PII. S0161171204401021 http://ijmms.hindawi.com © Hindawi Publishing Corp.
ON SMOOTH FUZZY SUBSPACES
S. E. ABBAS
Received 5 January 2004
We introduce a new concept of smooth topological subspaces, which coincides with the usual definition in the case whereµ=χY,Y⊂X. Also, we introduce some concepts such asq-nbd systems, continuity, separation axioms, compactness, and connectedness in this sense. Also, various characterization for some fuzzy topological concepts in this sense are given.
2000 Mathematics Subject Classification: 54A40.
1. Introduction and preliminaries. The concept of fuzzy topology was first defined in 1968 by Chang [2] and later redefined in somewhat different way by Lowen [8] and Hutton [7]. According to Šostak [11], these definitions, a fuzzy topology is a crisp sub-family of sub-family of fuzzy sets and fuzziness in the concept of openness of a fuzzy set has not been considered, which seems to be a drawback in the process of fuzzification of the concept of topological spaces. Therefore, Šostak introduced a new definition of fuzzy topology in 1985 [11], which we will call “smooth topology.” Later on he has de-veloped the theory of smooth topological spaces in [11,12]. After that, several authors [1,3,4,5,6,10] have reintroduced the same definition and studied smooth topologi-cal spaces being unaware of Šostak’s work. They referred to the fuzzy topology in the sense of Chang as the topology of fuzzy subsets.
Throughout this paper, letXbe a nonempty set,I=[0,1],I◦=(0,1], andI1=[0,1).
Forα∈I,α(x)=αfor allx∈X. The family of all fuzzy sets onXis defined byIX. For x∈Xandt∈I◦, a fuzzy pointxtis defined by
xt(y)=
t ify=x,
0 ify≠x. (1.1)
A fuzzy point xt is said to be quasicoincident with the fuzzy set U with respect to
µ∈IX if and only if t+U(x) > µ(x). We write this asx
tqU[µ]. ForU,V ∈IX,U is
quasicoincident withV with respect toµ. We denote this asUqV [µ], if there exists x∈Xsuch thatU(x)+V (x) > µ(x). Otherwise we denote the case asUqV [µ].
Let(X,T )be a Chang fuzzy topological space andxt∈µ. Then we say thatV∈Ꮽµ
is a fuzzyµ-q-nbd ofxt if there isU∈Tµsuch thatxtqU[µ]andU≤V [13].
A smooth topological space (STS) [10, 11] is an ordered pair (X,᐀), whereX is a nonempty set and᐀:IX→Iis a mapping satisfying the following conditions:
(O1) ᐀(0)=᐀(1)=1;
(O3) for every subfamily{Ai:i∈J} ⊆IX,᐀(i∈JAi)≥i∈J᐀(Ai).
The number᐀(A)is called the degree of openness ofA.
Let(X,᐀)be an STS andY⊆X. Then the mapping᐀Y:IY→Idefined by
᐀Y(U)=᐀(V ):V∈IX, V|Y=U (1.2)
is the induced smooth topology onYfrom᐀, and(Y ,᐀Y)is a subspace of(X,᐀)[10,11].
Let(X,᐀)and(Y ,᐀∗)be two STSs. A mappingf:X→Y is called fuzzy continuous
[10,11] if and only if᐀(f−1(A))≥᐀∗(A)for everyA∈IY.
2. Smooth topological subspaces. Forµ∈IXwe callᏭµ= {U∈IX:U≤µ}.
Definition2.1. Let(X,᐀)be an STS andµ∈IX. The mapping᐀µ:Ꮽµ→Idefined by
᐀µ(U)=᐀(V ):V∈IX, V∧µ=U (2.1)
is a smoothµ-topology induced over µ by᐀. For anyU∈Ꮽµ, the number᐀µ(U)is
called theµ-openness degree ofU.
It is easy to show that the above definition makes sense and to prove the following theorems.
Theorem2.2. ᐀µverifies the following properties:
(µO1) ᐀µ(0)=᐀µ(µ)=1;
(µO2) for allA,B∈Ꮽµ,᐀µ(A∧B)≥᐀µ(A)∧᐀µ(B);
(µO3) for every subfamily{Ai:i∈J} ⊆Ꮽµ,᐀µ(i∈JAi)≥i∈J᐀µ(Ai).
Remark2.3. IfY⊂X andµ=χY, we just have the usual concept of smooth
sub-space. Given᐀µandν∈Ꮽµwe can define(᐀µ)ν, the smoothν-topology induced over
νby᐀µ, in the obvious way. We have trivially᐀ν=(᐀µ)ν, that is, a smooth subspace
of a smooth subspace is also a smooth subspace.
Remark2.4. (1) Let(X,᐀)be an STS andµ∈IX. Then, for eachα∈I◦,᐀αµ= {U∈ Ꮽµ:᐀µ(U)≥α}is the fuzzyµ-topology in the sense of Macho Stadler and De Prada
Vicente [9]. Moreover,α1≤α2implies᐀α1
µ ≥᐀αµ2. Also,᐀µ(A)=sup{α:A∈᐀αµ}is a
smoothµ-topology.
(2) From a Chang fuzzy topological space (X,᐀α
µ), we can identify a smooth µ
-topology᐀αµ:Ꮽµ→I,
᐀αµ(A)=
1 ifA∈᐀α µ,
0 ifA∈᐀α µ,
(2.2)
for eachA∈Ꮽµ.
Theorem2.5. Let(X,᐀)be an STS andµ∈IX. Then, for eachα∈I
◦,U∈Ꮽµ, define an operatorClµ:Ꮽµ×I◦→Ꮽµas follows:
ForU1,U2∈Ꮽµandα,β∈I◦, the operatorClµsatisfies the following conditions:
(µC1) Clµ(0,α)=0;
(µC2) U1≤Clµ(U1,α);
(µC3) Clµ(U1,α)∨Clµ(U2,α)=Clµ(U1∨U2,α);
(µC4) Clµ(U1,α)≤Clµ(U1,β)ifα≤β;
(µC5) Clµ(Clµ(U1,α),α)=Clµ(U1,α).
Theorem2.6. Let(X,᐀)be an STS andµ∈IX. Then, for eachα∈I
◦,U∈Ꮽµ, define an operatorIntµ:Ꮽµ×I◦→Ꮽµas follows:
Intµ(U,α)=V∈Ꮽµ:U≥V ,᐀µ(V )≥α. (2.4)
ForU1,U2∈Ꮽµandα,β∈I◦, the operatorIntµsatisfies the following conditions:
(µI1) Intµ(µ−U1,α)=µ−Clµ(U1,α)andClµ(µ−U1,α)=µ−Intµ(U1,α);
(µI2) Intµ(µ,α)=µ;
(µI3) Intµ(U1,α)≤U1;
(µI4) Intµ(U1,α)∧Intµ(U2,α)=Intµ(U1∧U2,α);
(µI5) Intµ(U1,α)≥Intµ(U1,β)ifα≤β;
(µI6) Intµ(Intµ(U1,α),α)=Intµ(U1,α).
Theorem2.7. Let(X,᐀)be an STS,α∈I◦,µ∈IX,x
t∈µ, andU∈Ꮽµ. Thenxt∈
Clµ(U,α)if and only if for each V∈Ꮽµ such that᐀µ(V )≥αandxtqV [µ],UqV [µ] holds.
Proof. Letxt ∈Clµ(U,α), V ∈Ꮽµ such that᐀µ(V )≥α, xtqV [µ]. Suppose that
UqV [µ]which impliesU≤µ−V. FromxtqV [µ]we havext∈µ−V≥U. Since᐀µ(µ−
(µ−V ))≥α, thenxt∈Clµ(U,α)which is a contradiction. HenceUqV [µ].
Conversely, letV ∈Ꮽµ such that᐀µ(V )≥α, xtqV [µ], andUqV [µ]. Suppose that
xt∈Clµ(U,α). Then there isW∈Ꮽµ such that᐀µ(µ−W )≥α,W ≥U, andxt∈W.
Fromxt∈Wwe havextq(µ−W )[µ]. Then, from our hypothesesUq(µ−W )[µ]which
implies thatU≤W. This is a contradiction. Hencext∈Clµ(U,α).
3. Fuzzy µ-q-neighborhood systems. Here we build a fuzzy µ-q-neighborhood system of a fuzzy set in an STS and we introduce some of its properties. For a mapping ᏽ:Ꮽµ→IᏭµ,A∈Ꮽµ, andα∈I1, we define the familyᏽαA={B∈Ꮽµ:ᏽA(B)=ᏽ(A)(B)>α},
which will play an important role in this section.
Definition3.1. Let(X,᐀)be an STS,µ∈IX, andA∈Ꮽ
µ. Then the mappingᏽ: Ꮽµ→IᏭµ is called the fuzzyµ-q-neighborhood (µ-q-nbd, for short) ofAwith respect
to᐀µif and only if for eachα∈I1,
ᏽα A=
B∈Ꮽµ:∃C∈Ꮽµ:᐀µ(C)≥α AqC[µ]≤B. (3.1)
Remark 3.2. The real number ᏽA(B) is called the degree of µ-q-nbdness of the
fuzzy setBto the fuzzy setA. If the fuzzyµ-q-nbd system of a fuzzy setAhas the following property:ᏽA(Ꮽµ)⊆ {0,1}, thenᏽAis called theµ-q-nbd system ofA(given
Theorem3.3. Let(X,᐀)be an STS,µ∈IX, andA∈Ꮽ
µ. Then the mappingᏽA:Ꮽµ→I is the fuzzyµ-q-nbd system ofAwith respect to the᐀µif and only if
ᏽA(B)=
sup᐀µ(C):C∈Ꮽµ, AqC[µ]≤B, AqB[µ],
0, AqB[µ]. (3.2)
Proof. “If” part. Suppose that the mappingᏽA:Ꮽµ→Iis the fuzzyµ-q-nbd system
ofAwith respect to᐀µand consider the following cases.
(a) For the caseAqB[µ], suppose thatᏽA(B) >0. FromDefinition 3.1, there exists
C ∈Ꮽµ with ᐀µ(A)≥ α for allα∈ I◦ such thatAqC[µ]≤B, that is,AqB[µ] is a
contradiction. Thus,ᏽA(B)=0.
(b) For the caseAqB[µ], we may haveᏽA(B)=0 or ᏽA(B) >0. IfᏽA(B)=0, then
it is obvious thatᏽA(B)=0≤sup{᐀µ(C):C∈Ꮽµ, AqC[µ]≤B}; if sup{᐀µ(C):C∈ Ꮽµ, AqC[µ]≤B} =s > o, then there existsC∈Ꮽµsuch that᐀µ(C) >0 andAqC[µ]≤
B. We obtainᏽA(B) >0, which is a contradiction. Therefore,
ᏽA(B)=0=sup᐀µ(C):C∈Ꮽµ, AqC[µ]≤B. (3.3)
Now suppose thatᏽA(B)=s >0. For an arbitrary 0< ≤s, we haveᏽA(B) > s−, that
is,B∈ᏽs−
A . Since the mappingᏽA:Ꮽµ→Iis a fuzzyµ-q-nbd system ofA, there exists
C∈Ꮽµ with᐀µ(C)≥s−andAqC[µ]≤B, that is, sup{᐀µ(C):C∈Ꮽµ, AqC[µ]≤
B}> s−. Since >0 is arbitrary, we have
sup᐀µ(C):C∈Ꮽµ, AqC[µ]≤B≥s=ᏽA(B). (3.4)
On the other hand, let sup{᐀µ(C):C ∈Ꮽµ, AqC[µ]≤B} =n >0. Then for every
0< ≤n, there existsC∈Ꮽµ such that᐀µ(C) > n− and AqC[µ]≤B. Therefore
B∈ᏽn−
A , that is,ᏽA(B)≥n−. Sinceis arbitrary we have
ᏽA(B)≥n=sup᐀µ(C):C∈Ꮽµ, AqC[µ]≤B. (3.5)
Hence the inequality follows.
“Only if” part. For α∈I1, letB ∈ᏽα
A, that is,ᏽA(B)≥α. Then we can write α≤ ᏽA(B)=sup{᐀µ(C):C∈Ꮽµ, AqC[µ]≤B}. Then we have
ᏽα A⊆
B∈Ꮽµ:∃C∈Ꮽµ:᐀µ(C)≥α AqC[µ]≤B. (3.6)
By the same way we can show that
B
∈Ꮽµ:∃C∈Ꮽµ:᐀µ(C)≥α AqC[µ]≤B⊆ᏽαA. (3.7)
Remark3.4. InTheorem 3.3, the fuzzy subsetAofXcan be replaced by the fuzzy point onX, that is, by the special fuzzy subsetsxt. In this case,
ᏽxt(B)=
sup᐀µ(C):C∈Ꮽµ, xtqC[µ]≤B, xtqB[µ],
0, xtqB[µ].
(3.8)
Theorem3.5. Let(X,᐀)be an STS,µ∈IX, andA∈Ꮽ
µ. If the mappingᏽA:Ꮽµ→I is the fuzzyµ-q-nbd system ofAwith respect to᐀µ, then the following properties hold:
(µQ1) ᏽ0(0)=ᏽµ(µ)=1andᏽA(B) >0impliesA≤B; (µQ2) ifA1≤AandB≤B1, thenᏽA(B)≤ᏽA1(B1); (µQ3) ᏽA(B1)∧ᏽA(B2)≤ᏽA(B1∧B2);
(µQ4) ᏽA(B)≤supAqC[µ]≤B{ᏽA(C)∧ᏽC(B)},for allA,B∈Ꮽµ;
(µQ5) sup{ᏽA(U):U∈Ꮽµ} =1.
Proof. (µQ1), (µQ2), and (µQ5) follow directly fromDefinition 3.1andTheorem 3.3. (µQ3) Suppose thatᏽA(B1)=m >0 andᏽA(B2)=n >0. Then for a fixed >0 such
that≤m∧nimpliesᏽA(B1) > m−≥0 andᏽA(B2) > n−≥0. FromDefinition 3.1,
it is clear that there existsC1,C2∈Ꮽµ such that᐀µ(C1) > m−,᐀µ(C2) > n−and
AqC1[µ]≤B1,AqC2[µ]≤B2. Therefore,᐀µ(C1∧C2)≥᐀µ(C1)∧᐀µ(C2) > (m−)∧
(n−)=(m∧n)−andAq(C1∧C2)[µ]≤B1∧B2. Thus,ᏽA(B1∧B2)≥(m∧n)−.
Sinceis arbitrary, we find that
ᏽAB1∧B2≥ᏽAB1∧ᏽAB2. (3.9)
(µQ4) ᏽA(B)= sup{᐀µ(C):C ∈Ꮽµ, AqC[µ]≤B}. From Theorem 3.3, we obtain ᐀µ(C)≤ᏽA(C)and᐀µ(C)≤ᏽC(B). Thus,
sup᐀µ(C):C∈Ꮽµ, AqC[µ]≤B≤supᏽA(C)∧ᏽC(B). (3.10)
Hence
ᏽA(B)≤ sup AqC[µ]≤B
ᏽA(C)∧ᏽC(B). (3.11)
Theorem3.6. If the mappingᏽA:Ꮽµ→Isatisfies the conditions (µQ1)–(µQ5), then the mapping᐀µ:Ꮽµ→I, defined by
᐀µ(U)=
AqU[µ]
ᏽA(U), U≠0,
1, U=0,
(3.12)
whereU∈Ꮽµ, is a smoothµ-topology onX.
Proof. It is obvious that ᐀µ(0) = 1. Using (µQ2) and (µQ5) we obtain that
sup{ᏽA(B):B∈Ꮽµ} =ᏽA(µ)=1, for allA∈Ꮽµ, that is,᐀µ(µ)=1.
ForU1,U2∈Ꮽµ, ifU1∧U2=0, then it is clear that᐀µ(U1∧U2)=1≥᐀µ(U1)∧᐀µ(U2).
AqU2[µ], and applying (µQ3) we may write
᐀µU1∧U2=
Aq(U1∧U2)[µ]
ᏽAU1∧U2
≥
Aq(U1∧U2)[µ]
ᏽAU1∧ᏽAU2
=
Aq(U1∧U2)[µ] ᏽAU1
∧
Aq(U1∧U2)[µ] ᏽAU2
≥
AqU1[µ] ᏽAU1
∧
AqU2[µ] ᏽAU2
=᐀µU1∧᐀µU2.
(3.13)
Let{Ui:i∈J} ⊆Ꮽµ. Ifi∈JUi=0, then it is obvious that
᐀µ i∈J Ui =1≥
j∈J
᐀µUi. (3.14)
Now suppose thati∈JUi≠0. Considering (µQ4) and using the fact thatAq(i∈JUi)[µ]
if and only if there existsi◦∈Jsuch thatAqUi◦[µ]we observe that
ᏽA i∈J Ui
≥ᏽAUi◦
≥
AqUi◦[µ]
ᏽAUi◦
=᐀µUi◦ . (3.15) Hence, ᐀µ i∈J Ui =
Aq(i∈JUi)[µ] ᏽA i∈J Ui ≥ i∈J
᐀µUi. (3.16)
4. Fuzzyµ-continuity
Definition4.1. Let(X,᐀)and (Y ,ᐁ)be STSs,µ∈IX, andf:X→Y.f is fuzzy
µ-continuous if for eachA∈Ꮽf (µ),᐀µ(µ∧f−1(A))≥ᐁf (µ)(A)holds.
Remark4.2. Clearly, iff is fuzzy continuous, then it is also fuzzyµ-continuous, but the reciprocal is not in general true as shown by the following example.
Example4.3. LetX=Y=Iandµ=0.5. Consider the smooth topologies᐀,ᐁ:IX→I
as follows:
᐀(A)=
1 ifA=1,0,
0 otherwise, ᐁ(A)=
1 ifA=1,0, 1
2 ifA=0.5, 0 otherwise.
(4.1)
Then, the identity mapping idX:(X,᐀)→(X,ᐁ)is fuzzyµ-continuous. However it is
Lemma4.4. Let(X,᐀)and(Y ,ᐁ)be STSs andf:X→Y. Let{µj:j∈J} ⊂IXsuch that
j∈Jµj=1. Thenfisµj-continuous for eachj∈Jif and only iff is fuzzy continuous.
Proof. Due toRemark 4.2, it suffices to show that iff isµj-continuous for each
j∈J, thenfis fuzzy continuous. For eachB∈IY andj∈J, we have
᐀µj
f−1(B)∧µ
j≥ᐁf (µj)
B∧fµj (4.2)
then,{᐀(U):U∈IX, U∧µ
j=f−1(B)∧µj} ≥{ᐁ(V ):V∈IY, V∧f (µj)=B∧f (µj)}.
Byj∈Jµj=1 we haveU=f−1(B)andV=B, then
᐀f−1(B)≥ᐁ(B) ∀B∈IY. (4.3)
Hencef is fuzzy continuous.
Theorem 4.5. Let(X,᐀) and(Y ,ᐁ) be STSs,µ ∈IX, andf :X→Y an injective mapping. The following statements are equivalent.
(1)fis fuzzyµ-continuous.
(2)For eachB∈Ꮽf (µ),᐀µ(µ−(µ∧f−1(B)))≥ᐁf (µ)(f (µ)−B).
(3)For eachA∈Ꮽµandα∈I◦,f (Clµ(A,α))≤Clf (µ)(f (A),α).
(4)For eachB∈Ꮽf (µ)andα∈I◦,Clµ(µ∧f−1(B),α)≤f−1(Clf (µ)(B,α))∧µ.
(5)For eachB∈Ꮽf (µ)andα∈I◦,µ∧f−1(Intf (µ)(B,α))≤Intµ(µ∧f−1(B),α).
(6) For each xt ∈ µ and each B ∈ Ꮽf (µ), α ∈ I◦ such that ᐁf (µ)(B) ≥ α with
f (xt)qB[f (µ)], there isA∈Ꮽµsuch thatτµ(A)≥αwithxtqA[µ]andf (A)≤B.
(7)For eachxt∈µandB∈ᏽαf (xt),α∈I◦, there isA∈ᏽxαt such thatf (A)≤B. (8)For eachxt∈µand eachB∈ᏽαf (xt),f−1(B)∈ᏽαxt.
Proof. (1)⇒(2). For eachB∈Ꮽf (µ), we have
ᐁf (µ)f (µ)−B≤᐀µµ∧f−1f (µ)−B by(1)
=᐀µµ∧f−1f (µ)−f−1(B)
=᐀µµ−µ∧f−1(B).
(4.4)
(2)⇒(3). Suppose there existA∈Ꮽµandα∈I◦such that
fClµ(A,α)≤Clf (µ)f (A),α. (4.5)
There existy∈Y andt∈I◦such that
fClµ(A,α)(y) > t >Clf (µ)f (A),α(y). (4.6)
Iff−1({y})=φ, it is a contradiction becausef (Cl
µ(A,α))(y)=0. Iff−1({y})≠φ,
there existsx∈f−1({y})such that
Since Clf (µ)(f (A),α)(f (x)) < t, there existsB∈Ꮽf (µ) withᐁf (µ)(f (µ)−B)≥αand
f (A)≤Bsuch that
Clf (µ)f (A),αf (x)≤Bf (x)< t. (4.8)
Moreover, f (A)≤ B implies A ≤ f−1(B). From (2), ᐀
µ(µ−f−1(B))= ᐀µ(µ−(µ∧
f−1(B)))≥ᐁ
f (µ)(f (µ)−B)≥α. Thus,
Clµ(A,α)(x)≤f−1(B)(x)=Bf (x)< t. (4.9)
This is a contradiction for (4.7).
(3)⇒(4). For eachB∈Ꮽf (µ),α∈I◦. PutA=f−1(B)∧µ, and from (3), we have
fClµµ∧f−1(B),α≤Clf (µ)ff−1(B)∧µ,α≤Clf (µ)B∧f (µ),α. (4.10)
This implies
Clµµ∧f−1(B),α≤f−1fClµµ∧f−1(B),α≤f−1Clf (µ)(B,α)∧µ. (4.11)
Hence, Clµ(µ∧f−1(B),α)≤f−1(Clf (µ)(B,α))∧µ.
(4)⇒(5). This is easily proved fromTheorem 2.6(µI1).
(5)⇒(1). Suppose that ᐀µ(µ∧f−1(B))≥ᐁf (µ)(B), for each B ∈Ꮽf (µ). Then there
existsα∈I◦such that
᐀µµ∧f−1(B)< α≤ᐁf (µ)(B). (4.12)
ByTheorem 2.6,B=Intf (µ)(B,α). By (5),
µ∧f−1(B)=µ∧f−1Int
f (µ)(B,α)≤Intµµ∧f−1(B),α. (4.13)
On the other hand, byTheorem 2.6(µI3), we haveµ∧f−1(B)≥Int
µ(µ∧f−1(B),α). Thus,
µ∧f−1(B)=Int
µ(µ∧f−1(B),α), that is,᐀µ(µ∧f−1(B))≥α. This is a contradiction for
(4.12). Hencefis fuzzyµ-continuous.
(1)⇒(6). Letxt∈µ,α∈I◦, andB∈Ꮽf (µ)such thatᐁf (µ)(B)≥αwithf (xt)qB[f (µ)].
Then,xtqf−1(B)[µ]. By fuzzyµ-continuity off, we have᐀µ(µ∧f−1(B))≥ᐁf (µ)(B)≥α
and so,᐀µ(f−1(B))≥α. PutA=f−1(B). Then,τµ(A)≥αandf (A)=f (f−1(B))=B.
(6)⇒(3). LetA∈Ꮽµ,α∈I◦, xt ∈Clµ(A,α), andB∈Ꮽf (µ) such thatᐁf (µ)(B)≥α
with f (xt)qB[f (µ)]. By (6) there is C∈Ꮽµ such that ᐀µ(C)≥α withxtqC[µ] and
f (C)≤B. Since xt ∈Clµ(A,α), ᐀µ(C)≥α, and xtqC[µ], then by Theorem 2.7, we
have AqC[µ]which implies that f (A)qf (C)[f (µ)]and hencef (A)qB[f (µ)]. Thus, f (xt)∈Clf (µ)(f (A),α), andxt∈f−1(Clf (µ)(f (A),α))which implies that Clµ(A,α)≤
f−1(Cl
f (µ)(f (A),α)). Hencef (Clµ(A,α))≤Clf (µ)(f (A),α).
(6)⇒(7). Letxt∈µandB∈ᏽαf (xt). Then there existsC∈Ꮽf (µ)such thatᐁf (µ)(C)≥α andf (xt)qC[f (µ)]≤B. By (6) there isA∈Ꮽµsuch that᐀µ(A)≥αwithxtqA[µ]and
f (A)≤C≤B. Hence,A∈ᏽα
(7)⇒(8). Letxt∈µ and B∈ᏽαf (xt). By (7), there isC∈ᏽ
α
xt such thatf (C)≤B. So, there isA∈Ꮽµsuch thatxtqA[µ]andA≤C≤f−1(B). Hencef−1(B)∈ᏽαxt.
(8)⇒(6). Letxt∈µandB∈Ꮽf (µ)such thatᐁf (µ)(B)≥αwithf (xt)qB[f (µ)]. Then,
B∈ᏽα
f (xt). By (8),f−
1(B)∈ᏽα
xt and hence there isA∈Ꮽµ such that᐀µ(A)≥αwith xtqA[µ]≤f−1(B)andf (A)≤B.
Theorem4.6. Let(X,᐀),(Y ,ᐁ), and(Z,ᐂ)be STSs,µ∈IX,f:X→Y, andg:Y→Z. Iffis fuzzyµ-continuous andgis fuzzyf (µ)-continuous, theng◦fis fuzzyµ-continuous.
Definition4.7. Let(X,᐀)be an STS,µ∈IX, andA∈Ꮽµ. For eachα∈I◦,Ais said
to be
(1) α-fuzzyµ-regular open if and only ifA=Intµ(Clµ(A,α),α);
(2) α-fuzzyµ-regular closed if and only ifA=Clµ(Intµ(A,α),α).
Definition 4.8. Let (X,᐀) and (Y ,ᐁ)be STSs and let µ ∈IX, α∈I◦. Then, the
mapping f:X→Y is fuzzyµ-almost continuous if᐀µ(µ∧f−1(A))≥αfor each α
-fuzzyf (µ)-regular open setAinᏭf (µ).
Remark4.9. One may notice that, iffis fuzzy almost continuous [8], thenfis fuzzy µ-almost continuous, but the converse is not true in general as shown byExample 4.10. Also, iffis fuzzyµ-continuous, thenfis fuzzyµ-almost continuous, but the converse is not true in general as shown byExample 4.10.
Example4.10. We considerExample 4.3, and put
ᐂ(A)=
1 ifA=0,1, 1
2 ifA=0.5, 1
3 ifA=0.3, 0 otherwise.
(4.14)
For an STS(X,ᐂ)andµ=0.5, we have
(1) the identity mapping idX:(X,᐀)→(X,ᐁ)is fuzzyµ-almost continuous, but not
fuzzy almost continuous;
(2) the identity mapping idX:(X,᐀)→(X,ᐂ)is fuzzyµ-almost continuous, but not
fuzzyµ-continuous.
Theorem 4.11. Let(X,᐀) and(Y ,ᐁ)be STSs, µ∈IX, andf :X→Y an injective mapping. The following statements are equivalent.
(1)fis fuzzyµ-almost continuous.
(2)For eachα-fuzzyf (µ)-regular closedB∈Ꮽf (µ),α∈I◦, there exists᐀µ(µ−(µ∧
f−1(B)))≥α.
(3)For eachB∈Ꮽf (µ) andα∈I◦ such that ᐁf (µ)(B)≥αthere existsf−1(B)∧µ≤
Intµ(f−1(Intf (µ)(Clf (µ)(B,α),α))∧µ,α).
(4)For eachB∈Ꮽf (µ)andα∈I◦such thatᐁf (µ)(f (µ)−B)≥αthere existsf−1(B)∧
(5) For each xt ∈ µ and each B ∈ Ꮽf (µ), α ∈ I◦ such that ᐁf (µ)(B) ≥ α with
f (xt)qB[f (µ)], there is A ∈ Ꮽµ such that ᐀µ(A) ≥ α with xtqA[µ] and f (A) ≤
Intf (µ)(Clf (µ)(B,α),α).
(6) For each xt ∈ µ and B ∈ ᏽf (xα t), α∈ I◦ there is A ∈ ᏽ
α
xt such that f (A) ≤ Intf (µ)(Clf (µ)(B,α),α).
(7)For eachxt∈µand eachB∈ᏽαf (xt), there existsf− 1(Int
f (µ)(Clf (µ)(B,α),α))∈ᏽαxt.
Proof. (1)⇒(2) for eachα-fuzzyf (µ)-regular closedBandα∈I◦. Thenf (µ)−Bis
α-fuzzyf (µ)-regular open. Sincef is fuzzyµ-almost continuous,᐀µ(µ∧f−1(f (µ)−
B))≥αand hence᐀µ(µ∧(µ−f−1(B)))≥α. LetA(x)=(µ−(f−1(B)∧µ))(x)=µ(x)−
(f−1(B)∧µ)(x)= µ(x)−min{f−1(B)(x),µ(x)}. If µ(x)≤ f−1(B)(x), then A(x)= µ(x)−µ(x)=0 and᐀µ(A)=1≥αand hence ᐀µ(µ−(f−1(B)∧µ))≥α. If µ(x) >
f−1(B)(x), thenA=µ−f−1(B)=(µ−f−1(B))∧µand hence᐀
µ(A)=᐀µ(µ−(f−1(B))∧
µ)≥α. Thus, for eachα-fuzzyf (µ)-regular closed setB,᐀µ(µ−(f−1(B)∧µ))≥α.
(2)⇒(1). It is clear.
(1)⇒(3). LetB∈Ꮽf (µ),α∈I◦withᐁf (µ)(B)≥α. ThenB≤Intf (µ)(Clf (µ)(B,α),α)and
Intf (µ)(Clf (µ)(B,α),α)isα-fuzzyf (µ)-regular open. Sincefis fuzzyµ-almost
contin-uous,
f−1(B)∧µ≤f−1Int
f (µ)Clf (µ)(B,α),α∧µ, (4.15)
and᐀µ(µ∧f−1(Intf (µ)(Clf (µ)(B,α),α)))≥α. Thus
f−1(B)∧µ≤Int
µf−1Intf (µ)Clf (µ)(B,α),α∧µ,α. (4.16)
(3)⇒(4). This follows fromTheorem 2.6(µI1).
(4)⇒(2). LetBbeα-fuzzyf (µ)-regular closed,α∈I◦. Then by (4),
f−1(B)∧µ≥Cl
µf−1Clf (µ)Intf (µ)(B,α),α∧µ,α
=Clµf−1(B)∧µ,α.
(4.17)
Thus,᐀µ(µ−(µ∧f−1(B)))≥α(byTheorem 2.5(µC2)).
(1)⇒(5)⇒(3) and (5)⇒(6)⇒(7)⇒(5) are similar to that ofTheorem 4.5.
5. Fuzzyµ-separation axioms
Definition5.1. Let(X,᐀)be an STS,α∈I◦, andµ∈IX.µ is said to beα-fuzzy
µ-Hausdorff if for eachxt,ys(x≠y)∈µ, there areU1,U2∈Ꮽµ such that᐀µ(U1)≥α
and᐀µ(U2)≥αsuch thatxt∈U1,ys∈U2, andU1qU2[µ].
Proof. Letxt,ys(x≠y)∈µandm=µ(y)−s. Thenxt,ym(x≠y)∈µ. Sinceµis
α-fuzzyµ-Hausdorff,α∈I◦, there areU1,U2∈Ꮽµ with᐀µ(U1)≥α,᐀µ(U2)≥αsuch
thatxt∈U1,ym∈U2, andU1qU2[µ]and henceU1≤µ−U2and᐀µ(µ−(µ−U2))≥α,
which implies Clµ(U1,α)≤µ−U2. Sinceym ∈U2, s =µ(y)−m > µ(y)−U2(y)≥
(Clµ(U1,α))(y) and hence ys ∈ Clµ(U1,α). Thus, ys ∈ {Clµ(U1,α) : ᐀µ(U1) ≥α,
xt∈U1}.
Conversely, letxt,ys(x≠y)∈µ. Then,xt,yµ(y)−s(x≠y)∈µ. By hypothesis, there
is U1∈Ꮽµ with ᐀µ(U1)≥α such that xt ∈U1 and yµ(y)−s ∈Clµ(U1,α) and hence
µ(y)−s > (Clµ(U1,α))(y)which implies ys ∈µ−Clµ(U1,α)=U2 and ᐀µ(U2)≥α.
SinceU2=µ−Clµ(U1,α)≤µ−U1,U1qU2[µ]. Thereforeµisα-fuzzyµ-Hausdorff.
Definition5.3. Let(X,᐀)be an STS,α∈I◦, andµ∈IX.µ is said to beα-fuzzy
µ-regular space if for eachF∈Ꮽµ with᐀µ(µ−F)≥αand for each fuzzy pointxt∈µ
withxtqF[µ], there areU1,U2∈Ꮽµ such that᐀µ(U1)≥αand᐀µ(U2)≥αsuch that
xt∈U1,F≤U2, andU1qU2[µ].
Example5.4. Let X= {x,y,z}be a set. Define a smooth topology᐀:IX →I as follows:
᐀(U)=
1 ifU=0 or 1, 1
2 ifU=χ{x,y}, 1
2 ifU=χ{z}, 0 otherwise.
(5.1)
Thenµ=0.9 is 1/2-fuzzyµ-regular space.
Theorem5.5. Let(X,᐀)be an STS,α∈I◦, andµ∈IX. Then the following are equiv-alent.
(1)µisα-fuzzyµ-regular.
(2)For each fuzzy pointxt∈µand for eachU1∈Ꮽµwith᐀µ(U1)≥α,xt∈U1, there isU2∈Ꮽµwith᐀µ(U2)≥αsuch thatxt∈U2≤Clµ(U2,α)≤U1.
(3)For each fuzzy pointxt∈µand for eachF∈Ꮽµwith᐀µ(µ−F)≥αandxtqF[µ], there are U2,U3∈Ꮽµ with᐀µ(U2)≥α, ᐀µ(U3)≥αsuch thatxt∈U2, F ≤U3, and
Clµ(U2,α)qU3[µ].
Proof. (1)⇒(2). Letxt∈µ be a fuzzy point andU1∈Ꮽµwith᐀µ(U1)≥α,xt∈U.
Then, ᐀µ(µ−(µ−U1))≥α with xtq(µ−U1)[µ]. By (1), there are U2,U3 ∈Ꮽµ with ᐀µ(U2)≥α,᐀µ(U3)≥αsuch thatxt∈U3,µ−U1≤U2, andU2qU3[µ]. SinceU2qU3[µ],
U3≤µ−U2≤U1and hence Clµ(U3,α)≤µ−U2≤U1. Thusxt∈U3≤Clµ(U3,α)≤U1.
(2)⇒(3). Letxt∈µbe a fuzzy point,α∈I◦, andF∈Ꮽµwith᐀µ(µ−F)≥α,xtqF[µ].
Thenxt∈µ−F. By (2), there isU1∈Ꮽµ with᐀µ(U1)≥αsuch thatxt∈U1≤Clµ(U1,
α) ≤µ−F. By (2) again, there is U2 ∈ Ꮽµ with ᐀µ(U2)≥ α such that xt ∈ U2 ≤
Clµ(U2,α)≤U1≤Clµ(U1,α)≤µ−F. PutU3=µ−Clµ(U1,α). Hence there areU2,U3∈ Ꮽµwith᐀µ(U2)≥α,᐀µ(U3)≥αsuch thatxt∈U2,F≤U3, and Clµ(U2,α)qU3[µ].
6. Fuzzyµ-compactness
Definition 6.1. Let (X,᐀) be an STS,α∈I◦, and µ∈IX. Then, µ is α-fuzzy µ
-compact (resp.,α-fuzzy µ-almost compact) if and only if for each family{Ui∈Ꮽµ : ᐀µ(Ui)≥α, i∈Γ}such that(i∈ΓUi)(x)=µ(x)for all x∈X, there exists a finite
index setΓ◦⊂Γ such that(i∈Γ◦Ui)(x)=µ(x)(resp.,(
i∈Γ◦Clµ(Ui,α))(x)=µ(x)) for
allx∈X.
It is clear that ifµisα-fuzzyµ-compact, then it isα-fuzzyµ-almost compact. But the converse need not be true in general as shown by the following example.
Example6.2. LetXbe any nonempty set and let᐀:IX→I be a smooth topology
defined as
᐀(U)=
1 ifU=0 or 1, 1
3 ifU=α,for 0.4< α <0.8, 0 otherwise.
(6.1)
Then,µ=0.8 is 1/3-fuzzyµ-almost compact but not 1/3-fuzzyµ-compact.
In order to investigate for the condition under whichα-fuzzyµ-almost compact is α-fuzzyµ-compact, we set the following definition.
Definition6.3. Let(X,᐀)be an STS,α∈I◦, andµ∈IX. Then,µisα-fuzzyµ-regular
if and only if for eachU1∈Ꮽµwith᐀µ(U1)≥α,
U1=U2∈Ꮽµ:᐀µU2≥α,ClµU2,α≤U1. (6.2)
Theorem6.4. Let(X,᐀)be an STS,α∈I◦, andµ∈IX beα-fuzzyµ-regular. Then,
µisα-fuzzyµ-almost compact if and only ifµisα-fuzzyµ-compact.
Proof. Let{Ui∈Ꮽµ:᐀µ(Ui)≥α, i∈Γ}be a family such that(i∈ΓUi)(x)=µ(x)
for allx∈X. Since,µisα-fuzzyµ-regular, for each᐀µ(Ui)≥α,
Ui=
ik∈Ki
Uik∈Ꮽµ:᐀µ
Uik
≥α,ClµUik,α
≤Ui. (6.3)
Hence(i∈Γ(ik∈KiUik))(x)=µ(x)for allx∈X. Sinceµisα-fuzzyµ-almost compact, there exists a finite indexJ×KJsuch that
i∈J
ik∈KJ
ClµUik,α
(x)
=µ(x) ∀x∈X. (6.4)
Fori∈J, since(ik∈KjClµ(Uik,α))≤Ui we have(
i∈JUi)(x)=µ(x)for allx∈X.
Henceµisα-fuzzyµ-compact.
Definition 6.5. A collection σ ⊂Ꮽµ is said to be from a fuzzy µ-filterbasis, if
Theorem6.6. Let(X,᐀)be an STS,α∈I◦, andµ∈IX.µisα-fuzzyµ-compact if and only if for each fuzzyµ-filterbasisσ inµ,(U∈σClµ(U,α))(x) >0for somex∈X.
Proof. Letσ= {Ui∈Ꮽµ:᐀µ(Ui)≥α, i∈Γ}be a family such that(Ui∈σUi)(x)= µ(x)for allx∈X, and suppose that for each finite subcollection{U1,U2,...,Un}ofσ,
there isx∈Xsuch thatUi(x) < µ(x)for eachi=1,2,...,n. Thenµ(x)−Ui(x) >0 for
eachi=1,2,...,n. So,ni=1(µ(x)−Ui(x)) >0 and hence{µ−Ui:Ui∈σ}forms a fuzzy µ-filterbasis. Then,(Ui∈σClµ(µ−Ui,α))(x)=(
Ui∈σ(µ−Ui))(x)=0 for eachx∈X, which is a contradiction. Henceµisα-fuzzyµ-compact.
Conversely, suppose that there is fuzzy µ-filterbasis σ such that (U∈σClµ(U,
α))(x)=0 for eachx∈X, so that(U∈σ(µ−Clµ(U,α)))(x)=µ(x)for eachx∈X
and hence᐀µ(µ−Clµ(U,α))≥α. Sinceµ isα-fuzzyµ-compact, there is a finite
sub-set{U1,...,Un}(say) and hence(ni=1(µ−Clµ(Ui,α)))(x)=µ(x)for allx∈X, which implies(ni=1(µ−Ui))(x)=µ(x)for allx∈X. So that(
n
i=1Ui)(x)=0 for allx∈X which is a contradiction. Therefore(U∈σ(Clµ(Ui,α)))(x) >0 for eachx∈X.
Theorem6.7. Let(X,᐀)be an STS,α∈I◦, andµ∈IX.µisα-fuzzyµ-almost compact if and only if for eachα-fuzzyµ-filterbasisσ inµ,(U∈σClµ(U,α))(x) >0for some
x∈X.
Proof. Letσ= {Ui∈Ꮽµ:᐀µ(Ui)≥α, i∈Γ}be a family such that(Ui∈σUi)(x)= µ(x)for allx∈X, and suppose that for each finite subcollection{U1,U2,...,Un}of
σ,(ni=1Clµ(Ui,α))(x) < µ(x)for somex∈X. Then,ni=1(µ(x)−Clµ(Ui,α)(x)) >0 for somex∈X. Thus,β= {µ−Clµ(Ui,α):Ui∈σ}formsα-fuzzyµ-filterbasis. Since
(Ui∈σUi)(x)=µ(x)for allx∈X, hence(
Ui∈σClµ(µ−Clµ(Ui,α),α))(x)=0 for each x∈X, which is a contradiction. Hence(ni=1Clµ(Ui,α))(x)=µ(x)for somex∈X, and
µisα-fuzzyµ-almost compact.
Conversely, suppose that there isα-fuzzy µ-filterbasis σ such that (U∈σClµ(U,
α))(x)=0 for eachx∈X, so that(U∈σ(µ−Clµ(U,α)))(x)=µ(x)for eachx∈Xand
hence᐀µ(µ−Clµ(U,α))≥α. Sinceµisα-fuzzyµ-almost compact, there is a finite
sub-family{µ−Clµ(Ui,α):i=1,2,...,n}(say) such that(ni=1Clµ(µ−Clµ(Ui,α),α))(x)= µ(x)for allx∈X, which implies(ni=1µ−Clµ(µ−Clµ(Ui,α),α))(x)=0 for allx∈X. So
that(ni=1Ui)(x)=0 for allx∈X, which is a contradiction. Therefore(U∈σ(Clµ(Ui,
α)))(x) >0 for eachx∈X.
Theorem 6.8. Let(X,᐀)and(Y ,ᐁ)be STSs, α∈I◦, µ∈IX, andf :X→Y be a fuzzyµ-continuous bijective mapping. Ifµisα-fuzzyµ-compact, thenf (µ)isα-fuzzy f (µ)-compact.
Proof. Letσ= {Ui∈Ꮽf (µ):ᐁf (µ)(Ui)≥α}such that(Ui∈σUi)(y)=f (µ)(y)for ally∈Y. Sincefis fuzzyµ-continuous, then᐀µ(µ∧f−1(Ui))≥αfor eachUi∈σ. Since
f is injective map,(Ui∈σ(µ∧f− 1(U
i)))(x)=µ(x)for allx∈X. Sinceµ isα-fuzzy
µ-compact, there is a finite subfamily{µ∧f−1(U
i):i=1,2,...,n}such that(ni=1(µ∧ f−1(U
i)))(x)=µ(x)for allx∈Xand henceµ(x)=(µ∧ni=1f−1(Ui))(x)for allx∈X which impliesf (µ∧(ni=1f−1(Ui)))=f (µ)and hencef (µ)∧(
n
i=1f f−1(Ui))=f (µ). Sincef is bijective,f (µ)∧(ni=1Ui)=f (µ)and hence(
n
Theorem6.9. Let(X,᐀)and(Y ,ᐁ)be STSs,α∈I◦,µ∈IX, andf:X→Y be a fuzzy
µ-continuous bijective mapping. Ifµisα-fuzzyµ-almost compact, thenf (µ)isα-fuzzy f (µ)-almost compact.
Proof. Letσ = {Ui ∈Ꮽf (µ):ᐁf (µ)(Ui)≥α} such that (Ui∈σUi)(y)=f (µ)(y) for all y∈Y. Since f is fuzzyµ-continuous, then᐀µ(µ∧f−1(Ui))≥α for eachUi
∈σ. Sincef is injective map,(Ui∈σ(µ∧f− 1(U
i)))(x)=µ(x)for allx∈X. Sinceµ
isα-fuzzyµ-almost compact, there is a finite subfamily{µ∧f−1(U
i):i=1,2,...,n}
such that(ni=1Clµ(µ∧f−1(Ui),α))(x)=µ(x)for allx∈X, and hence byTheorem
4.5(4), we have µ(x)=(ni=1f−1(Clf (µ)(Ui,α))∧µ)(x) for allx ∈X which implies f (ni=1f−1(Cl
f (µ)(Ui,α))∧µ)=f (µ) and hence f (µ)∧(ni=1f f−1(Clf (µ)(Ui,α)))= f (µ). Sincefis bijective,f (µ)∧(ni=1Clf (µ)(Ui,α))=f (µ)and hence(ni=1Clf (µ)(Ui, α))(y)=f (µ)(y)for ally∈Y. Thusf (µ)isα-fuzzyf (µ)-almost compact.
7. Fuzzyµ-connected sets
Definition7.1. Let(X,᐀)be an STS,α∈I◦,µ∈IX, andU1,U2∈Ꮽµ. Then,U1,U2
are said to beα-fuzzyµ-separated ifU1qClµ(U2,α)[µ]andU2qClµ(U1,α)[µ].
Theorem7.2. LetU1,U2∈Ꮽµandα∈I◦. Then,
(1) ifU1 andU2 areα-fuzzyµ-separated and V1,V2∈Ꮽµ such thatφ≠V1≤U1,
φ≠V2≤U2, thenV1andV2areα-fuzzyµ-separated;
(2) ifU1qU2[µ], and either᐀µ(U1)≥α,᐀µ(U2)≥αor᐀µ(µ−U1)≥α,᐀µ(µ−U2)≥
α, thenU1andU2areα-fuzzyµ-separated;
(3) if either᐀µ(U1)≥α,᐀µ(U2)≥αor᐀µ(µ−U1)≥α,᐀µ(µ−U2)≥α, thenU1∧
(µ−U2)andU2∧(µ−U1)areα-fuzzyµ-separated.
Proof. (1) Since V1 ≤ U1. Then Clµ(V1,α)≤ Clµ(U1,α) hence U2qClµ(V1,α)[µ],
which impliesV2qClµ(U1,α)[µ]. ThusV1andV2areα-fuzzyµ-separated.
(2) Let᐀µ(µ−U1)≥α,᐀µ(µ−U2)≥α, and U1qU2[µ]. ThenU1=Clµ(U1,α) and
U2=Clµ(U2,α)sinceU1qU2[µ],U1qClµ(U2,α)[µ], andU2qClµ(U1,α)[µ]. ThusU1and
U2areα-fuzzyµ-separated.
Let᐀µ(U1)≥α,᐀µ(U2)≥α, andU1qU2[µ]. Then,U1≤µ−U2and hence Clµ(U1,α)≤
µ−U2which impliesU2qClµ(U1,α)[µ]. Similarly,U1qClµ(U2,α)[µ]. Thus,U1andU2
areα-fuzzyµ-separated.
(3) Let᐀µ(U1)≥α, ᐀µ(U2)≥α. Then,᐀µ(µ−(µ−U1))≥α,᐀µ(µ−(µ−U2))≥α,
and hence Clµ(U1∧(µ−U2),α)≤µ−U2. Thus Clµ(U1∧(µ−U2),α)qU2[µ]and hence
Clµ(U1∧(µ−U2),α)q(U2∧(µ−U1))[µ], sinceU2∧(µ−U1)≤U2. Similarly, Clµ(U2∧(µ−
U1),α)q(U1∧(µ−U2))[µ]. ThusU1∧(µ−U2)andU2∧(µ−U1)areα-fuzzyµ-separated. Let᐀µ(µ−U1)≥α,᐀µ(µ−U2)≥α. Then,U1=Clµ(U1,α),U2=Clµ(U2,α), and hence
Clµ(U2,α)q(U1∧(µ−U2))[µ]sinceU2∧(µ−U1)≤U2, Clµ(U2∧(µ−U1),α)q(U1∧(µ−
U2))[µ]. Similarly, Clµ(U1∧(µ−U2),α)q(U2∧(µ−U1))[µ]. Thus,U1∧(µ−U2) and
U2∧(µ−U1)areα-fuzzyµ-separated.
Theorem7.3. Let(X,᐀)be an STS,α∈I◦, andµ∈IX. ThenU1,U2∈Ꮽ
µareα-fuzzy
Proof. LetU1,U2∈Ꮽµbeα-fuzzyµ-separated. ThenU1≤µ−Clµ(U2,α)=V1,U2≤
µ−Clµ(U1,α)=V2,᐀µ(V1)≥α,᐀µ(V2)≥α,U2qV1[µ], andU1qV2[µ].
Conversely, letV1,V2∈Ꮽµwith᐀µ(V1)≥α,᐀µ(V2)≥αsuch thatU1≤V1,U2≤V2,
U1qV2[µ], andU2qV1[µ]. Then,᐀µ(µ−(µ−V1))≥α,᐀µ(µ−(µ−V2))≥αand hence
Clµ(U1,α)≤µ−V2≤µ−U2and Clµ(U2,α)≤µ−V1≤µ−U1. Thus, Clµ(U1,α)qU2[µ]
and Clµ(U2,α)qU1[µ]. HenceU1,U2areα-fuzzyµ-separated.
Definition7.4. U∈Ꮽµis said to beα-fuzzyµ-connected if it cannot be expressed
as the union of twoα-fuzzyµ-separated sets.
Example7.5. LetX= {a,b,c},µ∈IX, andU1,U2,U3,A∈Ꮽµbe defined as
µ(a)=0.9, µ(b)=0.8, µ(c)=0.7, U1(a)=0.5, U1(b)=0.2, U1(c)=0.6, U2(a)=0.0, U2(b)=0.4, U2(c)=0.0, U3(a)=0.0, U3(b)=0.0, U3(c)=0.1, A(a)=0.0, A(b)=0.4, A(c)=0.1.
(7.1)
Clearly᐀:IX→I, defined as
᐀(U)=
1 ifU=0 or 1, 1
2 ifU=U1, 0 otherwise,
(7.2)
is a smooth topology onX.
(1) We easily show that Clµ(U2,1/2)=Clµ(U3,1/2)=µ−U1. So,U2qClµ(U3,1/2)[µ]
andU3qClµ(U2,1/2)[µ]. ThusU2andU3are not 1/2-fuzzyµ-separated.
(2) We show thatAis 1/2-fuzzyµ-connected. In fact, letA=A1∨A2, whereA1,A2∈ Ꮽµ− {0}. Then eitherA1(b)=0.4 orA2(b)=0.4. IfA1(b)=0.4, then Clµ(A2,1/2)=
µ−U1. So,A1qClµ(A2,1/2)[µ]. IfA2(b)=0.6, similarly,A2qClµ(A1,1/2)[µ]. Thus,A1
andA2can not be 1/2-fuzzyµ-separated. HenceAis 1/2-fuzzyµ-connected.
Theorem7.6. Let(X,᐀)and(Y ,ᐁ)be STSs,α∈I◦,µ∈IXandf:X→Ybe a fuzzy
µ-continuous injective mapping. IfU∈Ꮽµisα-fuzzyµ-connected, thenf (U)isα-fuzzy
f (µ)-connected.
Proof. Suppose that there areα-fuzzyf (µ)-separated setsU1,U2∈Ꮽf (µ)such that
f (U)=U1∨U2. ByTheorem 7.3, there areV1,V2∈Ꮽf (µ)withᐁf (µ)(V1)≥α,ᐁf (µ)(V2)≥
α such thatU1≤V1, U2≤V2, U1qV2[f (µ)], and U2qV1[f (µ)]. Since f is fuzzy µ -continuous,᐀µ(µ∧f−1(V1))≥α,᐀µ(µ∧f−1(V2))≥α. Sincef is injective andV1≤
f−1(U1)andf−1(U2)areα-fuzzyµ-separated sets. Sincefis injective,U=f−1(f (U))= f−1(U1∨U2)=f−1(U1)∨f−1(U2), which is a contradiction with the fact thatU isα -fuzzyµ-connected. Hencef (U)isα-fuzzyf (µ)-connected.
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S. E. Abbas: Department of Mathematics, Faculty of Science, South Valley University, Sohag 82524, Egypt
