Generated I-fuzzy topological spaces

Fuzzy Sets and Systems154 (2005) 103–117www.elsevier.com/locate/fss

GeneratedI-fuzzy topological spaces�

Yue Yueli∗, Fang Jinming

Department of Mathematics, Ocean University of China, Qingdao 266071, People’s Republic of China

Received 23 October 2003; received in revised form 19 January 2005; accepted 9 March 2005Available online 30 March 2005

Abstract

In this paper, we extend Lowen functors� and� to I-fuzzy topological spaces (or Kubiak–Šostak fuzzy topologicalspaces) and study their properties. Then we introduce generatedI-fuzzy topological spaces and weakly generatedI-fuzzy topological spaces. Finally, we study the connections on a kind of compactness and separation propertiesbetween fuzzifying topological space(X, �) and its corresponding generatedI-fuzzy topological space(X,�(�)).© 2005 Elsevier B.V. All rights reserved.

Keywords:Lowen functors; GeneratedI-fuzzy topological space;I-fuzzy continuous function;I-fuzzy N-compactness;IFT 1;IFT 2

0. Introduction

Since Chang[1] introduced fuzzy theory into topology, many authors have discussed various aspects offuzzy topology. However, in a completely different direction, Höhle [3] created the notion of a topologybeing viewed as anL-subset of a powerset. Then Kubiak [5], Šostak [13] independently extended Höhle’snotion toL-subsets ofLX. In 1991, Ying [14] called Höhle’s topology fuzzifying topology.

In this paper,X is a nonempty set andI = [0,1]. The family of all fuzzy sets onX will be denotedby IX. By 0X (or 0) and 1X (or 1), we denote respectively the constant fuzzy set onX taking the value 0and 1. The set of all fuzzy pointsx� (i.e., a fuzzy setA ∈ IX such thatA(x) = � �= 0 andA(y) = 0 for

� Supported by Natural Science Foundation of China and Natural Science Foundation of Shandong province.∗ Corresponding author.E-mail address:[email protected](Y. Yueli).

0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2005.03.003

http://www.elsevier.com/locate/fss

mailto:[email protected]

104 Y. Yueli, F. Jinming / Fuzzy Sets and Systems 154 (2005) 103 – 117

y �= x) is denoted byM(IX). �r (A) = {x|A(x) > r}, wherer ∈ I , A ∈ IX. U ∈ P(X), 1U denotes thecharacteristic function ofU, i.e., 1U(x) = 1 whenx ∈ U and 1U(x) = 0 when others.

Let I-TOPdenote the category ofI-topological spaces (Chang–Goguen topological spaces) and continu-ous functions. By a fuzzifying topology[3,14,17]on a setXwe mean a function� : P(X)→ I such that (1)�(�) = �(X) = 1; (2) �(U ∩ V )� min{�(U), �(V )} for all U,V ∈ P(X); (3) �(

⋃t∈T Ut )� inf t∈T �(Ut )

for every family{Ut |t ∈ T } ⊆ P(X). A fuzzifying continuous function between fuzzifying topolog-ical spaces is a functionf : (X, �) → (Y, �) such that�(f←(U))��(U) for all U ∈ P(Y ). FYSdenotes the category of fuzzifying topological spaces and fuzzifying continuous functions (in fact,FYSis (2, I )-FTOP in [6]), andFYS(X) denotes all the fuzzifying topologies onX . An I-fuzzy topology(or Kubiak–Šostak fuzzy topology) on a setX is defined to be a functionT : IX → I satisfying thefollowing conditions (similar to the definition of fuzzifying topology): (1)T (0X) = T (1X) = 1; (2)T (A ∧ B)� min{T (A), T (B)} for all A,B ∈ IX; (3) T (∨t∈T At)� inf t∈T T (At ) for every family{At |t ∈ T } ⊆ IX. An I-fuzzy continuous function betweenI-fuzzy topological spaces is a functionf : (X, T1) → (Y, T2) such thatT1(f

←I (A))�T2(A) for all A ∈ IY (wheref←I (A)(x) = A(f (x)),

notation in [9,10]). LetI-FTOP denote the category ofI-fuzzy topological spaces andI-fuzzy continuousfunctions, andI-FTOP(X) denote all theI-fuzzy topologies onX.

According to their value ranges,L-topological spaces form different categories. Clearly, the investi-gation on their relationships is certainly important and necessary. Lowen was one of the first authorswho had studied the relation betweenI-TOP andTOP (the category of classical topological spaces). Heintroduced two well-known functors� and� (named Lowen functors). These functors later were extendedby different authors for various kinds of lattices studying the relation betweenL-TOP andTOP. Clearly,the categoryTOP can be regarded as a full subcategory ofFYSandI-TOP is full subcategory ofI-FTOP(see [16]). By the Lowen functor� : TOP → I -TOP, we know thatTOP can be regarded as a fullsubcategory ofI-TOP (see [8]). In the second section of this paper, we gain thatFYS is a full subcategoryof I-FTOP.

The aim of this paper is to establish the Lowen functors in Kubiak–Šostak’s sense. In the second sectionof this paper, we extend Lowen functors toI-fuzzy topological spaces and study their properties. In thethird section, we introduce the concepts of generatedI-fuzzy topological spaces, weakly generatedI-fuzzytopological spaces and stratifiedI-fuzzy topological spaces and study their mutual relations. Finally, weget the connections on one kind of compactness and separation axioms between fuzzifying topologicalspace(X, �) and its corresponding generatedI-fuzzy topological space(X,�(�)).

1. Preliminaries

Definition 1.1 (Ying[14] ). Let � be a fuzzifying topology onX andB : P(X) → I be a function withB��. ThenB is called a base of� if B satisfies the following condition:

∀U ∈ P(X),∀x ∈ X, Nx(U)� supx∈V⊆U

B(V ),

whereNx(U) = supx∈V⊆U �(V ).

Y. Yueli, F. Jinming / Fuzzy Sets and Systems 154 (2005) 103 – 117 105

Definition 1.2 (Ying[14] ). Let � be a fuzzifying topology onX and� : P(X)→ I be a function. Then� is called a subbase of� if �(�) : P(X)→ I is a base, where

∀U ∈ P(X), �(�)(U) = sup(�)�∈�V�=U

inf�∈� �(V�)

with (�) standing for “finite intersection”.

Lemma 1.3(Ying[14] ). A functionB : P(X)→ I is a base of� if and only if

�(U) = sup⋃�∈� V�=U

inf�∈� B(V�)

for U ∈ P(X), where the expression

sup⋃�∈� V�=U

inf�∈� B(V�)

will be denoted byB(�)(U), i.e., B(�) : P(X)→ I satisfyingB(�) = �.

Lemma 1.4(Ying[14] ). A function� : P(X)→ I is a subbase of� if and only if�(�)(X) = 1.

Definition 1.5 (Ying[15] ). Let {(Xj , �j )}j∈J be a collection of fuzzifying topological spaces andpj :∏j∈J Xj → Xj be a projection. Then the fuzzifying topology whose subbase is defined by

∀W ∈ P∏j∈J

Xj

, �(W) = sup

j∈Jinf

p−1j (U)=W

�j (U)

is called the product fuzzifying topology of{�j |j ∈ J }, denoted by∏j∈J �j , and(

∏j∈J Xj ,

∏j∈J �j )

is called the product space of{(Xj , �j )}j∈J .

Fang[2] extended the above definitions and results toI-fuzzy topological spaces. For convenience, welist them here

(1) Let T be anI-fuzzy topology onX andB : IX → I be a function withB�T . ThenB is called abase ofT if B satisfies the following condition:

∀A ∈ IX, ∀x� ∈ M(IX), Qx�(A)� supx�qB�A

B(B),

whereQx�(A) = supx�qB�A T (B), x�qV denotes thatx� is quasi-coincident withV, i.e.,V (x)+ � > 1(see[7]). Relation “is not quasi-coincident with” is denoted by¬q. A functionB : IX → I is a base ofT if and only if

T (A) = sup∨�∈� B�=A

inf�∈∧ B(B�) for all A ∈ IX,


where the expression

sup∨�∈∧ B�=A

inf�∈� B(B�)

will be denoted byB(�)(A), i.e.,B(�) : IX → I satisfyingB(�) = T .(2) Let� : IX → I be a function. Then� is called a subbase ofT if �(�) : IX → I is a base, where

�(�)(A) = sup(�)�∈�B�=A

inf�∈J �(B�) with (�)

standing for “finite intersection”. A function� : IX → I is a subbase ofT iff �(�)(1X) = 1.(3) Let {(Xj , Tj )}j∈J be a collection ofI-fuzzy topological spaces andpj :

∏j∈J Xj → Xj be a

projection. Then theI-fuzzy topology whose subbase is defined by

∀A ∈ I∏j∈J Xj , �(A) = sup

j∈Jsup

(pj )←I (B)=A

Tj (B)

is called the productI-fuzzy topology of{Tj |j ∈ J }, denoted by∏j∈J Tj , and(

∏j∈J Xj ,

∏j∈J Tj ) is

called the product space of{(Xj , Tj )}j∈J .In Ref. [11], the author gave the definitions of bases and subbases inL-fuzzy topological spaces. In

the following two examples, we will show that the definitions of bases and subbases above are differentfrom those given in [11] inI-fuzzy topological spaces whenL = I = [0,1].

Example 1.6. LetX = {x} be a single-point set,T : IX → I be defined as follows:

T (x�) ={

1, � = 0,�, others

and letB : IX → I be

B(x�) ={

�, � is rational number in[0,1],0, � is irrational number in[0,1].

ThenT is anI-fuzzy topology onX. Now we will show thatB is the base ofT in [2], while it is not thebase in [11]. It is easy to prove that

T (A) = sup∨�∈� B�=A

inf�∈� B(B�)

for all A ∈ IX, which shows thatB is the base ofT in [2]. We know that√

2/2 is an irrational number in[0,1] and we can take a sequence of rational numbers�1, �2, . . . , �n, . . . such that12 ��1 < �2 < · · · <�n < · · · <

√2/2 and�n →

√2/2 (n→∞). From the definition ofB andx√2/2 =

∨n∈N x�n , where

N denotes the set of all natural numbers, we haveB(∨n∈N x�n) = B(x√2/2) = 0 and infn∈N B(x�n) =inf n∈N �n� 1

2. HenceB does not satisfy the condition of base in [11]. ThereforeB is not the base in [11].


By the way, ifB is a base ofT in [2] and it also satisfies the following condition given in [11], i.e., for allindexing setJ, ∀{Aj : j ∈ J } ⊆ IX, inf j∈J B(Aj )�B(∨j∈J Aj ), then we haveB = T . In fact, from

T (A) = sup∨�∈� B�=A

inf�∈� B(B�)� sup∨

�∈� B�=AB(∨

�∈�B�

)= B(A).

HenceB = T .

Example 1.7. Let X be any nonempty set and� : IX → I be defined as�(A) = 12 for all A ∈ IX. We

know that� is subbase of oneI-fuzzy topology in[11] and thisI-fuzzy topology is justT : IX → I ,where

T (A) ={

1, A = 0X,1X,1/2, others.

But � is not the subbase in[2] since�(�)(1X) = 12 �= 1. We know that

�(A) ={

1, A = 1X,1/2, others

is the subbase ofT in [2].

Lemma 1.8(Höhle and Šostak[4] ). I-FTOP(X) is complete lattice.

Similar to that in[4], it is easy to prove thatFYS(X) is also a complete lattice. Furthermore, we havethe following lemma.

Lemma 1.9. Let {�t }t∈T ⊆ FYS(X). Then� : P(X) → I is the subbase of∨t∈T �t , where�(U) =

supt∈T (�t (U)) for everyU ∈ P(X), i.e.,∨t∈T �t = (�(�))(�).

Proof. From Lemma 1.4, it is easy to testify that� is a subbase of one fuzzifying topology, we denotethis fuzzifying topology as�. Now we prove

∨t∈T �t = �. From the definition of subbase, we know that

�(U) = sup⋃�∈� V�=U

inf�∈� sup

(�)∈��W�=V�

inf∈��

supt∈T

�t (W�)

for eachU ∈ P(X). Then we have�(U)� supt∈T (�t (U)), i.e.,�(U)��t (U) for all t ∈ T . Hence��tfor eacht ∈ T . Thus

∨t∈T �t��.

On the other hand, letU ∈ P(X) and < �(U). Then there exist{V�}�∈� such that

(i)⋃

�∈� V� = U ;(ii) for each� ∈ �, there exists{W�}∈�� satisfying(�)∈��W� = V�;

(iii) for each ∈ ��, there existst ∈ T such that < �t (W�).


Hence < (∨t∈T �t )(W�). We observe that

� inf∈��

(∨t∈T

�t

)(W�)�

(∨t∈T

�t

)(V�).

Then

� inf�∈�

(∨t∈T

�t

)(V�)�

(∨t∈T

�t

)(⋃�∈�

V�

)=(∨t∈T

�t

)(U),

i.e.,�(U)�(∨t∈T �j )(U). This is to say

∨t∈T �t��. �

Definition 1.10 (Rodabaugh[11] ). Given anI-fuzzy topological space(X, T ) and subsetY ⊆ X. Wecall (Y, T |Y ) (where(T |Y )(U) = sup{T (V )|V ∈ IX, V |Y = U}, which is anI-fuzzy topology onY)the subspace of(X, T ).

2. Lowen functors in I -fuzzy topological spaces

Lemma 2.1. Let(X, �) be a fuzzifying topological space and let�(�) : IX → I be defined by�(�)(A) =inf r∈I �(�r (A)) for A ∈ IX. Then�(�) is an I-fuzzy topology on X.

Proof. (i) �(�)(0X) = inf r∈I �(�r (0X)) = �(�) = 1, �(�)(1X) = inf r∈I �(�r (1X)) = 1.(ii) ∀A,B ∈ IX, we have

�(�)(A ∧ B) = infr∈I �(�r (A ∧ B)) = inf

r∈I �(�r (A) ∩ �r (B))

� infr∈I min{�(�r (A)), �(�r (B))} = min

{infr∈I(�(�r (A))), inf

r∈I(�(�r (B)))}

=min{�(�)(A),�(�)(B)}.

(iii) ∀At ∈ IX, t ∈ T , we have

�(�)

(∨t∈TAt

)= infr∈I �

(⋃t∈T

�r (At )

)� infr∈I inf

t∈T �(�r (At )) = inft∈T inf

r∈I �(�r (At )) = inft∈T �(�)(At ).

Therefore�(�) is anI-fuzzy topology onX. �

Lemma 2.2. LetU ∈ P(X) and� ∈ I . Then�(�)(�1U) = �(U).

Proof. Clear by definition of�(�). �


Theorem 2.3. f : (X,�(�1))→ (Y,�(�2)) is I-fuzzy continuous if and only iff : (X, �1)→ (Y, �2) isfuzzifying continuous.

Proof. Necessity: Let V ∈ P(Y ). Then

�1(f←(V )) = �(�1)(1f←(V )) = �(�1)(f

←I (1V )).

Sincef : (X,�(�1))→ (Y,�(�2)) is I-fuzzy continuous, we have

�(�1)(f←I (1V ))��(�2)(1V ) = �2(V ).

Then�1(f←(V ))��2(V ). Sof : (X, �1)→ (Y, �2) is fuzzifying continuous.

Sufficiency: LetB ∈ IY , we have

�(�1)(f←I (B)) = inf

r∈I �1(�r (f←I (B))) = inf

r∈I �1(f←(�r (B))).

Sincef : (X, �1)→ (Y, �2) is fuzzifying continuous, we get

∀r ∈ I, �1(f←(�r (B)))��2(�r (B)).

Then

infr∈I �1(f

←(�r (B)))� infr∈I �2(�r (B)) = �(�2)(B).

Hence,�(�1)(f←I (B))��(�2)(B). Sof : (X,�(�1))→ (Y,�(�2)) is I-fuzzy continuous. �

From Lemma 2.1 and Theorem 2.3, we know that for each fuzzifying topology we can construct anI-fuzzy topology, i.e.,� is a functor fromFYS to I-FTOP. The following theorem is true.

Theorem 2.4. � preserves arbitrary meets and nonempty joins.

Proof. We first prove� preserves arbitrary meets, i.e., let{�t }t∈T ⊆ FYS(X) be a collection of fuzzifyingtopologies, we need to prove�(

∧t∈T �t ) =∧t∈T �(�t ). ∀A ∈ IX,

�

(∧t∈T

�t

)(A) = inf

r∈I

(∧t∈T

�t

)(�r (A)) = inf

t∈T infr∈I �t (�r (A)) = inf

t∈T �(�t )(A) =(∧t∈T

�(�t )

)(A).

Now we prove� preserves nonempty joins, i.e., let{�t }t∈T ⊆ FYS(X) be a collection of nonemptyfuzzifying topologies, we need to prove�(

∨t∈T �t ) =∨t∈T �(�t ). From� preserving arbitrary meets,

we have�(∨t∈T �t )�

∨t∈T �(�t ). It is only to prove�(

∨t∈T �t )�

∨t∈T �(�t ). LetA ∈ IX and <

�(∨t∈T �t )(A) = inf r∈I (

∨t∈T �)(�r (A)). From Lemmas 1.9 and 2.2, we have

< infr∈I sup⋃

�∈� V�=�r (A)inf�∈� sup

(�)∈��W�=V�

inf∈��

supt∈T

�t (W�)

= infr∈I sup⋃

�∈� V�=�r (A)inf�∈� sup

(�)∈��W�=V�

inf∈��

supt∈T(�(�t )(1W�))

� infr∈I

(∨t∈T

�(�t )

)(1�r (A))


�(∨t∈T

�(�t )

)(∨r∈I

r1�r (A)

)

=(∨t∈T

�(�t )

)(A).

Then�(∨t∈T �t )�

∨t∈T �(�t ). Hence we get�(

∨t∈T �t ) =∨t∈T �(�t ). �

Now we consider the converse question, i.e. how to construct a fuzzifying topology from anI-fuzzytopology?

Lemma 2.5. Let T be an I-fuzzy topology on X and�T : P(X) → I be defined by�T (U) = supr∈Isup{T (B)|B ∈ IX, �r (B) = U} for U ∈ P(X). Then�T is the subbase of one fuzzifying topology,denoted by�(T ).

Proof. It is easy to testify it. �

Definition 2.6. Let T be anI-fuzzy topology onX. �(T ) is called generated fuzzifying topology byT .

Theorem 2.7. If f : (X, T1)→ (Y, T2) is I-fuzzy continuous, thenf : (X, �(T1))→ (Y, �(T2)) is fuzzi-fying continuous.

Proof. Sincef : (X, T1)→ (Y, T2) is I-fuzzy continuous, thenT2(B)�T1(f←I (B)) for all B ∈ IY . Let

U ∈ P(Y ). We have

�(T2)(U) = sup⋃�∈� V�=U

inf�∈� sup

(�)∈��W�=V�

inf∈��

supr∈I

sup{T2(D)|D ∈ IY , �r (D) = W�}

� sup⋃�∈� V�=U

inf�∈� sup

(�)∈��W�=V�

inf∈��

supr∈I

sup{T1(f←I (D))|D ∈ IY , �r (D) = W�}

� sup⋃�∈� B�=f←(U)

inf�∈� sup

(�)∈��C�=B�

inf∈��

supr∈I

sup{T1(E)|E ∈ IX, �r (E) = C�}= �(f←(U)).

Thereforef : (X, �(T1))→ (Y, �(T2)) is fuzzifying continuous. �

From Lemma 2.5 and Theorem 2.7, we know that� is a functor fromI-FTOP to FYS.

Theorem 2.8. (1) For every� ∈ FYS(X), �(�(�)) = �;(2) For everyT ∈ I -FTOP(X), �(�(T ))�T . Furthermore, if T = �(�), then�(�(T )) = T .

Proof. (1) For eachU ∈ P(X), we observe that

�(�(�))(U) = (�(�)�(�))(�)(U) = sup⋃

�∈� V�=Uinf�∈� �(�)�(�)(V�)

= sup⋃�∈� V�=U

inf�∈� sup

(�)∈��W�=V�

inf∈��

��(�)(W�)


= sup⋃�∈� V�=U

inf�∈� sup

(�)∈��W�=V�

inf∈��

supr∈I

sup{�(�)(D)|D ∈ IX, �r (D) = W�}

� supr∈I

sup{�(�)(D)|D ∈ IX, �r (D) = U}� �(U).

Then�(�(�))��.Conversely, letU ∈ P(X) and < �(�(�))(U), i.e.,

< sup⋃�∈� V�=U

inf�∈� sup

(�)∈��W�=V�

inf∈��

supr∈I

sup{�(�)(D)|D ∈ IX, �r (D) = W�}.

Then there exist{V�}�∈� such that

(i)⋃

�∈� V� = U ;(ii) for each� ∈ �, there exists{W�}∈�� such that(�)∈��W� = V�;

(iii) for each ∈ ��, there existr ∈ I andD ∈ IX such that�r (D) = W� and < �(�)(D) =inf r∈I �(�r (D)).

We have < �(W�), then� inf ∈�� (W�)��(V�). Hence� inf �∈� �(V�)��(U). Thus�(�(�))��.So�(�(�)) = �.

(2) LetA ∈ IX, ∈ I and�(�(T ))(A) < . Then

> infr∈I sup⋃

�∈� V�=�r (A)inf�∈� sup

(�)∈��W�=V�

inf∈��

sup�∈I

sup{T (D)|��(D) = W�}� infr∈I sup

�∈Isup{T (D)|��(D) = �r (A)}

� infr∈I T (A) = T (A).

Therefore�(�(T ))�T . Moreover, ifT = �(�), from (1), we can get�(�(T )) = T . �

Corollary 2.9. Both� : FYS(X)→ �(FYS(X)) and � : �(FYS(X))→ FYS(X) are complete latticeisomorphisms. Furthermore, we can get thatFYS is a full subcategory of I-FTOP.

Example 2.10. (1) Let X be any nonempty set and� : P(X)→ I be defined by

�(U) ={

1, U = ∅, X,0, others.

It is easy to testify that� is a fuzzifying topology onX. From the definition of�(�), we know that

�(�)(A) ={

1, A = 0X,1X and�,0, others,

where� is the constant function fromX to I.(2) LetT : IX → I be defined as follows:

T (A) ={

1, A = 0X,1X,1/2, others.


It is easy to testify thatT is anI-fuzzy topology onX. We can get that

�(T )(U) ={

1, U = ∅, X,1/2, others.

Furthermore, we have

�(�(T ))(A) ={

1, A = 0X,1X and�,1/2, others,

where� is the constant function fromX to I.

Theorem 2.11.Let � ∈ FYS(X) andY ⊆ X be nonempty. Then�(�|Y ) = �(�)|Y .

Proof. Let A ∈ IY , ∈ I and < �(�|Y )(A) = inf r∈I sup{�(U)|U ∈ P(X),U ∩ Y = �r (A)}. Foreachr ∈ I , there existsUr ∈ P(X) with Ur ∩ Y = �r (A) such that < �(Ur). Then < �(Ur) =�(�)(r1Ur ), and then��(�)(

∨r∈I r1Ur ). We know that

∨r∈I r1Ur |Y = A. Thus�(�(�)|Y )(A).

Therefore�(�|Y )��(�)|Y .Conversely, assume that

< (�(�)|Y )(A) = sup{�(�)(B)|B ∈ IX, B|Y = A} = sup

{infr∈I �(�r (B))|B ∈ IX, B|Y = A

}.

Then there existsB ∈ IX with B|Y = A such that < �(�r (B)) for eachr ∈ I . From�r (B) ∩ Y =�r (B|Y ) = �r (A) for eachr ∈ I , we have < (�|Y )(�r (A)). Then� inf r∈I (�|Y )(�r (A)) = �(�|Y )(A).Thus�(�|Y )��(�)|Y . Finally, we obtain�(�|Y ) = �(�)|Y as desired. �

Theorem 2.12.Let {(Xt , �t )}t∈T be a collection of fuzzifying topological spaces andX = ∏t∈T Xt .

Then�(∏t∈T �t ) =∏t∈T �(�t ).

Proof. LetA ∈ IX. By Definition 1.5, we have

�

(∏t∈T

�t

)(A)= inf

r∈I sup⋃�∈� V�=�r (A)

inf�∈� sup

(�)∈��W�=V�

inf∈��

supt∈T

sup{�t (D)|p←t (D) = W�}

= infr∈I sup⋃

�∈� V�=�r (A)inf�∈� sup

(�)∈��W�=V�

inf∈��

supt∈T

sup{�(�t )(1D)|p←t (D) = W�}

and (∏t∈T

�(�t )

)(A) = sup∨

�∈� B�=Ainf�∈� sup

(�)∈��C�=B�

inf∈��

supt∈T

sup{�(�t )(U)|(pj )←I (U) = C�}.

Comparing the two expressions above, we know�(∏t∈T �t )�

∏t∈T �(�t ).

On the other hand, let <∏t∈T �(�t )(A). Then there exist{B�}�∈� such that

(i)∨

�∈� B� = A;(ii) for each� ∈ �, there exist{C�}∈�� such that(�)∈��C� = B�;


(iii) for each ∈ ��, there existt ∈ T andU ∈ IXt such that(pt )←I (U) = C� and < �(�t )(U) =inf r∈I �t (�r (U)), i.e., < �t (�r (U)) for eachr ∈ I .

From⋃

�∈� �r (B�) = �r (A), (�)∈��r (C�,) = �r (B�) and�r (C�,) = �r ((pt )←I (U)) = p←t (�r (U))for eachr ∈ I , we know that��(

∏t∈T �t )(A). Hence�(

∏t∈T �t )�

∏t∈T �(�t ) from the arbitrariness

of . So�(∏t∈T �t ) =∏t∈T �(�t ). �

3. GeneratedI -fuzzy topological spaces

In this section, we discuss some relations between�, � and generatedI-fuzzy topological spaces, weaklygeneratedI-fuzzy topological spaces and stratifiedI-fuzzy topological spaces.

Lemma 3.1. Let (X, T ) be an I-fuzzy topological space on X and[T ] : P(X) → I be defined by[T ](U) = T (1U) for U ∈ P(X). Then[T ] is a fuzzifying topology on X.

Proof. Omitted.

Definition 3.2. Let(X, T )beI-fuzzy topological space onX. (X, [T ]) is called the fuzzifying backgroundspace of(X, T ). If T (A) = inf r∈I T (1�r (A)) for all A ∈ IX, then(X, T ) is called generatedI-fuzzytopological space. If we haveT (A)� inf r∈I T (1�r (A)) for all A ∈ IX, then(X, T ) is called weaklygeneratedI-fuzzy topological space. IfT (�) = 1 for all �, which is a constant function fromX to I, then(X, T ) is called stratifiedI-fuzzy topological space.

From the definitions of� and generatedI-fuzzy topological space, we know that(X, T ) is generatedI-fuzzy topological space if and only ifT = �([T ]).

Theorem 3.3. (X, T ) is stratified I-fuzzy topological space if and only if�([T ])�T .

Proof. Necessity: Let (X, T ) be stratifiedI-fuzzy topological space.∀A ∈ IX, we have

T (A)= T(∨r∈I

r1�r (A)

)� infr∈I min{T (r), T (1�r (A))}

= infr∈I T (1�r (A)) = inf

r∈I [T ](�r (A))=�([T ])(A).

Therefore�([T ])�T .Sufficiency: Let � be any constant function fromX to I, then

T (�)��([T ])(�) = min

{infr<�[T ](X), inf

r��[T ](�)

}= 1.

So(X, T ) is stratifiedI-fuzzy topological space.�


Theorem 3.4. If (X, T ) is weakly generated I-fuzzy topological space, then[T ] = �(T ).

Proof. [T ]��(T ) is obvious and the converse inequality can be easily obtained from the definition of�and Theorem 2.8(i). �

Theorem 3.5. (X, T ) is weakly generated I-fuzzy topological space if and only if�([T ])�T .

Proof. Omitted.

From the theorems above, we can get the theorem as follows.

Theorem 3.6. (X, T ) is generated I-fuzzy topological space if and only if(X, T ) is weakly generatedI-fuzzy topological space and(X, T ) is also stratified I-fuzzy topological space.

4. Applications

In this section, we discuss some relations in the aspects of one kind of compactness and separationaxioms between fuzzifying topological spaces and generatedI-fuzzy topological spaces.

Definition 4.1 (Shen[12] ). Let(X, �)be a fuzzifying topological space. Degrees that(X, �)areT0, T1, T2defined as follows:

T0(X, �) = inf

{max

{supy �∈A

Nx(A), supx �∈B

Ny(B)

}∣∣∣∣∣ x, y ∈ X, x �= y},

T1(X, �) = inf

{min

{supy �∈A

Nx(A), supx �∈B

Ny(B)

}∣∣∣∣∣ x, y ∈ X, x �= y},

T2(X, �) = inf

{sup

U∩V=∅min{Nx(U),Ny(V )}|x, y ∈ X, x �= y

}.

Definition 4.2. Let (X, T ) be anI-fuzzy topological space. Degrees that(X, T ) areIFT0, IFT1, IFT2defined as follows:

IFT0(X, T ) = inf

{max

{supx� ¬qA

Qy�(A), supy� ¬qB

(Qx�(B))

}∣∣∣∣∣ �, � ∈ (0,1], x �= y},

IFT1(X, T ) = inf

{min

{supx�¬qA

Qy�(A), supy�¬qB

(Qx�(B))

}∣∣∣∣∣ �, � ∈ (0,1], x �= y},

IFT2(X, T ) = inf

{sup

A∧B=0max{Qy�(B),Qx�(A)}

∣∣∣∣ �, � ∈ (0,1], x �= y}.


Definition 4.3 (Zhao[18] ). Let (X, �) be anI-topological space,A ∈ IX, ⊆ �′ and ∈ (0,1]. If foreachx�A, there existsP ∈ such thatx�P , then is called an−remote family ( − RF) of A. Ifthere existsr ∈ (0, ) such that is r − RF ofA, then is called− − RF ofA.

Definition 4.4 (Zhao[18] ). Let (X, �) beI-topological space andA ∈ IX. If there exists a finite subfam-ily � of such that� is − −RF ofA for all −RF of A, A is calledN-compact. If 1X is N-compact,(X, �) is calledN-compact.

Definition 4.5. Let (X, �) be a fuzzifying topological space andA ⊆ X. If A is a compact set in(X, �r )(where�r = {B|B ⊆ X, �(B)�r} and we know that�r is a topology onX) for all r > 0, thenA iscalled fuzzifying compact in(X, �). If X is fuzzifying compact in(X, �), then(X, �) is called fuzzifyingcompact.

Definition 4.6 (Zhou[19] ). Let (X, T ) be anI-fuzzy topological space andA ∈ IX. If A is N-compactin (X, T r ) (whereT r = {B|B ∈ IX, T (B)�r} andT r is anI-topology onX) for all r > 0, thenA iscalledI-fuzzy N-compact in(X, T ). If 1X is I-fuzzy N-compact in(X, T ), then(X, T ) is calledI-fuzzyN-compact.

Theorem 4.7. (X, �) is fuzzifying compact if and only if(X,�(�)) is I-fuzzy N-compact.

Proof. Sufficiency: ∀r ∈ (0,1]. LetU = {Us ∈ �r |s ∈ S} be an open cover ofX. We know that�(Us)�rfor all s ∈ S. Then�(�)(1Us ) = �(Us)�r, i.e., 1Us ∈ �(�)r or 1U ′s ∈ (�(�)r )′. Taking > 0, we knowthat� = {1U ′s } is a − RF of 1X. Since(X,�(�)r ) is N-compact, there exists a finite setS0 of Ssuchthat�0 = {1U ′s |s ∈ S0} is − − RF of 1X, i.e., there exists� ∈ (0, ) such that�0 is � − RF of 1X.Then there existss ∈ S0 such that� > 1U ′s (x) for all x ∈ X, i.e., x ∈ Us . Hence{Us |s ∈ S0} is afinite open cover ofX. Therefore(X, �r ) is compact, and then(X, �) is fuzzifying compact from thearbitrariness ofr.

Necessity: ∀r ∈ (0,1]. Let ∈ (0,1] and� ⊆ (�(�)r )′ be any − RF of 1X. Then there existsAx ∈ � such thatx�Ax for all x ∈ X, i.e.,Ax(x) < . Takes(x) ∈ (0,1) with Ax(x) < s(x) < .Thenx ∈ �1−s(x)(A′x). Because of(Ax)′ ∈ �(�)r , i.e.,�(�)(A′x)�r, we have�(�t (A′x))�r for all t ∈ I .Then�1−s(x)(A′x) ∈ �r . Now we know that{�1−s(x)(A′x)|x ∈ X} is an open cover ofX. Since(X, �r ) iscompact, there exists a finite subfamily{�1−s(x1)(A

′x1), . . . , �1−s(xn)(A′xn)} of {�1−s(x)(A′x)|x ∈ X} such

that{�1−s(x1)(A′x1), . . . , �1−s(xn)(A′xn)} is an open cover ofX. Then we can chooses < satisfys(xi)�s

for all i = 1, . . . , n. Hence{�1−s(A′x1), . . . , �1−s(A′xn)} is s −RF of 1X. Thus(X,�(�)r ) is N-compact.

Furthermore,(X,�(�)) is I-fuzzy N-compact from the arbitrariness ofr. �

Theorem 4.8. If (X, T ) is generated I-fuzzy topological space, i.e., there exists one fuzzifying topology� such thatT = �(�), then

(1) IFT0(X, T ) = T0(X, �); (2) IFT1(X, T ) = T1(X, �); (3) IFT2(X, T ) = T2(X, �).

Proof. We prove (2) and (3), the proof of (1) is similar to that of (2).


We firstly proveIFT1(X, T )�T1(X, �). Let < T1(X, �). For any two fuzzy pointsx�, y� with x �= y,there existA,C ⊆ X such thaty �∈ A, x ∈ C ⊆ A, < �(C) and there also existB,D ⊆ X such thatx �∈ B, y ∈ D ⊆ B and < �(D) from the definition ofT1(X, �). Since(X, T ) is generatedI-fuzzytopological space, we have

< �(C) = T (1C)�Qx�(1C)� supy� ¬qU

Qx�(U)

and

< �(D) = T (1D)�Qy�(1D)� supx� ¬qV

Qy�(V ).

Then

< min{Qx�(1C),Qy�(1D)}� min

{supy� ¬qU

Qx�(U), supx� ¬qV

Qy�(V )

}.

Hence� IFT1(X, T ). ThereforeIFT1(X, T )�T1(X, �).On the other hand, let < IFT1(X, T ) and take� ∈ (0,1]. For anyx, y ∈ X with x �= y, there

existsA ∈ IX such thatx�¬qA and < Qy�(A) and there also existsB ∈ IX such thaty�¬qB and < Qx�(B). Furthermore, there existC,D ∈ IX with y�qC�A andx�qD�B such that < T (C) =inf r∈I �(�r (C)) and < T (D) = inf r∈I �(�r (D)). Now letV = �1−�(C) andU = �1−�(D). Then wehavex �∈ V , x ∈ U , y �∈ U , y ∈ V and < �(U)�Nx(U), < �(V )�Ny(V ). Thus�T1(X, �). SoIFT1(X, T )�T1(X, �).

Now we proveIFT2(X, T ) = T2(X, �). Let < IFT2(X, T ) andx, y ∈ X with x �= y. Take� ∈(0,1]. Then there existA,B ∈ IX withA∧B = 0 such that < Qx�(A) and < Qy�(B). Furthermore,there existC,D ∈ IX with x�qC�A andy�qD�B such that < T (C) = inf r∈I �(�r (C)) and <T (D) = inf r∈I �(�r (D)). Now letU = �1−�(C) andV = �1−�(D). Then we havex ∈ U , y ∈ V , <�(U)�Nx(U), < �(V )�Ny(V )andU∩V = ∅. Hence�T2(X, �). ThereforeIFT2(X, T )�T2(X, �).

Conversely, let < T2(X, �). For any two fuzzy pointsx�, y� with x �= y, there existU,V ⊆ X withU ∩V = ∅ such that < Nx(U) and < Ny(V ). From the definition ofNx , there existW,H ⊆ X withx ∈ W ⊆ U andy ∈ H ⊆ V such that < �(W) and < �(H). Thenx�q1W andy�q1H . Since(X, T )is generatedI-fuzzy topological space, we have 1W ∧ 1H = 0, < �(W) = T (1W) and < �(H) =T (1H). Hence�Qx�(1W) and �Qy�(1H). Then� IFT2(X, T ). Thus IFT2(X, T )�T2(X, �). SoIFT2(X, T ) = T2(X, �). �

Acknowledgements

We would like to thank Prof. S.E. Rodabaugh and the anonymous referees for their useful commentsand valuable suggestions.

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