Fuzzy Almost e*-continuous Mappings and Fuzzy e*-compactness in Smooth Topological Spaces

ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 11 December 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/ijfma.v14n2a17

347

International Journal of

Fuzzy Almost e*-continuous Mappings and Fuzzy

e*-compactness in Smooth Topological Spaces

A.Prabhu1, A. Vadivel2 and B. Vijayalakshmi3 1

Department of Mathematics

Annamalai University, Annamalainagar, Tamil Nadu-608 002, India. 2

Department of Mathematics

Government Arts College (Autonomous) Karur, Tamil Nadu-639 005, India. 3

Department of Mathematics

Government Arts college, Chidambaram, Tamilnadu, India. Email: [email protected]

Received 25 November 2017; accepted 07 December 2017

Abstract. In this paper, we introduce the concept of fuzzy almost e*-continuous, fuzzy weakly e*-continuous, r-fuzzy e*-compact, r-fuzzy almost e*-compactness and r -fuzzy near e*-compactness in S⌣ostak’s fuzzy topological spaces. We study some properties of them under several types of fuzzy e*-continuous mappings and fuzzy e* -regular spaces.

Keywords: fuzzy (almost, weakly)e*-continuous, fuzzy e*-regular spaces, r-fuzzy (almost, near) e*-compactness.

AMS Mathematics Subject Classification (2010): 54A40, 54C05, 03E72

1. Introduction

Chang [1] introduced and studied the notion of fuzzy topology. Hutton [18], Lowen [20] and others discussed respectively various aspects of fuzzy topology. In these authors’s papers, a fuzzy topology is defined as a classical subset of the fuzzy power set of a nonempty classical set. It is easy to see that they have always investigated fuzzy objects with crisp methods. However, 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, S⌣ostak [27] introduced a new definition of fuzzy topology as an extension of both crisp topology and Chang’s fuzzy topology. S⌣

A.Prabhu, A. Vadivel and B. Vijayalakshmi

348

of r-fuzzy

e

-open sets and r-fuzzy

e

-continuity in S⌣os tak’s fuzzy topological spaces. Vadivel et.al [35, 36] introduced the concept of fuzzy almost

e

-continuity, fuzzy

e-compactness, r-fuzzy e*-open and r-fuzzy e*-closed sets in a fuzzy topological space in the sense of Sˆostak [27]. In this paper, we introduce the concept of fuzzy almost e*-continuous, r-fuzzy e*-compact, r-fuzzy almost e*-compactness and r -fuzzy near e*-compactness in S⌣ostak’s fuzzy topological spaces in a different view point [12,27]. We study some properties of them under several types of fuzzy

e

-continuous mappings of fuzzy topological spaces.

2. Preliminaries

Throughout this paper, nonempty sets will be denoted by X ,Y etc., I =[0,1] and

1] (0, =

0

I . For

α

∈I,

α

(x)=

α

for all x∈X. A fuzzy point xt for t∈I0 is an element of IX such that xt(y)=

{

ty=x ; 0ify ≠ x.

The set of all fuzzy points in X is denoted by P_t(X). A fuzzy point x_t∈

λ

iff

). ( < x

t

λ

A fuzzy set

λ

is quasi-coincident with

µ

, denoted by

λ

q

µ

, if there exists

X

x∈ such that

λ

(x)+

µ

(x)>1. If

λ

is not quasi-coincident with

µ

, we denoted

.

µ

λ

q If A⊂ X, we define the characteristic function χ_A on X by χ_A(x)=

{

1x in A ,

if

0 x not in A.

Now, we introduce some basic notions and definitions that are used in the sequel.

Definition 2.1. [27] A function

τ

:IX →I is called a smooth topology on X if it satisfies the following conditions:

1.

τ

(0)=

τ

(1)=1, 2. ( ) ( _i)

i i i

µ τ µ

τ

∨

∧

Γ ∈ Γ

∈ ≥

, for any {

µ

i}i∈Γ ⊂ IX,

3. τ(µ₁∧µ₂)≥τ(µ₁)∧τ(µ₂), for any

µ

1,

µ

2∈IX.

The pair (X,

τ

) is called a smooth topological space or fuzzy topological space( for short, fts).

Definition 2.2. [24] Let (X,

τ

) be a fuzzy topological space. Then for each r∈I0, }

) ( : {

= IX r

r

µ

∈

τ

µ

≥

τ

is a Chang’s fuzzy topology on X .

Theorem 2.1. [3] Let (X,

τ

) be a fts. Then for each

λ

∈IX,r∈I₀ we define an

Topological Spaces

349

Theorem 2.2. [17, 19] Let (X,

τ

) be a fts. Then for each r∈I₀,

λ

∈IX we define an operator I_τ :IX ×I₀→IX as follows I_τ(

λ

,r)=

∨

{

µ

∈IX :

λ

≥

µ

,

τ

(

µ

)≥r}. For

λ

,

µ

∈IX and r,s∈I0, the operator Iτ satisfies the following conditions: 1. I_τ( r1, )=1, 2.

λ

≥I_τ(

λ

,r), 3. I_τ(

λ

,r)∧I_τ(

µ

,r)=I_τ(

λ

∧

µ

,r), 4. I_τ(

λ

,r)≤I_τ(

λ

,s) if s≤r, 5. I_τ(I_τ(

λ

,r),r)=I_τ(

λ

,r),

6. I_τ(1−

λ

,r)=1−C_τ(

λ

,r)andC_τ(1−

λ

,r)=1−I_τ(

λ

,r). 7. If S =

∨

{

r∈I₀:I_τ(

µ

,r)=

µ

}

,then I_τ(

µ

,s)=

µ

.

8. I_τ(C_τ(I_τ(C_τ(

λ

,r),r),r),r)=I_τ(C_τ(

λ

,r),r).

Definition 2.3. [32] Let (X,

τ

) be a fts. For

λ

,

µ

∈IX and r∈I₀,

λ

is called r -fuzzy regular open (for short, r-fro) (resp. r-fuzzy regular closed (for short, r-frc)) if

) ), , ( (

=I C

λ

r r

λ

τ τ (resp.

λ

=Cτ(Iτ(

λ

,r),r)).

Definition 2.4. [32] Let (X,

τ

) be a fts. Then for each

µ

∈IX,x_t∈P_t(X) and r∈I0,

1.

µ

is called r-open Q_τ-neighborhood of xt if xtq

µ

with

τ

(

µ

)≥r.

2.

µ

is called r-open R_τ-neighborhood of x_t if x_tq

µ

with

µ

=I_τ(C_τ(

λ

,r),r). We denote Q_τ(x_t,r)={

µ

∈IX :x_tq

µ

,

τ

(

µ

)≥r},

)}. ), , ( ( = : {

= ) ,

(x r I xq I C r r

R_{τ t}

µ

∈ X _t

µ

_{τ τ}

λ

Definition 2.5. [32] Let (X,

τ

) be a fts. Then for each

λ

∈IX,x_t∈P_t(X) and r∈I0,

1. x_t is called r-

τ

cluster point of

λ

if for every

µ

∈Q_τ(x_t,r), we have

µ

q

λ

.

2. x_t is called r-

δ

cluster point of

λ

if for every

µ

∈R_τ(x_t ,r), we have

µ

q

λ

.

3. An

δ

-closure operator is a mapping D_τ :IX ×I →IX defined as follows:

δ

C_τ(

λ

,r)

or D_τ(

λ

,r)=

∨

{x_t∈P_t(X):x_t is r-

δ

-cluster point of

λ

}. Equivalently,

δ

C_τ(

λ

,r)=

∧

{

µ

∈IX :

µ

≥

λ

,

µ

isar−frcset}and

} froset isa

, : {

= ) ,

( r

∨

∈I ≤ r−

I

λ

µ

X

µ

λ

µ

δ

τ .

Definition 2.6. [19] Let (X,

τ

) be a fuzzy topological space. For

λ

∈IX and r∈I0, 1.

λ

is called r-fuzzy

δ

-closed iff

λ

=D_τ(

λ

,r).

A.Prabhu, A. Vadivel and B. Vijayalakshmi

350

Definition 2.7. [36] Let (X,

τ

) be a fuzzy topological space. For

λ

,

µ

∈IX and r∈I₀,

λ

is called an r-fuzzy e*-open (resp. r-fuzzy e*-closed) set if

λ

≤C (_τ I_τ (

δ

-) ), ), ,

( r r r

C_τ

λ

(resp. I (_τ C_τ(

δ

-I_τ(

λ

,r),r),r)≤

λ

).

Definition 2.8. [36] Let (X,

τ

) be a fuzzy topological space.

λ

,

µ

∈IX and r∈I₀,

1. e*I_τ(

λ

,r)=

∨

{

µ

∈IX :

µ

≤

λ

,

µ

is a r- fe*o set } is called the r-fuzzy e* -interior of

λ

2. * _τ(

λ

, )= {

µ

∈ :

µ

≥

λ

,

µ

∧

X

I r

C

e is a r- fe*c set } is called the r -fuzzy e*-closure of

λ

.

Definition 2.9. [37] Let (X,

τ

) and (Y,

η

) be fts’s. A mapping f :X →Y is called a fuzzy e*-continuous iff f−1(

µ

) is r-f *o

e set of X for each

µ

∈IY with

η

(

µ

)≥r.

Theorem 2.3. [36] Let (X,

τ

) be a fts. For

λ

∈IX and r∈I0 we have 1. _e*_I (1

λ

,_r)=1 (_e*_C (

λ

,_r)).

τ

τ − − 2. e*Cτ(1−

λ

,r)=1−(e*Iτ(

λ

,r)).

Theorem 2.4. [34] Let (X,

τ

) be a smooth topological space. Then for each

λ

∈IX

and r∈I₀ 1. e*I_τ(e*C_τ(e*I_τ(e*C_τ(

λ

,r),r),r),r)=e*I_τ(e*C_τ(

λ

,r),r). 2. * ( * ( * ( * ( , ), ), ), )= * ( * ( , ), )

r r I e C e r r r r I e C e I e C

e _{τ τ τ τ}

λ

_{τ τ}

λ

.

3. Fuzzy almost e*-continuous mappings

Now, we introduce the following definitions.

Definition 3.1. Let (X,

τ

) and (Y,

η

) be a fuzzy topological spaces and r∈I₀. A mapping f :X →Y is called

1. fuzzy almost e*-continuous if f−1(

λ

) is a r-fe*o set in X for each

λ

∈IY with

) ), , ( (

=I C

λ

r r

λ

η η or equivalently, f−1(

λ

) is a r-fe*c set in X for each

λ

∈IY

with

λ

=C_η(I_η(

λ

,r),r).

2. fuzzy almost e*-open map if f(

λ

) is a r-fe*o set in Yfor each r-froset

λ

inX.

3. fuzzy almost e*-closed map if f(

λ

) is a r-fe*c set in Yfor each r-frcset

λ

in X.

Remark 3.1. Clearly, a fuzzy

e

-continuous mapping is fuzzy almost

e

-continuous.

Example 3.1. Let

λ

,

µ

,

γ

,

δ

and

α

be fuzzy subsets of X =Y =

{

a,b,c

}

defined as follows:

λ

(a)=0.5,

λ

(b)=0.3,

λ

(c)=0.2;

µ

(a)=0.4,

µ

(b)=0.6,

µ

(c)=0.1;

γ

(a)=0.5,

γ

(b)=0.6,

γ

(c)=0.2;

δ

(a)=0.4,

δ

(b)=0.3,

δ

(c)=0.1;

α

(a)=0.2,

α

(b)=0.4,

α

(c)=0.5.

Topological Spaces

351

     

     

. otherwise 0,

, = if , 2 1

, 1 or 0 = if 1,

= ) (

, otherwise 0,

, , , , = if , 2 1

, 1 or 0 = if 1,

= )

(

λ

α

λ

λ

η

δ

γ

µ

λ

λ

λ

λ

τ

are fuzzy topologies on X. For r=

2 1

, then for any fuzzy regular open set

α

in Y, the

identity mapping f :(X,

τ

)→(Y ,

η

) is fuzzy almost e*-continuous.

Example 3.2. Let

λ

,

µ γ

, and

δ

be fuzzy subsets of X =Y =

{

a,b,c

}

defined as follows:

0.4, = ) (a

λ

λ

(b)=0.5,

λ

(c)=0.2;

µ

(a)=0.2,

µ

(b)=0.3,

µ

(c)=0.2;

0.4, = ) (a

γ

γ

(b)=0.6,

γ

(c)=0.5;

δ

(a)=0.3,

δ

(b)=0.5,

δ

(c)=0.3. Then

τ η

, :IX →I defined as

     

     

. otherwise 0,

, , = if , 2 1

, 1 or 0 = if 1,

= ) (

, otherwise 0,

, , = if , 2 1

, 1 or 0 = if 1,

= )

(

λ

γ

δ

λ

λ

η

µ

λ

λ

λ

λ

τ

are fuzzy topologies on X. For r=

2 1

, then for any fuzzy regular open set

λ

in X, the

identity mapping f :(X,

τ

)→(Y ,

η

) is fuzzy almost e*-open map.

Theorem 3.1. Let (X,

τ

) and (Y,

η

) be fts’s. Let f :(X ,

τ

)→(Y ,

η

) be a mapping. The following statements are equivalent:

1. f is fuzzy almost e*-continuous. 2. f−1(

µ

) is r-f_e*o

set, for each

µ

∈IY, r∈I₀ with

µ

=I_η(C_η(

µ

,r),r).

3. f−1(

µ

)≤e*I_τ(f−1(I_η(C_η(

µ

,r),r)),r), for each

µ

∈IY, r∈I0 with

η

(

µ

)≥r. 4. * ( 1( ( (

µ

, ), )), ) 1(

µ

),

η η

τ f− C I r r r ≤ f−

C

e for each

µ

∈IY, r∈I0 with

η

(1−

µ

)≥r. Proof: (1) ⇔(2): Follows from Definition 2.1

(1) ⇒ (3): Let f be almost e*-continuous and

µ

be any r-fuzzy open set in Y. Then

µ

=I_η(

µ

,r)≤I_η(C_η(

µ

,r),r).

By Definition 1.3, I_η(C_η(

µ

,r),r) is a r-fuzzy regular open set in Y. Since f is fuzzy almost e*-continuous, f−1(I_η(C_η(

µ

,r),r)) is a r-fuzzy e*-open set in X.

Hence f−1(

µ

)≤ f−1(I_η(C_η(

µ

,r),r))=e*I_τ(f−1(I_η(C_η(

µ

,r),r)),r).

A.Prabhu, A. Vadivel and B. Vijayalakshmi

352

By (3),f−1(

µ

c)≤e*I_τ(f−1(I_η(C_η(

µ

c,r),r)),r). Hence,

). )), ), , ( ( ( ( = )) )), ), , ( ( ( ( ( )) ( ( = )

( 1 * 1 * 1

1

r r r I C f C e r r r C I f I e f

f−

µ

−

µ

c c≥ _τ − _{η η}

µ

c c _τ − _{η η}

µ

(4) ⇒ (1): Let

µ

be a r-fuzzy regular closed set in Y. Then

µ

is a r-fuzzy closed set in Y and hence f−1(

µ

)≤e*I_τ(f−1(I_η(C_η(

µ

,r),r)),r)=e*I_τ(f−1(

µ

),r).

Thus f−1(

µ

)=e*C_τ(f−1(

µ

),r) and hence f−1(

µ

) is a r-fuzzy e*-closed set in X.

Therefore, f is a fuzzy almost e*-continuous map.

Definition 3.2. A fts is called r-fuzzy e*-regular iff for each

τ

(

µ

)≥r,

{

| ( ) , ( ( ( , ), ), )

}

.

=

ρ

τ

ρ

δ

ρ

µ

µ

∨

∈IX ≥r Iτ Cτ Iτ r r r ≤

A fuzzy topological space (X,

τ

) is called fuzzy e*-regular iff it is r-fuzzy e*-regular for each r∈I0.

Example 3.3. Let

λ

,

µ γ

, and

δ

be fuzzy subsets of X =

{

a,b,c

}

defined as follows:

0.8, = ) (a

λ

λ

(b)=0.6,

λ

(c)=0.6;

0.6, = ) (b

µ

µ

(c)=0.5;

0.2, = ) (a

γ

γ

(b)=0.4,

γ

(c)=0.4; 0.6,

= ) (b

δ

δ

(c)=0.5.

Then

τ

:IX →I defined as

     

, otherwise 0,

, , , , = if , 2 1

, 1 or 0 = if 1, = )

( λ λ µγ δ

λ λ

τ

is a fuzzy topology on X. For r=

2 1

, then the topological space (X,

τ

) is both

2 1

-fuzzy e*-regular and

2 1

-fuzzy regular.

Remark 3.2. Clearly, r-fuzzy regular space is r-fuzzy e*-regular space, but the converse is false, in general, is shown by the following example.

Example 3.4. Let

λ

,

µ γ

, and

δ

be fuzzy subsets of X =

{

a,b,c

}

defined as follows:

0.5, = ) (a

λ

λ

(b)=0.6,

λ

(c)=0.5;

0.4, = ) (b

µ

µ

(c)=0.4;

0.2, = ) (a

γ

γ

(b)=0.6,

γ

(c)=0.5;

0.7, = ) (b

Topological Spaces

353 Then

τ

:IX →I

defined as

    

     

 

, otherwise 0,

, = if , 2 1

, = if , 2 1

, = if , 2 1

, = if , 2 1

, 1 or 0 = if 1,

= ) (

δ λ

γ λ

µ λ

λ λ λ

λ τ

is a fuzzy topology on X. For r=

2 1

, then the topological space (X,

τ

) is

2 1

-fuzzy e*

-regular but not

2 1

-fuzzy regular. Since for any

ω

=(0.8_a,0.7_b,0.6_c)∈

τ

,

. > 1 = ) 2 1 ,

(

λ

ω

τ

C

Theorem 3.2. Let (X,

τ

) and (Y,

η

) be fts’s. Let f :(X,

τ

)→(Y ,

η

) be a mapping and (Y,

η

) be a fuzzy e*-regular space. Then f is fuzzy almost e*-continuous iff f is fuzzy e*-continuous.

Proof:(⇐) It is trivial from Remark 2.1.

)

(⇒ Let

λ

∈IY. By the fuzzy e*-regularity of (Y,

η

), for we have

λ

=

∨

λ

_α, where

r ≥

) (

λ

_α

η

and I_η(C_η(

δ

I_η(

λ

_α,r),r),r)≤

λ

.

By Theorem 2.1(3), ( )= ( ) 1 1

α

λ

λ

−

−

∨

f f

≤

∨

e*I_τ(f−1(I_η(C_η(

δ

I_η(

λ

_α,r),r),r)),r)

≤e*I_τ(f−1(

λ

),r).

But * ( 1(

λ

), ) 1(

λ

).

τ f− r ≤ f−

I

e Thus, _f 1(

λ

)=_e*_I (_f 1(

λ

),_r).

τ −

−

Hence f is fuzzy e* -continuous.

Definition 3.3. Let (X,

τ

) and (Y,

η

) be fts’s. A mapping f :(X ,

τ

)→(Y ,

η

) is called fuzzy weakly e*-continuous if for each

µ

∈IY and r∈I0 with

η

(

µ

)≥r, we

have f−1(

µ

)≤e*I_τ(f−1(e*C_η(

µ

,r)),r).

Example 3.5. Let

α

,

β

,

γ

,

δ

and

ω

be fuzzy subsets of X =Y =

{

a,b,c,d

}

defined as follows:

α

(a)=0.5,

α

(b)=0.4,

α

(c)=0.6,

α

(d)=0.4;

0.4, = ) (

β

a

β

(b)=0.6,

β

(c)=0.4,

β

(d)=0.5;

0.5, = ) (

A.Prabhu, A. Vadivel and B. Vijayalakshmi 354 0.4, = ) (

δ

a

δ

(b)=0.4,

δ

(c)=0.4,

δ

(d)=0.4;

0.3, = ) (

ω

a

ω

(b)=0.4,

ω

(c)=0.5,

ω

(d)=0.5.

Then

τ η

, :IX →I defined as

            . otherwise 0, , = if , 2 1 , 1 or 0 = if 1, = ) ( , otherwise 0, , , , , = if , 2 1 , 1 or 0 = if 1, = ) ( λ ω λ λ η δ γ β α λ λ λ τ

are fuzzy topologies on X. For r=

2 1

, then the identity mapping f :(X,

τ

)→(Y ,

η

) is

fuzzy weakly e*-continuous since, for any r-fuzzy open set

ω

in Y,

ω

is r-fe*c in Y

and the inverse image of

ω

is r-fe*o set in X. Also

). )), , ( ( ( )

( * 1 *

1 r r C e f I e

f−

ω

≤ _τ − _η

ω

Theorem 3.3. Let f :(X ,

τ

)→(Y ,

η

) be a mapping. Then the following are equivalent: 1. f is fuzzy weakly e*-continuous.

2. e*C_τ(f−1(e*I_η(

µ

,r)),r)≤ f−1(

µ

) for each

µ

∈IY, r∈I0 with

η

(1−

µ

)≥r. 3. 1( ( , )) * ( 1( * ( , )), ),

r r C e f I e r I

f− _η

µ

≤ _τ − _η

µ

for each

µ

∈IY, r∈I0.

4. e*C_τ(f−1(e*I_η(

µ

,r)),r)≤ f−1(C_η(

µ

,r)), for each

µ

∈IY,r∈I₀.

Proof:(1)⇒(2): Let

µ

∈IY, r∈I0 with

η

(1−

µ

)≥r. By (1) and Theorem 1.3,

) 1 ( = ) (

1− f−1

µ

f−1 −

µ

≤e*I_τ(f−1(e*C_η(1−µ)),r)

= ( (1 ( , )), )

* 1 * r r I e f I

e _τ − − _η µ =e*I_τ(1− f−1(e*I_η(µ,r)),r)

=1 ( ( ( , )), )

* 1

*_{C f e I r r} e _τ − _η µ

− Thus, * ( 1( * (

µ

, )), ) 1(

µ

).

η

τ f− e I r r ≤ f−

C e

(2)⇒(1): It is analogous to the proof of (1)⇒ (2).

(1)⇒(3): Let

µ

∈IY and r∈I₀. Since

η

(I_η(

µ

,r))≥r, by (1),

). )), , ( ( ( ) )), ), , ( ( ( ( )) , (

( * 1 * * 1 *

1 r r C e f I e r r r I C e f I e r I

f− _τ

µ

≤ _τ − _{η η}

µ

≤ _τ − _η

µ

(3)⇒(4) and (4)⇒(2): Obvious.

(2)⇒(4): It is analogous to the proof of (1) ⇒ (3).

Theorem 3.4. A mapping f :(X ,

τ

)→(Y ,

η

) is fuzzy weakly e*-continuous iff

)) , ( ( ) ), (

( 1 1 *

* r C e f r f C

e _τ −

µ

≤ − _η

µ

for each

µ

∈IY, r∈I0 with

η

(

µ

)≥r. Proof: Let

µ

∈IY, r∈I₀ with

η

(

µ

)≥r. Then, by the definition of e*C_η,

. )) , ( 1

( −e*C µ r ≥r

η _η By the fuzzy weakly e*-continuity of f we have, ) )), ), , ( 1 ( ( ( )) , ( 1

( * * 1 * *

1 r r r C e C e f I e r C e

Topological Spaces

355

Since _e*_I (_f 1(_e*_C (1 _e*_C (µ,_r),_r)),_r)=1 _e*_C (_f 1(_e*_I (_e*_C (µ,_r),_r)),_r).

η η τ

η η

τ − − − −

We have, 1− f−1(e*C_η(µ,r))≤1−e*C_τ(f−1(e*I_η(e*C_η(µ,r),r)),r). It implies, * ( 1( * ( * ( , ), )), ) 1( * ( , )).

r C e f r r r C e I e f C

e _τ − _{η η}

µ

≤ − _η

µ

Since

µ

≤e*I_η(e*C_η(

µ

,r),r), we have, e*C_τ(f−1(

µ

),r)≤ f−1(e*C_η(

µ

,r)).

Conversely, let

µ

∈IY, r∈I0 with

η

(

µ

)≥r. Then (1 ( , )) .

*

r r C

e ≥

− µ

η _η

By hypothesis, e*C_τ(f−1(1−e*C_η(µ,r)),r)≤ f−1(e*C_η(1−e*C_η(µ,r),r))

⇒e*C_τ(1− f−1(e*C_η(µ,r)),r)≤ f−1(1−e*I_η(e*C_η(µ,r),r))

⇒1 _e*_I (_f 1(_e*_C (µ,_r)),_r) 1 _f 1(_e*_I (_e*_C (µ,_r),_r))

η η η

τ − ≤ − −

−

⇒ f−1(e*I_η(e*C_η(

µ

,r),r))≤e*I_τ(f−1(e*C_η(

µ

,r)),r)

⇒ f−1(

µ

)≤e*I_τ(f−1(e*C_η(

µ

,r)),r).

Hence f is fuzzy weakly e*-continuous.

3.1. Fuzzy almost e*-compactness

The study of fuzzy almost compact spaces was initiated in [2]. The same concept was taken up for further study in [21] and in [22]. The notion was extended to arbitrary fuzzy sets in a fts. Now we introduce the following definitions.

Definition 3.1.1. Let (X,

τ

) be a fts and r∈I0. A fuzzy set

X I ∈

µ

is called r-fuzzy e*

-compact in (X,

τ

) iff for each family {

λ

_i∈IX |

λ

_i is r-fe*o, i∈Γ} such that

i i

λ

µ

_∨

Γ ∈

≤ there exists a finite index set Γ₀ ⊂Γ such that .

0 i i

λ

µ

_∨

Γ ∈

≤ (X,

τ

) is called r

-fuzzy e*-compact iff 1is r-fuzzy e*-compact in (X,

τ

).

Definition 3.1.2. Let (X,

τ

) be a fts and r∈I0. A fuzzy set

X I ∈

µ

is called r-fuzzy almost e*-compact in (X,

τ

) iff for each family {

λ

_i∈IX |

λ

_i is r-fe*o, i∈Γ} such that _i

i λ

µ

∨

Γ ∈

≤ there exists a finite index set Γ0 ⊂Γ such that ( , ). *

0

r C

e i

i

λ

µ

∨

τ

Γ ∈

≤

) ,

(X

τ

is called r-fuzzy almost e*-compact iff 1 is r-fuzzy almost e*-compact in

). , (X

τ

Theorem 3.1.1. Let (X,

τ

) be a fts and r∈I0. Then the following statements are

A.Prabhu, A. Vadivel and B. Vijayalakshmi

356

2. For every collection

{

∈IX eC e I _i r r _i i∈Γ

}

i | ( ( , ), )= ,

*

*

λ

λ

λ

τ τ with i =0,

i

λ

∧

_∈Γ there exists a finite subset Γ0 ⊆Γ with ( , )=0.

* 0 r I e _i i

λ

τ

∧

_Γ ∈

3.

∧

* ( , )≠0

Γ ∈ r C e i i

λ

τ holds for every collection of {

λ

i∈IX |

} ), ), , ( (

=_e*_{I e}*_{C r r i}∈Γ i

i

λ

λ

τ τ with the finite intersection property (FIP, for short), that

is, for each finite subset Γ₀⊂Γ with

∧

≠0.

Γ ∈ i

i

λ

4. Every family of

{

λ

_i∈IX |e*I_τ(e*C_τ(

λ

_i,r),r)=

λ

_i,i∈Γ

}

such that i =1,

i

λ

∨

_∈Γ there exists a finite subset Γ₀ ⊆Γ such that e*C ( i,r)=1.

i

λ

τ

∨

_∈Γ

Proof:(1)⇒(2): Let (X,

τ

) be r-fuzzy almost e*-compact and the collection

{

λ

_i∈IX |e*C_τ(e*I_τ(

λ

_i,r),r)=

λ

_i,i∈Γ

}

with i =0.

i

λ

∧

_∈Γ Then (1 i)=1.

i

λ

−

∨

_∈Γ Since

), ), , 1 ( ( =

1−

λ

i e*Iτ e*Cτ −

λ

i r r we have ( (1 , ), )=1 * * r r C e I e _i i

λ

τ τ −

∨

_∈Γ and

) ), , 1 ( ( * * r r C e I

e τ τ −

λ

i is r-f o. *

e From the r-fuzzy almost e*-compactness, it follows that there exists a finite subset Γ₀ of Γ such that e*C (e*I (e*C (1 _i,r),r),r)=1

i

λ

τ τ τ −

∨

_∈Γ and then ) ), ), , 1 ( ( ( 1 =

0 e*C e*I e*C i r r r

i

λ

τ τ τ − −

∨

Γ ∈ ) ), ), , 1 ( ( ( 1

= e*C e*I e*C i r r r

i

λ

τ τ τ − −

∧

_∈Γ ) ), ), , ( ( (

= e*I e*C e*I _i r r r i

λ

τ τ τ

∧

_∈Γ = e*I ( _i,r).

i

λ

τ

∧

_∈Γ : (3)

(2)⇒ Let

{

λ

_i∈IX |e*I_τ(e*C_τ(

λ

_i,r),r)=

λ

_i, i∈Γ

}

with FIP. Suppose that

) , ( * r C e i i

λ

τ

∧

_∈Γ =0. Now e*C_τ(

λ

_i,r)=e*C_τ(e*I_τ(e*C_τ(

λ

_i,r),r),r) and by assumption, there exists a finite subset Γ0 of Γ such that

. 0 = ) ), , ( ( = * * 0 0 r r C e I e i i i i

λ

λ

∈Γ τ τ Γ ∈

∧

∧

It is a contradiction.

: (4)

(3)⇒ Let

{

∈IX e I eC _i r r _i i∈Γ

}

i | ( ( , ), )= ,

*

*

λ

λ

λ

τ τ such that i =1.

i

λ

∨

_∈Γ Suppose

1 ) , ( * 0 ≠

∨

_Γ ∈ r C e _i i

λ

τ for every finite subset Γ0 of Γ.

Then e*Iτ(e*Cτ(1−e*Cτ(

λ

i,r),r),r)=1−e*Cτ(

λ

i,r),i∈Γ with FIP,

Since e*I_τ(e*C_τ(1−e*C_τ(

λ

_i,r),r),r)=e*I_τ(1−e*I_τ(e*C_τ

λ

_i,r),r),r) )

, 1 (

=e*Iτ −

λ

i r =1 ( , ) *

Topological Spaces

357 By(3), we obtain

∧

* (1− * ( , ), )≠0

Γ ∈ r r C e C e i i

λ

τ

τ ⇒

∧

1− * ( * ( , ), )≠0.

Γ ∈ r r C e I e i i

λ

τ τ ⇒ . 1 ) ), , ( ( = _∈Γ * * ≠ Γ ∈

∨

∨

_{i i} e I eC _i r r i

λ

λ

_{τ τ} It is a contradiction.

(4)⇒(1): Let {

λ

_i∈IX :

λ

_i is r-fe*o, i∈Γ} with _i

i λ

∨

_∈Γ =1. It implies

) ), , ( ( * * 0 r r C e I e i i

λ

τ τ

∨

_Γ ∈

=1 with by Theorem 1.4,

) ), , ( ( = ) ), ), ), , ( ( ( ( * * * * * * r r C e I e r r r r C e I e C e I

e _{τ τ τ τ}

λ

_{i τ τ}

λ

_i

From the hypothesis, it follows that there exists a finite subset Γ₀ of Γ with

. 1 = ) ), ), , ( ( ( * * * 0 r r r C e I e C e i i

λ

τ τ τ

∨

_Γ ∈

Since _e*_I (_e*_C ( ,_r),_r) _e*_C ( ,_r),

i i

i

λ

λ

λ

≤ τ τ ≤ τ then

). , ( = ) ), ), , ( ( ( * * *

*_{C e I eC r r r eC r}

e _{τ τ τ}

λ

_{i τ}

λ

_i Hence * ( , )=1,

0 r C e _i i

λ

τ

∨

_∈Γ i.e X is r-fuzzy almost e*-compact.

Theorem 3.1.2. A fuzzy weakly e*-continuous image of r-fuzzy e*-compact set is r -fuzzy almost e*-compact.

Proof: Let

µ

be r-fuzzy e*-compact in (X,

τ

) and f :(X ,

τ

)→(Y ,

η

) be a fuzzy weakly e*-continuous mapping. If {

λ

_i∈IY |

λ

_i is r-f_e*o

, i ∈Γ} with ( ) _i,

i

f µ

∨

λ Γ ∈

≤

then 1( i).

i f

λ

µ

− Γ ∈

∨

≤ Since f is fuzzy weakly e*-continuous, then for each i∈Γ,

). )), , ( ( ( )

( * 1 *

1 r r C e f I e

f−

λ

_i ≤ _τ − _η

λ

_i Hence e*I (f 1(e*C ( _i,r)),r),

i

λ

µ

τ − η Γ ∈

∨

≤ ) )), , ( (

( 1 *

* r r C e f I

e τ η

λ

i

−

is r-f_e*o.

Since

µ

is r-fuzzy e*-compact in (X,

τ

), there exists a finite subset Γ₀ of Γ with * ( 1( * ( , )), ).

0 r r C e f I e _i i

λ

µ

_τ − _η Γ ∈

∨

≤ It follows that

        ≤ − Γ ∈

∨

( ( ( , )), ) )

( * 1 *

0 r r C e f I e f f _i i λ µ τ η

(

( ( ( , )), )

)

= * 1 *

0 r r C e f I e f i i λ η τ − Γ ∈

∨

(

1( * ( , ))

)

0 r C e f f _i i λ η − Γ ∈

∨

≤ ) , ( * 0 r C e i i

λ

η

∨

_Γ ∈

≤ Hence f(

µ

) is r-fuzzy almost e*-compact in (Y,

η

).

A.Prabhu, A. Vadivel and B. Vijayalakshmi

358

Proof: Similar to the proof of Theorem 3.2.

Definition 3.1.3. A mapping f :(X,

τ

)→(Y ,

η

) is said to be fuzzy strongly e* -continuous iff ( * (

λ

, )) (

λ

),

τ r f

C e

f ≤ for all

λ

∈IX and r∈I₀.

Theorem 3.1.4. A fuzzy strongly e*-continuous image of r-fuzzy almost e*-compact set is r-fuzzy e*-compact.

Proof: Let

µ

∈IX be r-fuzzy almost e*-compact in (X,

τ

) and f :(X ,

τ

)→(Y ,

η

) a fuzzy strongly e*-continuous mapping. If {

λ

_i∈IY |

λ

_i is r-f_e*o

, i∈Γ} with

, )

( _{i i}

f

µ

≤

∨

_∈Γ

λ

then 1( i).

i

f

λ

µ

−

Γ ∈

∨

≤ Since f is fuzzy strongly e*-continuous and

hence e*-continuous, f−1(

λ

_i) is r-fe*o. Since

µ

is r-fuzzy almost e*-compact, there exists a finite subset Γ0 of Γ with ( ( ), ).

1 *

0

r f

C

e i

i

λ

µ

τ −

Γ ∈

∨

≤ From the fuzzy strong e*

-continuity of f , we deduce

    

  

≤ −

Γ ∈

∨

( ( ), )

)

( * 1

0

r f

C e f

f i

i

λ

µ

τ

(

( ( ), )

)

= * 1

0

r f

C e

f i

i

λ

τ −

Γ ∈

∨

(

1( )

)

0

i i

f

f −

λ

Γ ∈

∨

≤

i i

λ

∨

_Γ

∈

≤

0

Hence f(

µ

) is r-fuzzy e*-compact in (Y,

η

).

3.2. Fuzzy near e*-compactness

In this section, we introduce fuzzy nearly e*-compactness and investigate the properties and some of their characterizations.

Definition 3.2.1. Let (X,

τ

) be a fts and r∈I₀. A fuzzy set

µ

∈IX is called r-fuzzy nearly e*-compact in (X,

τ

) iff for each family {

λ

_i∈IX |

λ

_i is r-fe*o, i∈Γ} such that _i,

i λ

µ

_∨

Γ ∈

≤ there exists a finite index set Γ₀ ⊂Γ such that

). ), , ( ( *

*

0

r r C e I

e i

i

λ

µ

_∨

_{τ τ}

Γ ∈

≤ (X,

τ

) is called r-fuzzy nearly e*-compact iff 1is r-fuzzy

nearly e*-compact in (X,

τ

).

Topological Spaces

359

r-fuzzy e*-compactness ⇒ r-fuzzy nearly e*-compactness ⇒ r-fuzzy almost e* -compactness.

Theorem 3.2.1. Let (X,

τ

) be r-fuzzy e*-regular. It is r-fuzzy almost e*-compact iff it is r-fuzzy e*-compact.

Proof: Let {

λ

_i∈IX |

λ

_i is r-fe*o, i∈Γ} be a family such that with _i =1.

i

λ

∨

_∈Γ Since

) ,

(X

τ

is r-fuzzy e*-regular, for each

τ

(

λ

i)≥r,

{

| ( ) , ( ( ( , ), ), )

}

.

= _i

k i k

i k i i K k i

i

λ

τ

λ

r I C

δ

I

λ

r r r

λ

λ

∨

≥ τ τ τ ≤

∈

Hence _=1.

  

  

∨

∨

_{∈Γ ∈} ik

i K k i i

λ

Since (X,

τ

) is r-fuzzy almost e*-compact, there exists a

finite index J∈KJ such that1= ( , ) . *

    

  

∨

∨

_{∈ ∈} eC r

k j J

K k j J j

λ

τ

For j∈J, since * ( j_k, ) j,

J K k j

r C

e _τ

λ

≤

λ

∨

_∈ we have _j =1.

J j

λ

∨

_∈ Hence (X,

τ

) is r-fuzzy *

e -compact.

Corollary 3.2.1. Let (X,

τ

) be r-fuzzy e*-regular. It is r-fuzzy nearly e*-compact iff it is r-fuzzy e*-compact.

Proof: It is straightforward.

Theorem 3.2.2. A fts (X,

τ

) is r-fuzzy nearly e*-compact iff every collection

{

∈I eC e I i r r i i∈Γ

}

X

i | ( ( , ), )= ,

*

*

λ

λ

λ

τ τ

having the FIP, we have

∧

≠0.

Γ ∈ i

i

λ

Proof: Let

{

λ

_i∈IX |e*C_τ(e*I_τ(

λ

_i,r),r)=

λ

_i, i∈Γ

}

with FIP. If _i =0,

i

λ

∧

_∈Γ then

1 = ) 1

( i

i

λ

−

∨

_∈Γ and

λ

i is r-f o *

e set. From the r-fuzzy near e*-compactness of (X,

τ

),

there exists a finite subset Γ₀ of Γ with * ( * (1 , ), )=1.

0

r r C

e I

e i

i

λ

τ

τ −

∨

_Γ

∈

Since

. 1 = ) ), , ( ( 1

= ) ), , ( 1 ( = ) ), , 1 (

( * * * * *

*

i i

i

i r r e I e I r r eC e I r r

C e I

e _{τ τ} −

λ

_τ − _τ

λ

− _{τ τ}

λ

−

λ

It follows that 1 =1.

0 i i

λ

−

∨

_∈Γ Hence =0.

0 i i

λ

∧

_Γ

∈

A.Prabhu, A. Vadivel and B. Vijayalakshmi

360 Conversely, let {

λ

_i∈IX :

λ

_i is r-f *o

e , i∈Γ} with i =1.

i

λ

∨

_∈Γ Suppose that

, 1 ) ), , ( ( * * 0 ≠

∨

_Γ ∈ r r C e I e _i i

λ

τ

τ for every finite subset Γ0 of Γ. Then by Theorem 1.4

), ), , ( ( 1 = ) ), ), ), , ( ( 1 ( ( * * * * * * r r C e I e r r r r C e I e I e C

e τ τ − τ τ

λ

i − τ τ

λ

i

for each i∈Γ with the FIP. From the hypothesis, it follows that

0 ) ), , ( (

1− * * ≠

∧

_∈Γ e I eC _i r r i

λ

τ

τ * ( * ( , ), ) 1.

0 ≠ ⇒

∨

Γ ∈ r r C e I e i i

λ

τ τ

Hence

∨

≤

∨

* ( * ( , ), )≠1.

Γ ∈ Γ ∈ r r C e I e i i i i

λ

λ

τ τ It is a contradiction.

Theorem 3.2.3. The image of a r-fuzzy nearly e*-compact set under a fuzzy almost e* -continuous and fuzzy almost e*-open mapping is r-fuzzy nearly e*-compact.

Proof: Let

µ

∈IX be r-fuzzy nearly e*-compact in (X,

τ

) and f :(X,

τ

)→(Y ,

η

)

be a fuzzy almost e*-continuous and fuzzy almost e*-open mapping. If i X

i I

λ

λ

|

{ ∈ is

r-f_e*o

, i∈Γ} with ( ) _i,

i

f µ

∨

λ Γ ∈

≤ then for each i∈Γ, e*Iη(e*Cη(

λ

i,r),r) is r-f o

* e

set. From the fuzzy almost e*-continuity of f , it follows that

)), ), , ( ( (

(f 1 e*I e*C _i r r i

λ

µ

− _{η η} Γ ∈

∨

≤ f−1(e*I_η(e*C_η(

λ

_i,r),r))

is r-fe*o in X. Since

µ

is r-fuzzy nearly e*-compact in (X,

τ

), there exists a finite subset Γ₀ of Γ such that * ( * ( 1( * ( * ( , )), ), ), ).

0 r r r r C e I e f C e I e _i i

λ

µ

_{τ τ} − _{η η} Γ ∈

∨

≤

Thus _

      ≤ − Γ ∈

∨

( ( ( ( ( , ), )), ), ) )

( * * 1 * *

0 r r r r C e I e f C e I e f f i i

λ

µ

_{τ τ η η}

(

( ( ( ( ( , ), )), ), )

)

= * * 1 * *

0 r r r r C e I e f C e I e f _i i

λ

η η τ τ − Γ ∈

∨

From e*I_η(e*C_η(

λ

_i,r),r)≤e*C_η(

λ

_i,r) for every i∈Γ and the almost e* continuity of

,

f it follows that f−1(e*I_η(e*C_η(λ_i,r),r))≤ f−1(e*C_η(λ_i,r)),(1− f−1(e*C_η(λ_i,r))) is r-fe*o in X .

)), , ( ( ) )), ), , ( ( (

( 1 * * 1 *

* r C e f r r r C e I e f C

e _τ − _{η η}

λ

_i ≤ − _η

λ

_i

⇒ )). , ( ( ) ), )), ), , ( ( ( (

( * 1 * * 1 *

* r C e f r r r r C e I e f C e I

e τ τ η η

λ

i η

λ

i

−

− _≤

⇒

Since f is fuzzy almost e*-open and

). ), ), ), )), ), , ( ( ( ( ( ( ( = ) ), )), ), , ( ( ( (

( * 1 * * * * * 1 * *

* r r r r r r C e I e f C e I e C e I e r r r r eC I e f C e I

e τ τ η η

λ

i τ τ τ τ η η

λ

i

− −

We have f

(

e*Iτ(e*Cτ(f 1(e*Iη(e*Cη(

λ

i,r),r)),r),r)

)

−

Topological Spaces

361 Now from

f

(

e*Iτ(e*Cτ(f 1(e*Iη(e*Cη(

λ

i,r),r)),r),r)

)

≤ f(f 1(e*Cη(

λ

i,r)))≤e*Cη(

λ

i,r).

− −

We deduce for each i∈Γ,

(

e*I (e*C (f 1(e*I (e*C ( ,r),r)),r),r)

)

e*I (e*C ( ,r),r).

f τ τ η η

λ

i ≤ η η

λ

i

−

Hence

(

( ( ( ( ( , ), )), ), )

)

( ( , ), ). )

( * *

0 *

* 1 * *

0

r r C e I e r

r r r C e I e f C e I e f

f i

i i

i

λ

λ

µ

∨

τ τ η η

∨

η η

Γ ∈ −

Γ

∈ ≤

≤

Thus f(

µ

) is r-fuzzy nearly e*-compact in (Y ,

η

).

Theorem 3.2.4. The image of a r-fuzzy e*-compact set under a fuzzy almost

e

-continuous mapping is r-fuzzy nearly e*-compact.

Proof: Let

µ

∈IX be r-fuzzy e*-compact in (X,

τ

) and f :(X ,

τ

)→(Y ,

η

) be a fuzzy almost e*-continuous mapping. If {

λ

_i∈IX |

λ

_i is r-fe*o, i∈Γ} with

, )

( _i

i

f µ

∨

λ Γ ∈

≤ then f( ) e*I (e*C ( _i,r),r)

i

λ

µ

∨

η η

Γ ∈

≤ such that, by Theorem 1.4, for each

,

Γ ∈

i e*Iη(e*Cη(e*Iη(e*Cη(

λ

i,r),r),r),r)=e*Iη(e*Cη(

λ

i,r),r).

From the almost e*-continuity of f , it follows that f 1(e*I (e*C ( _i,r),r)),

i

λ

µ

− η η

Γ ∈

∨

≤

)) ), , ( (

( * *

1

r r C e I e

f η η

λ

i

−

is r-f *o

e in X. Since

µ

is r-fuzzy e*-compact in (X,

τ

),

there exists a finite subset Γ0 of Γ such that

). ), )), ), , ( ( ( (

( * 1 * *

*

0

r r r r C e I e f C e I

e i

i

λ

µ

τ τ − η η

Γ ∈

∨

≤

Thus, _

  

  

≤ −

Γ ∈

∨

( ( ( , ), )) )

( 1 * *

0

r r C e I e f f

f _i

i

λ

µ

η η =

(

1( * ( * ( , ), ))

)

0

r r C e I e f

f _i

i

λ

η η − Γ ∈

∨

). ), , ( ( *

*

0

r r C e I

e i

i

λ

η η

∨

_Γ

∈

≤ Thus f(

µ

) is r-fuzzy nearly e*-compact in (Y,

η

).

4. Conclusion

S⌣ostak’s fuzzy topology has been recently of major interest among fuzzy topologies. In this paper, we have introduced fuzzy almost e*-continuous, fuzzy weakly e*-continuous,

r-fuzzy e*-compact, r-fuzzy almost e*-compactness and r-fuzzy near compactness in

A.Prabhu, A. Vadivel and B. Vijayalakshmi

362

Acknowledgement. The authors express their earnest gratitude to the reviewers, editor-in-chief and managing editors for their constructive suggestions and comments which helped to improve the present paper.

REFERENCES

1. C.L.Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968) 182-190. 2. A.Di Concilio and G.Gerla, Almost compactness in fuzzy topological spaces, Fuzzy

Sets and Systems, 13 (1984), 187-192.

3. K.C.Chattopadhyay and S.K.Samanta, Fuzzy topology : fuzzy closure operator, fuzzy compactness and fuzzy connectedness, Fuzzy sets and systems, 54(2) (1993) 207-212.

4. K.C.Chattopadhyay, R.N.Hazra and S.K.Samanta, Gradation of openness : Fuzzy topology, Fuzzy Sets and Systems, 49(2) (1992) 237-242.

5. M.Demirci, Almost compactness and near compactness in three different topological structures of smooth topological spaces, Bull. Cal. Math. Soc., 88 (1996) 79-86. 6. M. Demirci, On several types of compactness in smooth topological spaces, Fuzzy

Sets and Systems, 90(1) (1997) 83-88.

7. M.Demirci, Neighborhood structures of smooth topological spaces, Fuzzy Sets and

Systems, 92 (1997) 123-128.

8. M.Demirci, Three topological structures of smooth topological spaces, Fuzzy Sets

and Systems, 101(1) (1999) 185-190.

9. M.K.EI Gayyar, E.E.Kerre and A.A.Ramadan, Almost compactness and near compactness in Smooth topological spaces, Fuzzy Sets and Systems, 62 (1994) 193-202.

10. M.K.EI Gayyar, E.E.Kerre and A.A.Ramadan, On smooth topological spaces II: separation axioms, Fuzzy Sets and Systems, 119(3) (2001) 459-504.

11. R.Ertuɺɺrk and M.Demirci, On the compactness in fuzzy topological spaces in S⌣

ostak’s sense, Mathematiki Vesnik, 50 (1998) 75-81.

12. A.H.Es and D.Coker, On several types of degree of fuzzy compactness in fuzzy topological spaces in S⌣ostak’s sense, J. Fuzzy Math., 3(3) (1995) 481-491.

13. A.H.Es and D.Coker, On several types of degree of fuzzy compactness, Fuzzy sets

and Systems, 87 (1997) 349-359.

14. V.Gregori and A.Vidal, Gradation of openness and Chang’s fuzzy topologies, Fuzzy

Sets and Systems, 109(2) (2000) 233-244.

15. R.N.Hazra, S.K.Samanta and K.C.Chattopadhyay, Fuzzy topology redefined, Fuzzy

Sets and Systems, 45(1) (1992) 79-82.

16. U.Hoɺɺhle and A.P.S⌣ostak, A general theory of fuzzy topological spaces, Fuzzy Sets

and Systems, 73 (1995) 131-149.

17. U.Hoɺɺhle and A.P.S⌣ostak, Axiomatic Foundations of Fixed-Basis fuzzy topology,

The Handbooks of Fuzzy sets series, vol. 3, Kluwer Academic Publishers, Dordrecht

(Chapter 3) (1999) 123-272.

18. B.Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl., 50 (1975) 74-79.

Topological Spaces

363

fuzzy topology, Indian J. Pure and Appl. Math., 34(2) (2003) 311-333.

20. R.Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56 (1976) 621-633.

21. M.N.Mukherjee and S.P.Sinha, Almost compact fuzzy sets in fuzzy topological spaces, Fuzzy Sets and Systems, 38 (1990) 389-396.

22. M.N.Mukherjee and S.P.Sinha, Almost compact fuzzy topological spaces, Mat.

Vesnik, 41 (1989) 89-97.

23. W.Peeters, Subspaces of smooth fuzzy topologies and initial smooth fuzzy structures, Fuzzy Sets and Systems, 104 (1999) 423-433.

24. A.A.Ramadan, Smooth topological spaces, Fuzzy Sets and Systems, 48 (1992) 371-375.

25. A.A.Ramadan, On Smooth topological spaces III, J. Fuzzy Math., 8(1) (2000) 53-64. 26. A.A.Ramadan, S.N.El-Deeb and M.A.Abdel-Sattar, On smooth topological spaces

IV, Fuzzy Sets and Systems, 119 (3) (2001) 473-482.

27. A.P.S⌣ostak, On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palerms Ser II, 11(1985) 89-103.

28. A.P.S⌣ostak, On some modifications of fuzzy topologies, Matematicki Vesnik, 41 (1989) 20-37.

29. A.P.S⌣ostak, Basic structures of fuzzy topology, J. Math. Sciences, 78(6) (1996) 662-701.

30. R.Srivastava, On separation axioms in newly defined fuzzy topology, Fuzzy Sets and

Systems, 62 (1994) 341-346.

31. V.Seenivasan and K.Kamala, Fuzzy e-continuity and fuzzy e-open sets, Annals of

Fuzzy Mathematics and Informatics, 8(1) (2014) 141–148.

32. Y.C.Kim and J.W.Park, r-fuzzy

δ

-closure and r-fuzzy

θ

-closure sets, J. Korean

Institute of Intelligent Systems, 10(6) (2000) 557-563.

33. D.Sobana, V.Chandrasekhar and A.Vadivel, Fuzzy

e

-continuity in S⌣ostak’s fuzzy topological spaces, (submitted).

34. A.Vadivel and B.Vijayalakshmi, Fuzzy e-irresolute mappings and fuzzy e -connectedness in smooth topological spaces, (submitted).

35. A.Vadivel and B.Vijayalakshmi, Fuzzy Almost e-continuous mappings and fuzzy

e

-connectedness in smooth topological spaces, (submitted).