzy soft closed supersets of fA , i.e.
F cl ( fA ) = {hC ; hC ∈ τ c and fA ⊆ hC }. Clearly, Fcl(fA ) is the smallest
fuzzy soft closed set over X which contains fA , and Fcl(fA ) is fuzzy
soft closed set.

Aktas¸ and Çag˘ man [1] showed that every fuzzy set may be considered as a soft set. That is, fuzzy sets are a special class of soft
sets.

Definition 2.14. [11,13] The fuzzy soft set fA ∈ FSS(X)E is called
fuzzy soft point if there exist x ∈ X and e ∈ E such that μef (x ) = α ;

Definition 2.3. [7] Let A⊆E. A pair (f, A), denoted by fA , is called
fuzzy soft set over X , where f is a mapping given by f : A −→ IX
defined by fA (e ) = μef ; where μef = 0 if e ∈ A, and μef = 0 if e ∈
A

A

A

A, where 0(x ) = 0∀ x ∈ X . The family of all these fuzzy soft sets
over X denoted by FSS(X)E .

A


(0 ≤ α ≤ 1) and μef (y ) = 0 ∀y ∈ X − {x} and this fuzzy soft point
A

is denoted by xeα or fe . The class of all fuzzy soft points of X, denoted by FSP(X)E .
Definition 2.15. [6] The fuzzy soft point xeα is said to be belonging
to the fuzzy soft set fA , denoted by xeα ∈ fA , if for the element e ∈ A,
α ≤ μef (x ). If xeα is not belong to fA , we write xeα ∈/ fA and implies
A

Definition 2.4. [3,7,10,11,13,14] The complement of a fuzzy soft set
(f, A) , denoted by (f, A)c , and defined by (f, A)c = ( f c , A ) , fAc :
A −→ IX is a mapping given by μef c = 1 − μef ∀e ∈ A. Clearly,

( fAc )c

A

= fA .

A

Definition 2.5. [7,10,11,13,14] A fuzzy soft set fE over X is said to be
a null-fuzzy soft set, denoted by 0E , if for all e ∈ E, fE (e ) = 0.
Definition 2.6. [7,10,11,13,14] A fuzzy soft set fE over X is said to
be an absolute fuzzy soft set, denoted by 1E , if fE (e ) = 1 ∀e ∈ E.
Clearly, we have (0E )c = 1E and (1E )c = 0E .
Definition 2.7. [3,7,10,11,13,14] Let fA , gB ∈ FSS(X)E . Then fA is fuzzy
soft subset of gB , denoted by fA ⊆ gB , if A⊆B and μef (x ) ≤


that α > μef (x ).
A

Definition 2.16. [11,13] A fuzzy soft point xeα is said to be a quasicoincident with a fuzzy soft set fA , denoted by xeα q fA , if α +
μef (x ) > 1. Otherwise, xeα is non-quasi-coincident with fA and deA

noted by xeα q fA .
Definition 2.17. [11,13] A fuzzy soft set fA is said to be quasicoincident with gB , denoted by fA q gB , if there exists x ∈ X such
that μef (x ) + μegB (x ) > 1, for some e ∈ A ∩ B. If this is true we can
A

say that fA and gB are quasi-coincident at x. Otherwise, fA and gB
are not quasi-coincident and denoted by fA q gB .

μegB (x )∀x ∈ X, ∀e ∈ E. Also, gB is called fuzzy soft superset of fA de-

Proposition 2.1. [11, 13] Let fA and gB be two fuzzy soft sets. Then,
fA ⊆ gB if and only if fA q (gB )c . In particular, xeα ∈ fA if and only if
xeα q (fA )c .

Definition 2.8. [3,7,10,11,13,14] Two fuzzy soft sets fA and gB on X
are called equal if fA ⊆ gB and gB ⊆ fA .

Definition 2.18. [10] Let FSS(X)E and FSS(Y)K be families of fuzzy
soft sets over X and Y, respectively. Let u : X −→ Y and p : E −→ K
be mappings. Then the map fpu is called fuzzy soft mapping from
FSS(X)E to FSS(Y)K , denoted by fpu : FSS(X)E −→ FSS(Y)K , such that:

A


noted by gB ⊇ fA . If fA is not fuzzy soft subset of gB , we written as
fA
gB .

Definition 2.9. [7,10,11,13,14] The union of two fuzzy soft sets fA
and gB over the common universe X, denoted by fA ࣶgB , is also a
fuzzy soft set hC , where C = A ∪ B and for all e ∈ C, hC (e ) = μeh =

μef
A

∨

μegB ∀e

∈ E.

C

Definition 2.10. [7,10,11,13,14] The intersection of two fuzzy soft
sets fA and gB over the common universe X, denoted by fA ࣵgB , is
also a fuzzy soft set hC , where C = A ∩ B and for all e ∈ C, hC (e ) =
μeh = μef ∧ μegB ∀e ∈ E.
C

A

Definition 2.11. [14] Let FSS(X)E be a collection of fuzzy soft sets
over a universe X with a fixed set of parameters E. Then τ ⊆FSS(X)E
is called fuzzy soft topology on X if

1. 0E , 1E ∈ τ , where 0E (e ) = 0 and 1E (e ) = 1∀e ∈ E,
2. The union of any members of τ belongs to τ .
3. The intersection of any two members of τ belongs to τ .
The triplet (X, τ , E) is called fuzzy soft topological space over X.
Also, each member of τ is called fuzzy soft open set in (X, τ , E).
Definition 2.12. [14] Let (X, τ , E) be a fuzzy soft topological space.
A fuzzy soft set fA over X is said to be fuzzy soft closed set in X, if
its relative complement fAc is fuzzy soft open set.

1. If gB ∈ FSS(X)E , then the image of gB under the fuzzy soft mapping fpu is a fuzzy soft set over Y defined by fpu (gB ) where ∀k ∈
p(E), ∀y ∈ Y,

f pu (gB )(k )(y ) =

∨ [ ∨ (gB (e ))](x ) if x ∈ u−1 (y ),0

u(x )=y p(e )=k

2. If hC ∈ FSS(Y)K , then the pre-image of hC under the fuzzy soft
−1
mapping fpu , f pu
(hC ) is a fuzzy soft set over X defined by ∀e ∈
−1
p ( K ), ∀x ∈ X,
−1
f pu
(hC )(e )(x ) = hC ( p(e ))(u(x ))

for p(e ) ∈ C,0


Definition 2.19. [10] The fuzzy soft mapping fpu is called surjective
(resp. injective) if p and u are surjective (resp. injective), also fpu is
said to be constant if p and u are constant.
Definition 2.20. [10] Let (X, τ 1 , E) and (Y, τ 2 , K) be two fuzzy soft
topological spaces and fpu : FSS(X)E −→ FSS(Y)K be a fuzzy soft mapping. Then fpu is called:
−1
1. Fuzzy soft continuous if f pu
(hC ) ∈ τ1 ∀ hC ∈ τ 2 .
2. Fuzzy soft open if fpu (gB ) ∈ τ 2 ∀ gB ∈ τ 1 .

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Definition 2.21. [5] Two non-null fuzzy soft sets fE and gE are said
to be fuzzy soft Q-separated in a fuzzy soft topological space (X, τ ,
E) if Fcl(fE )ࣵ gE = fE F cl (gE ) = 0E .
Definition 2.22. [5] Let (X, τ , E) be a fuzzy soft topological space
and fE ∈ FSS(X)E . Then, fE is called:
FSC1 -connected: if does not exist two non-null fuzzy soft open
sets hE and sE such that fE ⊆ hE ࣶsE , hE ࣵsE ⊆ fEc , fE hE = 0E , and
f E sE = 0E .
FSC2 -connected: if does not exist two non-null fuzzy soft open
sets hE and sE such that fE ⊆ hE ࣶsE , fE hE sE = 0E , fE hE = 0E ,

and fE sE = 0E .
FSC3 -connected: if does not exist two non-null fuzzy soft open
sets hE and sE such that fE ⊆ hE ࣶsE , hE ࣵsE ⊆ fEc , hE
fEc , and sE
c
fE .
FSC4 -connected: if does not exist two non-null fuzzy soft open
sets hE and sE such that fE ⊆ hE ࣶsE , fE hE sE = 0E , hE
fEc , and
fEc .
Otherwise, fE is called FSCi -disconnected set for i = 1, 2, 3, 4.
In the above definition, if we take 1E instead of fE , then the
fuzzy soft topological space (X, τ , E) is called FSCi -connected space
( i = 1, 2, 3, 4 ).
sE

1. FSCM -disconnected set if there exist two non-null fuzzy soft Qseparated sets hE , sE in X such that fE = hE sE . Otherwise, fE is
called FSCM -connected set.
2. FSCS -disconnected set if there exist two non-null fuzzy soft
weakly-separated sets hE , sE in X such that fE = hE sE . Otherwise, fE is called FSCS -connected set.
3. FSO-disconnected (respectively, FSOq -disconnected) set if there
exist two non-null fuzzy soft separated (respectively, strongly
separated) sets hE , sE in X such that fE = hE sE . Otherwise, fE
is called FSO-connected (respectively, FSOq -connected) set.
4. FSC5 -connected set in X if there does not exist any non-null
proper fuzzy soft clopen set in (fE , τ fE , E). Note that, this kind
of fuzzy soft connectedness was studied by Mahanta and Das
[6], Shabir and Naz [12].
In the above definitions, if we take 1E instead of fE , then the
fuzzy soft topological space (X, τ , E) is called FSCM -connected (respectively, FSCS -connected, FSO-connected, FSOq -connected, FSC5 connected) space.

Remark 2.3. [4] In a fuzzy soft topological space (X, τ , E). The
classes of FSO-connected, FSOq -connected, and FSCi -connected sets
for i = 1, 2, 3, 4, S, M can be described by the following diagram.

Remark 2.1. [5] The relationship between FSCi -connectedness (i =
1, 2, 3, 4 ) can be described by the following diagram:

F SC1

⇓

F SC 3

⇒
⇒

F SC 2

F SC 1

↓

F SC S
F SC 3

⇓

F SC 2

↓


F SC 4

F SC M

Definition 2.24. [4] Let fE ∈ FSS(X)E . The support of fE (e), denoted
by S(fE (e)), is the set, S( fE (e )) = {x ∈ X;fE (e)(x) > 0}.
Definition 2.25. [4] Two fuzzy soft sets fE and gE are said to be
quasi-coincident with respect to fE if μef (x ) + μegE (x ) > 1 for every
E

x ∈ S(fE (e)).

↓

F SOq

3. Equivalence relations and components
In disconnected fuzzy soft topological space (X, τ , E), the universe fuzzy soft set 1E can be decomposed into several pieces of
fuzzy soft sets, each of which is connected. As in general topological space, the whole space is decomposed into components.
In fuzzy soft setting, this decomposition is obtained in form of
unions of equivalence classes of a certain equivalence relation, defined on the set of fuzzy soft points in X. The union of equivalence
classes turns out to be a maximal fuzzy soft connected set. Accordingly, we have many types of notions of components in fuzzy soft
setting.
e

Definition 2.26. [4] Two non-null fuzzy soft sets fE and gE are said
to be fuzzy soft strongly separated in a fuzzy soft topological space
(X, τ , E) if there exist hE and sE ∈ τ such that fE ⊆ hE , gE ⊆ sE ,
fE sE = gE hE = 0E , fE , hE are fuzzy soft quasi-coincident with

respect to fE , and gE , sE are fuzzy soft quasi-coincident with respect
to gE .
Remark 2.2. [4] In fuzzy soft topological space (X, τ , E) the relationship between different notions of fuzzy soft separated sets can
be described by the following diagram.

fuzzy soft strongly separated

⇓

fuzzy soft separated
fuzzy soft Q -separated

←→ F SO

F SC 4

Definition 2.23. [4] Two non-null fuzzy soft sets fE and gE are said
to be:
1. Weakly separated sets in a fuzzy soft topological space (X, τ , E)
if Fcl(fE ) q gE and fE q Fcl(gE ).
2. Separated sets in a fuzzy soft topological space (X, τ , E) if there
exist non-null fuzzy soft open sets hE and sE such that fE ⊆ hE ,
gE ⊆ sE and fE sE = gE hE = 0E .

3

⇒

⇓


fuzzy soft weakly separated

e

e

connected set fA such that xα1 ∈ fA and yβ2 ∈ fA for i = 1 , 2, S, M,
O, Oq }
Then, Ei is an equivalence relation on FSP(X)E .
Proof. As a sample we will prove the case of i = 1. Reflexivity follows from the fact that for each fuzzy soft point xeα in X, there
exists a fuzzy soft point xe1 in X, which is a FSC1 -connected and obviously contains xeα . Symmetry is obvious. To show transitivity, let
e
e
e
e
e
xα1 , yβ2 and zγ3 be fuzzy soft points in X such that (xα1 , yβ2 ) ∈ E1
e

e

and (yβ2 , zγ3 ) ∈ E1 . Then, there exist FSC1 -connected sets fA and gB
e
e
e
e
in X such that xα1 ∈fA , yβ2 ∈ fA and yβ2 ∈ gB , zγ3 ∈ gB . Therefore,

β ≤ μef2 (y ) and β ≤ μeg2B (y ). Hence, fA gB = 0E . So by Theorem
A


Definition 2.27. [4] A fuzzy soft set fE in a fuzzy soft topological
space (X, τ , E) is called:

e

Proposition 3.1. For fuzzy soft points xα1 and yβ2 in X define a relation Ei as follows:
e
e
e
e
Ei = {(xα1 , yβ2 ); xα1 , yβ2 ∈ FSP(X)E and there exists a FSCi -

e

4.10 in [9], fA ࣶgB is a FSC1 -connected. Also, we have xα1 ∈ fA ࣶgB
e
and zγ3 ∈ fA ࣶgB . Therefore, E1 is an equivalence relation. Similarly,
Ei is an equivalence relation for i = 2, S, M, O, Oq .

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Let xeα be a fuzzy soft point in X and Ei be the equivalence relation on FSP(X)E , described as above. Then the equivalence class
determined by xeα , is denoted by Ei (xeα ) for i = 1, 2, S, M, O, Oq .
Definition 3.1. The union ࣶ Ei (xeα ) of all fuzzy soft points contained in the equivalence class Ei (xeα ) is called a Ci -component of
the universe fuzzy soft set 1E , determined by xeα . We denoted it by
Ci (xeα ) for i = 1, 2, S, M, O, Oq .
Theorem 3.1. For each fuzzy soft point xeα ∈ FSP(X)E , the component
Ci (xeα ) is the maximal FSCi -connected (respectively, FSO-connected,
FSOq -connected) set in X containing xeα for i = 1, 2, S, M.
Proof. As a sample we will prove the case of C1 (xeα ). Let xeα ∈
FSP(X)E and let {(fA )i ; i ∈ I} be the family of FSC1 -connected sets
in X, containing xeα . We claim C1 (xeα ) = ( fA )i .
i∈I

Firstly, we show that

μe

( f A )i ( y ) =

i∈I

( fA )i ⊆ C1

( xe

α ). Let y ∈ X, e ∈ E,

βi for each i ∈ I and supβi = β . Then, μe
i∈I


supβi = β .

i∈I

( f A )i ( y ) =

i∈I

Now, if β = 0, we have nothing to prove. Suppose β = 0. Then
for every real number > 0, there exists i ∈ I such that μe( f ) (y ) =
A i

βi > β − . Therefore, for each fuzzy soft point yeβ − where 0 <
< β , there exists a fuzzy soft set (fA )i such that yeβ − ∈ (fA )i .
Since (fA )i is a FSC1 -connected set, containing xeα , it follows that
(xeα , yeβ − ) ∈ E1 and hence yeβ − ∈ E1 (xeα ) for every 0 < < β .
Now, let {(yeβ ) j ; j ∈ J } be the family of all fuzzy soft points in

X with support y which are E1 -related to xeα . Then {yeβ − }0< <β
⊆{(yeβ ) j ; j ∈ J } ⊆ E1 (xeα ). Therefore,
yeβ − ∈
(yeβ ) j ∈ E1 (xeα ).
j∈J
ye
0< <β β −
C1 (xeα ).

But
⊆


0< <β
E1 (xeα )

= yeβ . Hence, yeβ ∈

Conversely, we show that C1 (xeα ) ⊆

and

{ ( ye

= C1 (xeα ) and so

i∈I

i∈I

( f A )i

( fA )i . Let y ∈ X, e ∈ E

β ) j ; j ∈ J } be the family of all fuzzy soft points in X with

support y such that (yeβ ) j ∈ E1 (xeα ). Suppose, supβ j = β . Then,

μe E

1


( xe

α

( y ) = μe
)

j∈J

( ye

)
β j

j∈J

(y ) = β .

Now, since (yeβ ) j ∈ E1 (xeα ) , there exists for every j ∈ J a
FSC1 -connected set (fA )j such that xeα ∈ (fA )j and (yeβ ) j ∈ (fA )j .

Hence, the family of fuzzy soft sets {(fA )j ; j ∈ J }⊆ {(fA )i ; i ∈
I}. Therefore, sup{μe( f ) (y )} = μe ( f ) (y ) ≤ μe ( f ) (y ). But, β =
A j

j∈J

j∈J

A j


i∈I

A i

supβ j ≤ sup{μe( f ) (y )}. Therefore, β ≤ μe ( f ) (y ). Hence, C1 (xeα ) ⊆
A j
A i
j∈J

i∈I

( f A )i .

j∈J

i∈I

In analogy with the general topological spaces, in an indiscrete
fuzzy soft topological space, 1E is the only C1 -components (C2 components). In a discrete general topological space, singletons are
connected sets and hence components. This feature is too is retained in the fuzzy soft setting but with an interesting departure
in the case of FSC1 -connectedness, as reflected in the following results.
Theorem 3.3. In a discrete fuzzy soft topological space, the only
FSC1 -connected sets are fuzzy soft points with value one.
Proof. Let xe1 be a fuzzy soft point in a discrete fuzzy soft topological space (X, τ , E). Let hC and sD be fuzzy soft open sets in X such
that xe1 ∈ hC ࣶsD , hC ࣵsD ⊆ (xe1 )c . Then we have either (μeh (x ) = 1
C

and μesD (x ) = 0) or (μeh (x ) = 0 and μesD (x ) = 1). Therefore, xe1
C


hC = 0E or xe1 sD = 0E . Hence, xe1 is a FSC1 -connected.
Next, to show that each fuzzy soft point xeα , where 0 < α < 1,
has a FSC1 -disconnection, what is required, is the construction of
two fuzzy soft open sets uN and jL in X satisfying xeα ∈ uN ࣶjL , uN ࣵjL
⊆ (xeα )c and xeα uN = 0E = xeα jL . Now, consider any fuzzy soft
sets uN and jL in X such that μeuN (x ) = max{α , 1 − α} and μej (x ) =
L

min{α , 1 − α}. Then, uN and jL are FSC1 -disconnection of xeα .
Finally, we construct a FSC1 -disconnection for any fuzzy soft set
in X, which is not a fuzzy soft point. Let fA be any fuzzy soft set
which takes non-zero values at least at two distinct points y and z
in X. Suppose μef (y ) = α and μef (z ) = β . Now, define fuzzy soft
A

μehC (y ) = α , μehC (z ) = 0 and μehC (x ) = μefA (x ) ∀x ∈ X − {y, z}
μesD (y ) = 0, μesD (z ) = β and μesD (x ) = 1 − μefA (x ) ∀x ∈ X − {y, z}
It is clear that, hC and sD form FSC1 -disconnection of fA .
Theorem 3.4. In a discrete fuzzy soft topological space, fuzzy soft
points are only FSC2 -connected sets.
Proof. Let xeα be a fuzzy soft point in discrete fuzzy soft topological space. Let hC and sD be fuzzy soft open sets in X such
that xeα ∈ hC ࣶsD , xeα hC sD = 0E . Then, we have either (μeh (x ) ≥
C

α , μesD (x ) = 0) or (μeh (x ) = 0, μesD (x ) ≥ α ). Therefore, xeα hC = 0E
C

or xeα sD = 0E . Hence, xeα is a FSC2 -connected.
Next, we construct a FSC2 -disconnection for any fuzzy soft in X,

which is not a fuzzy soft point. Let fA be any fuzzy soft set which
takes non-zero values at least at two distinct points y and z in X.
Suppose μef (y ) = β and μef (z ) = γ . Now, define fuzzy soft sets hC
A

( xe

That C1 α ) is the maximal FSC1 -connected set containing α ,
now follows from the fact that, if gB is any FSC1 -connected set in
X containing xeα , then gB ∈ {(fA )i ; i ∈ I} and hence gB ⊆ ( fA )i =
xe

i∈I

C1 (xeα ).

Theorem 3.2. In a fuzzy soft topological space (X, τ , E), the universe
fuzzy soft set 1E is the disjoint union of its Ci -components for i = 1,
2, S, M, O, Oq .
Proof. As a sample we will prove the case of C1 -component.
Let {C1i (xeα ); i ∈ I} be the family of C1 -components of 1E in X.
Then
C1i (xeα ) ⊆ 1E . Since each fuzzy soft point xe1 ∈ E1 (xeα )
i∈I

⊆ C1i (xeα ), then 1E ⊆

i∈I

C1i (xeα ). Moreover, if two C1 -components


C1 (xeα ) and C1 (ytβ ) are intersecting, then C1 (xeα )
connected set in X. Hence C1
of Theorem 3.1

( xe

α ) and C1

(yt

C1 (ytβ ) is a FSC1 -

β ) are identical in view

A

sets hC and sD , as follows:

and sD , as follows:

A

μehC (y ) = β , μehC (z ) = 0 and μehC (x ) = μefA (x ) ∀x ∈ X − {y, z}
μesD (y ) = 0, μesD (z ) = γ and μesD (x ) = 0 ∀x ∈ X − {y, z}
It is clear that, hC and sD form FSC2 -disconnection of fA .
Corollary 3.1. In a discrete fuzzy soft topological space, fuzzy soft
points with value 1 are the only C1 -components (C2 -components).
Let β 1 be the set of all fuzzy soft points in a fuzzy soft topological
2


space (X, τ , E), whose values are greater than

1
2.

Proposition 3.2. For fuzzy soft points xeα and ytβ in β 1 define a re2
lation E3∗ (E4∗ ) as follows:
E3∗ (E4∗ ) = {(xeα , ytβ ); there exists a FSC3 -connected (FSC4 connected) set fA in X such that xeα ∈ fA and ytβ ∈ fA }
Then E3∗ (E4∗ ) is an equivalence relation on β 1 .
2

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Proof. Reflexivity follows from the fact that each fuzzy soft point
xeα in β 1 is a FSC3 -connected set and obviously contains xeα . Sym2

e

e

e


metry is obvious. To show transitivity, let xα1 , yβ2 and zγ3 be fuzzy
e
e
e
e
soft points in β 1 such that (xα1 , yβ2 ) ∈ E3∗ and (yβ2 , zγ3 ) ∈ E3∗ .
2

e

Then, there exist FSC3 -connected sets fA and gB in X such that xα1
e
e
e
e
∈ fA , yβ2 ∈ fA and yβ2 ∈ gB , zγ3 ∈ gB . Therefore, β ≤ μ f2 (y ) and
A

β ≤ μeg2B (y ). Hence, fA and gB are overlapping at y. So by Theorem
e

4.12 in [9], fA ࣶgB is a FSC3 -connected. Also, we have xα1 ∈ fA ࣶgB
e
and zγ3 ∈ fA ࣶgB . Therefore, E3∗ is an equivalence relation. Similarly,
∗
E4 is an equivalence relation.
Here, the equivalence relation E3∗ (E4∗ ) partitions the set of fuzzy
soft points β 1 into equivalence classes. As usual, we shall de2


note an equivalence class containing the fuzzy soft point xeα by
E3∗ (xeα )(E4∗ (xeα )).
Definition 3.2. Let xeα be a fuzzy soft point in β 1 and {(fA )i ; i ∈
2

I} be the family of FSC3 -connected (FSC4 -connected) sets in X containing xeα . Then, the union ( fA )i is called a C3 -quasicomponent
i∈I

(respectively, C4 -quasicomponent) of 1E containing xeα and is denoted by C3∗ (xeα ) (respectively, C4∗ (xeα )).
Theorem 3.5. For each fuzzy soft point xeα in β 1 , the quasicompo2
nent C3∗ (xeα )(C4∗ (xeα )) is a FSC3 -connected (FSC4 -connected) set in X,
∗
e
∗
e
containing the union ࣶ E3 (xα )( E4 (xα )).
Proof. In view of Corollary 4.2 in [9], C3∗ (xeα ) is a FSC3 -connected
set in X, since xeα ∈ ( fA )i . Now, E3∗ (xeα ) ⊆ C3∗ (xeα ) follows from
i∈I
the fact that if ytβ E3∗ xeα ,
containing xe and yt .

α

there is a FSC3 -connected set ( fA )i◦ in X

β

5


there exist FSC3 -connected sets fA and gB in ψ such that xeα ∈ fA ,
ytβ ∈ fA and ytβ ∈ gB , zγs ∈ gB . Now, two cases arise:
Case I. β > 12 . Then, the fuzzy soft sets overlap at y. Hence, by
Theorem 4.12 in [9], fA ࣶ gB is a FSC3 -connected set in ψ such
that xeα ∈ fA ࣶ gB and zγs ∈ fA ࣶgB . So, xeα E3 zγs .

Case II. β ≤ 12 . Choose the C3 -quasicomponent C3∗ (yt1 ), which
contains yt1 , hence also ytβ . Now, the fuzzy soft sets gB and C3∗ (yt1 )

overlap at y, so gB C3∗ (yt1 ) is a FSC3 -connected set. By the same argument fA gB C3∗ (yt1 ) is also FSC3 -connected set in ψ , containing
both xeα and zγs . Hence, E3 is an equivalence relation.
Now, we attain the desired objective of decomposing 1E into
disjoint, maximal FSC3 -connected (FSC4 -connected) sets via the
equivalence classes defected by the equivalence relation E3 ( E4 ), as
defined in Proposition 3.2 Let E3 (xeα )(E4 (xeα )) denoted the equivalence class containing the fuzzy soft point xeα .
Definition 3.3. The union ࣶ E3 (xeα )( E4 (xeα )) of all fuzzy soft
points contained in the equivalence class E3 (xeα )(E4 (xeα )) is
called C3 -component (C4 -component) of 1E , and is denoted by
C3 (xeα )(C4 (xeα )).
Theorem 3.7. For each fuzzy soft point xeα ∈ FSP(X)E , the C3 - component (C4 -component) C3 (xeα )(C4 (xeα )) is the maximal FSC3 - connected
(FSC4 -connected) set in X, containing xeα .
Proof. We claim that, for each fuzzy soft point xeα , C3 (xeα ) =
( fA )i , where {(fA )i ; i ∈ I} is the family of those members of ψ
i∈I

which are FSC3 -connected, and contains the fuzzy soft point xe1 .
The family {(fA )i ; i ∈ I} is non-empty since C3 -quasicomponent
C3∗ (xeα ) ∈ ψ .
Firstly, we show that ( fA )i ⊆ C3 (xeα ). Let y ∈ X, t ∈ E and
i∈I


Theorem 3.6. In a fuzzy soft topological space (X, τ , E), the universe
fuzzy soft set 1E is the overlapping union of its C3 -quasicomponents
(C4 -quasicomponents).
Proof. Let {C3∗i (xeα ); i ∈ I} be the family of C3 -quasicomponents of
1E in X. Then
C3∗i (xeα ) ⊆ 1E . Since each fuzzy soft point xe1 ∈
i∈I

E3∗ (xeα ) ⊆ C3∗i (xeα ), then 1E ⊆

i∈I

C3∗i (xeα ). Moreover, let C3∗ (xeα ) be the
xe

yt

C3 -quasicomponents of 1E containing α , and β be a fuzzy soft
point in β 1 such that ytβ ∈
/ E3∗ (xeα ) . Now, if the quasicomponent
2

C3∗ (xeα ) and C3∗ (ytβ ) are overlapping, then C3∗ (xeα )

C3∗ (ytβ ) is a FSC3 -

connected set in X, by Theorem 3.6 and Theorem 4.12 in [9]. Hence,
ytβ E3∗ xeα and so ytβ ∈ E3∗ (xeα ) which is a contradiction.
Now, in order to introduce the concept of C3 -components (respectively, C4 -components), we begin with the following notions.

Let ϕ be the family of C3 -quasicomponents (C4 -quasicomponents)
of 1E and let ψ be the family of arbitrary unions of members of ϕ .
Then, we prove the following proposition.

Proposition 3.3. For any fuzzy soft points xeα and ytβ in FSP(X)E , define a relation E3 ( E4 ), as follows: xeα E3 ytβ (xeα E4 ytβ ) iff there exists
a FSC3 -connected (FSC4 -connected) set fA in ψ such that α ∈ fA and
ytβ ∈ fA . Then E3 (E4 ) is an equivalence relation on FSP(X)E .
xe

xe

Proof. Let α be a fuzzy soft point in X. Then there exists a C3 component, in particular C3∗ (xe1 ), which contains xeα . Hence, the relation E3 is reflexive. Symmetry is obvious. Next, let xeα , ytβ and zγs
be fuzzy soft points in β 1 such that xeα E3 ytβ and ytβ E3 zγs . Then,
2

suppose μt ( f ) (y ) = β . If β = 0, we have nothing to prove. If β
A i
i∈I

= 0, suppose μt( f ) (y ) = βi for each i ∈ I. Now, the fuzzy soft
A i
point ytβ ∈ (fA )i for each i ∈ I . Therefore, ytβ E3 xe1 , since the
i

i

C3 -quasicomponent C3∗ (xe1 ) is a FSC3 -connected set such that xeα
∈ C3∗ (xe1 ) ∈ ψ . Therefore, ytβ E3 xeα for each i ∈ I . Hence, ytβ ∈
i


i

E3 (xeα ), for each i ∈ I and so βi ≤ μCt (xe ) (y ) for each i ∈
3 α
I. Then β = sup{βi } ≤ μCt (xe ) (y ) implies ytβ ∈ C3 (xeα ). Hence ( fA )i

C3 (xeα ) =

α

3

i∈I

⊆ C3 (xeα ).
Conversely, we show that C3 (xeα ) ⊆
E and suppose

μCt (xe ) (y )
3 α

i∈I

i∈I

( fA )i . Let y ∈ X, t ∈

= γ . Again if γ = 0, we have noth-

ing to prove. Suppose γ = 0, and {ytγi ; i ∈ I1 } be the family of

fuzzy soft points such that ytγi E3 xeα . Then, clearly, μCt (xe ) (y ) =
3

μt

i∈I1

ytγ

i

α

(y ) = sup{γi } = γ . Since ytγi E3 xeα for each i ∈ I1 , there exi∈I1

ists a FSC3 -connected set ( fA )γi ∈ ψ such that xeα ∈ ( fA )γi and ytγi
∈ ( fA )γi . Now, for each i ∈ I1 , consider the fuzzy soft set (fA )i , define as follows: ( fA )i = ( fA )γi C3∗ (xe1 ). Then, (fA )i ∈ ψ is a FSC3 connected set, such that xe1 ∈ (fA )i and ytγi ∈ (fA )i . Therefore,
ytγi
∈
∈

i∈I1
i∈I

μt
i∈I

( fA )i , but {(fA )i ; i ∈ I1 } ⊆ {(fA )i ; i ∈ I} and so, we have

( fA )i . Now, γ = μt

i∈I1

ytγ

i

(y ) ≤ μt
i∈I

( fA )i (y ). Therefore,

μCt

i∈I1
i∈I1

e
3 ( xα )

ytγi

(y ) ≤

e
( fA )i (y ) and so, C3 (xα ) ⊆ i∈I ( f A )i .

In view of Corollary 4.2 in [9], the C3 -component C3 (xeα ) is a
FSC3 -connected set, since xe1 ∈ ( fA )i . To show that C3 (xeα ) is a
i∈I


maximal FSC3 -connected set containing xeα , let gB be any FSC3 connected set containing xeα , such that C3 (xeα ) ⊆ gB . Then, the fuzzy
soft set, defined as gB C3∗ (ye1 ) for every y ∈ S(gB (e)). Therefore, gB

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A. Kandil et al. / Journal of the Egyptian Mathematical Society 000 (2017) 1–7

⊆ E3∗ (xe1 ) ⊆ C3∗ (xe1 ) . But, C3∗ (xe1 ) ⊆ C3 (xeα ) as C3∗ (xe1 ) ∈ ψ is a FSC3 connected set, containing xe1 . Thus, we have gB = C3 (xeα ).

Definition 3.5. For any fuzzy soft points xeα and ytβ in FSP(X)E ,

Theorem 3.8. In a fuzzy topological space (X, τ , E), the universe
fuzzy soft set 1E is the disjoint union of its C3 -components (respectively, C4 -components).

connected set fA such that xeα ∈ fA and ytβ ∈ fA .
Example 3.1 shows that E5 may not be reflexive. E5 is obviously
symmetric. By using Theorem 4.10 in [9], it can be readily verified
that E5 is transitive.

Proof. As a sample, we prove the case of C3 -components. Let
{C3i (xeα ); i ∈ I} be the family of C3 -components of 1E in X. Then,
it can be verified that
C3i (xeα ) = 1E . Next, suppose the C3 -


Theorem 3.11. Let (X, τ , E) be a fuzzy soft topological space and xeα
∈ FSP(X)E . Then, xeα is not a FSC5 -connected iff there exists a 0 = β
< α , fA and gB ∈ τ such that μef (x ) = β and μegB (x ) = 1 − β .

i∈I

components C3 (xeα ) and C3 (ytβ ) intersect at a point z. Then

μCs

e
3 ( xα )

C3 (yt )
β

(z ) = 0. Hence, μCs

e
3 ( xα )

(z ) = γ1 and μCs

t
3 ( yβ )

(z ) = γ2

where γ 1 = 0 and γ 2 = 0. Now, consider the C3 -quasicomponent

C3∗ (z1s ), containing the fuzzy soft point z1s , which overlaps with
C3 -component C3 (xeα ) and C3 (ytβ ) at z. Therefore, C3 (xeα ) C3∗ (z1s )
and C3 (ytβ )

C3∗ (z1s ) are FSC3 -connected sets, containing the fuzzy

xe

yt

soft points α and β respectively, and also the fuzzy soft point
z1s . Since C3 (z1s ) is the maximal FSC3 -connected set containing
the fuzzy soft point z1s . Therefore, C3 (xeα ) C3∗ (z1s ) ⊆ C3 (z1s ) and
C3 (ytβ ) C3∗ (z1s ) ⊆ C3 (z1s ) so that C3 (xeα ) ⊆ C3 (z1s ) and C3 (ytβ ) ⊆
C3 (z1s ). Now, as C3 (xeα ) and C3 (ytβ ) are maximal FSC3 -connected
sets containing the fuzzy soft points xeα and ytβ respectively, C3 (xeα )
and C3 (ytβ ) are identical.
Theorem 3.9. For each fuzzy soft point xeα in X, the C3 -component
(C4 -component) C3 (xeα ) (C4 (xeα )) is a fuzzy soft closed set in X.
Proof. In view of Theorem 4.15 in [9], F cl (C3 (xeα )) is a FSC3 connected set in X. Moreover, xeα is contained in F cl (C3 (xeα )), as
xeα ∈ C3 (xeα ) ⊆ F cl (C3 (xeα )). Since C3 (xeα ) is the maximal FSC3 connected set containing xeα , it follows that C3 (xeα ) and F cl (C3 (xeα ))
are identical.
Again, it is obvious that an indiscrete fuzzy soft topological
space, 1E is the only C3 -component (C4 -component). Moreover,
when the fuzzy soft topological space is discrete, we state the following result:
Theorem 3.10. In a discrete fuzzy soft topological space, fuzzy soft
points are the only FSC3 -connected (FSC3 -connected) sets.
Proof. Immediate.
Therefore, in a discrete fuzzy soft topological space, the C3 component (C4 -component) are only the fuzzy soft points with
value 1.

Definition 3.4. Let fA be a fuzzy soft set in a fuzzy soft topological
space (X, τ , E). The maximal FSC5 -connected set containing fA is
called the C5 -component of fA .
Remark 3.1. The C5 -component of a fuzzy soft set may not exist
as shown by the following example:
Example 3.1. Let X = {a, b}, E = {e1 , e2 } and τ = {1E , 0E , {(e1 ,
{a 1 } ), (e2 , {b 21 } )}, {(e1 , {b 1 } ), (e2 , {a 12 } )}, {(e1 , {a 12 , b 1 } ), (e2 , {a 12 ,
2

2

2

b 1 } )}} be a fuzzy soft topology defined on X. Let fA = {(e1 , {b0.7 })}.

define a relation E5 , as follows: xeα E5 ytβ iff there exists a FSC5 -

A

Proof. Let xeα be not FSC5 -connected. Then, xeα contains a non-null
proper fuzzy soft clopen set xeβ (say). Therefore, there exist fuzzy
soft sets fA ∈ τ , gB ∈ τ c such that fA
∈

τ c,

gcB

then


∈ τ and

μegc (x )
B

xeα = gB

= 1 − β.

xeα = xeβ . Since gB

Conversely, let there exist a 0 = β < α such that there exist
fA and gB ∈ τ satisfying μef (x ) = β and μegB (x ) = 1 − β . Then, gcB ∈
A

τ c and μegc (x ) = β . Also, fA xeα = gcB xeα = xeβ and so xeβ is nonB

null proper fuzzy soft clopen set in xeα . Therefore, xeα is not FSC5 connected.
Remark 3.2. E5 is an equivalence relation iff 1E is a FSC5 connected set and then it is the only C5 -component of (X, τ , E).
Remark 3.3. The C5 -component of a fuzzy soft set if it exists, may
not be fuzzy soft closed as shown by the following example:

Example 3.2. Consider the fuzzy soft topological space (X, τ , E)
defined in Example 3.1. fE = {(e1 , {a 1 } ), (e2 , {b 12 } )} is a FSC5 2

connected.

Solution. Let gE be any fuzzy soft subset of X containing fE .
Then, gE is of the form fE ⊆ gE = {(e1 , {aα , bβ }), (e2 , {aγ , bδ })}
where α , δ ≥


and γ , β > 0. Then, {(e1 , {a 1 , bβ } ), (e2 , {aγ , b 1 } )}

1
2

2

2

or {(e1 , {a 12 , b 1 } ), (e2 , {a 12 , b 1 } )} is a non-null proper fuzzy soft
2

2

clopen set in gE according as γ , β < 12 or γ , β ≥ 12 . So, gE is not
FSC5 -connected. Therefore, fE is the C5 -component of fE and it is
not fuzzy soft closed.
4. Conclusion
In this paper, we define on the set of fuzzy soft points in X an
equivalence relation. The union of equivalence classes turns out to
be a maximal fuzzy soft connected set which is called a fuzzy soft
connected component. According to Remark 2.3, we have many
types of connected components in fuzzy soft setting. The universe
fuzzy soft set 1E is the disjoint union of its Ci -components for
i = 1, 2, S, M, O, Oq . Furthermore, we introduced some very interesting properties for fuzzy soft connected components in discrete
fuzzy soft topological spaces which is a departure from the general topology such that in a discrete fuzzy soft topological space,
the C3 -component (C4 -component) are only the fuzzy soft points
with value 1. Also, we find that: for each fuzzy soft point xeα in X,
the C3 -component (C4 -component) C3 (xeα ) (C4 (xeα )) is a fuzzy soft

closed set in X. Moreover, we prove that the C5 -component of a
fuzzy soft set may not exist and the C5 -component of a fuzzy soft
set if it exists, may not be fuzzy soft closed set.

2

Since {(e1 , {b 12 } )} is a non-null proper fuzzy soft clopen set in fA ,
then fA is not a FSC5 -connected set. Also, there does not exist any
FSC5 -connected set containing fA . So, fA has no C5 -component.

Acknowledgment
The author would like to thank the referees for their useful
comments and valuable suggestions given to this paper.

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