regular semi continuous and open mappings on intuitionistic fuzzy … · 2018-09-30 ·...

© 2018 JETIR September 2018, Volume 5, Issue 9 www.jetir.org (ISSN-2349-5162)

JETIRA006352 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 193

REGULAR SEMI CONTINUOUS AND OPEN

MAPPINGS ON INTUITIONISTIC FUZZY

TOPOLOGICAL SPACES IN S OSTAK’S SENSE

G. Saravanakumar1

Research Scholar, Department of Mathematics, Annamalai University

S. Tamilselvan2

Mathematics Section (FEAT), Annamalai University

A. Vadivel3

Department of Mathematics, Government Arts College (Autonomous), Karur -5

ABSTRACT: We introduce the concepts of fuzzy ( , )r s -regular semi (resp. ( , )r s - , ( , )r s -pre and ( , )r s - )

continuous mappings and fuzzy ( , )r s -regular semi (resp. ( , )r s - , ( , )r s -pre and ( , )r s - ) open mappings on

intuitionistic fuzzy topological spaces in the sense of S ostak's. Further we investigate some of their characteristic and respective

properties.

KEYWORDS AND PHRASES: fuzzy ( , )r s -regular semi (resp. ( , )r s - , ( , )r s -pre and ( , )r s - ) continuous

mappings and fuzzy ( , )r s -regular semi (resp. ( , )r s - , ( , )r s -pre and ( , )r s - ) open mappings.

AMS (2000) SUBJECT CLASSIFICATION: 54A40.

1. Introduction The concept of fuzzy sets was introduced by Zadeh [13]. Chang [2] defined fuzzy topological spaces. These

spaces and its generalizations are later studied by several authors one of which, developed by S ostak [12], used the idea of

degree of openness. This type of generalization of fuzzy topological spaces was later rephrased by Chattopadhyay, Hazra and

Samanta [3], and by Ramadan [10]. As a generalization of fuzzy sets, the concept of intuitionistic fuzzy sets was introduced by

Atanassov [1]. Recently, Coker and his collegues [4,6,7] introduced intuitionistic fuzzy topological spaces using intuitionistic

fuzzy sets. Using the idea of degree of openness and degree of nonopenness, Coker and Demirci [5] defined

Intuitionistic fuzzy topological spaces in S ostak's sense as a generalization of smooth fuzzy topological spaces and

intuitionistic fuzzy topological spaces. In this paper, we introduce the concepts of fuzzy ( , )r s -regular semi (resp. ( , )r s - ,

( , )r s -pre and ( , )r s - ) continuous mappings and fuzzy ( , )r s -regular semi (resp. ( , )r s - , ( , )r s -pre and ( , )r s - )

open mappings on intuitionistic fuzzy topological spaces in the sense of S ostak's. Further we investigate some of their

characteristic and respective properties.

2. Preliminaries Let I be the unit interval [0,1] of the real line. A member of XI is called a fuzzy set of X . By 0 and

1 we denote constant maps on X with value 0 and 1, respectively. For any XI ,

c denotes the complement of 1 .

All other notations are standard notations of fuzzy set theory. Let X be a nonempty set. An intuitionistic fuzzy set A is an

ordered pair ( , )A AA where the functions :A X I and :A X I denote the degree of membership and degree of

non-membership, respectively, and 1A A .Obviously every fuzzy set on X is an intuitionistic fuzzy set of the form

( ,1 ) . [1] Let ( , )A AA and ( , )B BB be intuitionistic fuzzy sets on X . Then[(i)] A B iff A B and

A B ,[(ii)] A B iff A B and B A ,[(iii)] ( , )c

A AA ,[(iv)] ( , )A B A BA B ,

[(v)]

( , )A B A BA B ,[(vi)] ~0 (0,1) and ~1 (1,0) . [1] Let f be a map from a set X to a set Y . Let

( , )A AA be a intuitionistic fuzzy set of X and ( , )B BB an intuitionistic fuzzy set of Y . Then:[(i)]The image of A

under f , denoted by ( )f A is an intuitionistic fuzzy set in Y defined by ( ) ( ( ),1 (1 ))A Af A f f .[(ii)]The inverse

image of B under f , denoted by 1( )f B

is an intuitionistic fuzzy set in X defined by1 1 1( ) ( ( ), ( ))B Bf B f f . [10]

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A smooth fuzzy topology on X is a map : XT I I which satisfies the following properties:[(i)] (0) (1) 1T T ,[(ii)]

1 2 1 2( ) ( ) ( )T T T ,[(iii)] ( ) ( )i iT T The pair ( , )X T is called a smooth fuzzy topological space. [5] An

intuitionistic fuzzy topology on X is a family T of intuitionistic fuzzy sets in X which satisfies the following properties:[(i)]

~ ~0 ,1 T ,[(ii)]If 1 2,A A T , then 1 2A A T ,[(iii)]If iA T for all i , then iA T .The pair ( , )X T is called an

intuitionistic fuzzy topological space. Let ( )I X be a family of all intuitionistic fuzzy sets of X and let I I be the set of the

pair ( , )r s such that ,r s I and 1r s . [6] Let X be a nonempty set. An intuitionistic fuzzy topology in S ostak's sense

(SoIFT for shor) 1 2( , )T T T on X is a map : ( )T I X I I which satisfied the following properties:[(i)]

1 ~ 1 ~(0 ) (1 ) 1T T and 2 ~ 2 ~(0 ) (1 ) 1T T ,[(ii)] 1 1 1( ) ( ) ( )T A B T A T B and 2 2 2( ) ( ) ( )T A B T A T B ,[(iii)]

1 1( ) ( )i iT A T A and 2 2( ) ( )i iT A T A . The 1 2( , ) ( , , )X T X T T is said to be an intuitionistic fuzzy topological

space in S ostak's sense (SoIFTS for short). Also, we call 1( )T A a gradation of openness of A and 2( )T A a gradation of

nonopenness of A [8] Let A be an intuitionistic fuzzy set in a SoIFTS 1 2( , , )X T T and ( , )r s I I . Then A is said to be

[(i)]fuzzy ( , )r s -open if 1( )T A r and 2( )T A s ,[(ii)]fuzzy ( , )r s -closed if 1( )cT A r and

2( )cT A s . [8] Let

1 2( , , )X T T be a SoIFTS. For each ( , )r s I I and for each ( )A I X , the fuzzy ( , )r s -interior is defined by

1 2, ( , , ) { ( ) | , is fuzzy( , )-open}T Tint A r s B I X A B B r s .and the fuzzy ( , )r s -closure is defined by

1 2, ( , , ) { ( ) | , is fuzzy( , )-closed}T Tcl A r s B I X A B B r s .The operators 1 2, : ( ) ( )T Tint I X I I I X and

1 2, : ( ) ( )T Tcl I X I I I X are called the fuzzy interior operator and fuzzy closure operator in 1 2( , , )X T T , respectively.

[8] For an intuitionistic fuzzy set A in a SoIFTS 1 2( , , )X T T and ( , )r s I I [(i)]1 2 1 2, ,( , , ) ( , , )c c

T T T Tint A r s cl A r s ,[(ii)]

1 2 1 2, ,( , , ) ( , , )c c

T T T Tcl A r s int A r s . [8] Let ( , )X T be an intuitionistic fuzzy topological space in S ostak's sense. Then it is

easy to see that for each ( , )r s I I , the family ( , )r sT defined by

( , ) 1 2{ ( ) | ( ) and ( ) }r sT A I X T A r T A s is an intuitionistic fuzzy topology on X

[8] Let ( , )X T be an intuitionistic fuzzy topological space ( , )r s I I . Then the map ( , ) : ( )r sT I X I I defined

by( , )

(1,0), if 0,1

( ) ( , ), if {0,1}

(0,1), otherwise,

r s

A

T A r s A T

becomes an intuitionistic fuzzy topology in S ostak's sense on X . [9] Let A be

an intuitionistic fuzzy set in a SoIFTS 1 2( , , )X T T and ( , )r s I I . Then A is said to be [(i)]fuzzy ( , )r s -semi open if

there is a fuzzy ( , )r s -open set B in X such that 1 2, ( , , )T TB A cl B r s , [ii)]fuzzy ( , )r s -semi closed if there is a fuzzy

( , )r s -regular B in X such that 1 2, ( , , )T Tint B r s A B .[11] Let A be an intuitionistic fuzzy set in a SoIFTS 1 2( , , )X T T

and ( , )r s I I . Then A is said to be[(i)]fuzzy ( , )r s -regular open if 1 2 1 2, ,( ( , , ), , )T T T TA int cl A r s r s ,[(ii)]fuzzy ( , )r s -

regular closed if 1 2 1 2, ,( ( , , ), , )T T T TA cl int A r s r s , [(iii)]fuzzy ( , )r s - open if

1 2 1 2 1 2, , ,( ( ( , , ), , ) , )T T T T T TA int cl int A r s r s r s ,[(iv)]fuzzy ( , )r s - closed if 1 2 1 2 1 2, , ,( ( ( , , ), , ) , )T T T T T TA cl int cl A r s r s r s

,[(v)]fuzzy ( , )r s -pre open if 1 2 1 2, ,( ( , , ), , )T T T TA int cl A r s r s ,[(vi)]fuzzy ( , )r s -pre closed if

1 2 1 2, ,( ( , , ), , )T T T TA cl int A r s r s ,

[(vii)]fuzzy ( , )r s - open if 1 2 1 2 1 2, , ,( ( ( , , ), , ) , )T T T T T TA cl int cl A r s r s r s ,[(viii)]fuzzy ( , )r s - closed if

1 2 1 2 1 2, , ,( ( ( , , ), , ) , )T T T T T TA int cl int A r s r s r s , [(ix)]fuzzy ( , )r s -regular semi open if there is a fuzzy ( , )r s -regular open set

B in X such that 1 2, ( , , )T TB A cl B r s ,[(x)]fuzzy ( , )r s -regular semi closed if there is a fuzzy ( , )r s -regular closed set

B in X such that 1 2, ( , , )T Tint B r s A B .[11] Let 1 2( , , )X T T be a SoIFTS. For each ( , )r s I I and for each

( )A I X , the fuzzy ( , )r s -regular semi (resp. ( , )r s - , ( , )r s -pre and ( , )r s - ) - interior is defined by

1 2 1 2 1 2 1 2, , , ,( , , ) (resp. ( , , ), ( , , ) and ( , , ))

{ ( ) | , is fuzzy ( , )- regular semi (resp. , pre and ) -open}

T T T T T T T Trsint A r s int A r s pint A r s int A r s

B I X A B B r s




and the fuzzy ( , )r s -regular semi (resp. ( , )r s - , ( , )r s -pre and ( , )r s - ) - closure is defined by

1 2 1 2 1 2 1 2, , , ,( , , ) (resp. ( , , ), ( , , ) and ( , , ))

{ ( ) | , is fuzzy ( , )-regular semi(resp. , pre and ) -closed}

T T T T T T T Trscl A r s cl A r s pcl A r s cl A r s

B I X A B B r s

3. Fuzzy ( , )r s -regular semi (resp. , pre and ) continuous mappings

Definition 3.1 Let 1 2 1 2: ( , , ) ( , , )f X T T Y W W be a mapping from a SoIFTS X to another SoIFTS Y and

( , )r s I I . Then f is said to be fuzzy ( , )r s -regular semi (resp. regular, , pre and ) continuous if 1( )f B

is a fuzzy

( , )r s -regular semi (resp. regular, , pre and ) open set of X for each fuzzy ( , )r s -open set B of Y .

Fuzzy (r,s)-regular continuous

Fuzzy (r,s)-continuous Fuzzy (r,s)- regular semi continuous

Fuzzy (r,s)- continuous Fuzzy (r,s)- semi continuous Fuzzy

(r,s)- pre continuous Fuzzy (r,s)- continuous

From the above definitions it is clear that the above implications are true for ( , )r s I I .

The converses of the above implications are not true as the following examples show:

Example 3.1 Let { , }X a b Y , and let 1 2, ( )A A I X , 1 ( )B I Y defined as

1 1 1 1( ) ( ) (0.6,0.3), ( ) ( ) (0.5,0.3)A x B x A y B y , 2 2( ) (0.8,0.1), ( ) (0.5,0.1)A x A y

Define : ( )T I X I I and : ( )W I Y I I by

~ ~

1 11 2 1 22 2

(1,0) if 0 ,1 ,

( ) ( ( ), ( )) ( , ) if ,

(0,1) otherwise

A

T A T A T A A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

B

B B

For 1/ 2, 1/ 2r s . Then the identity mapping

1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) -continuous which is not fuzzy (1/ 2,1/ 2) -regular continuous. Since 1B

is fuzzy (1/ 2,1/ 2) -open in W but 1

1( )f B is not fuzzy (1/ 2,1/ 2) -regular open in T .

Example 3.2 Let { , }X a b Y , and let 1 2, ( )A A I X , 1 ( )B I Y defined as

1 1( ) (0.6,0.3), ( ) (0.5,0.3)A x A y , 2 1 2 1( ) ( ) (0.8,0.1), ( ) ( ) (0.5,0.1)A x B x A y B y


~ ~

1 11 2 1 22 2

(1,0) if 0 ,1 ,

( ) ( ( ), ( )) ( , ) if ,

(0,1) otherwise

A

T A T A T A A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

B

B B


1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) -semi continuous which is not fuzzy (1/ 2,1/ 2) -regular semi continuous.

Since 1B is fuzzy (1/ 2,1/ 2) - open in W but 1

1( )f B is not fuzzy (1/ 2,1/ 2) -regular semi open in T .

Example 3.3 Let { , }X a b Y , and let 1 ( )AinI X , 1 2, ( )B B I Y defined as

1 1 1 1( ) ( ) (0.3,0.6), ( ) ( ) (0.3,0.5)A x B x A y B y , 2 2( ) (0.4,0.3), ( ) (0.4,0.3)B x B y





~ ~

1 11 2 12 2

(1,0) if 0 ,1 ,

( ) ( ( ), ( )) ( , ) if

(0,1) otherwise

A

T A T A T A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 11 22 2

(1,0) if 0 ,1 ,

( , ) if ,

(0,1) otherwise

B

B B B


1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) - semi continuous, regular semi continuous which is not fuzzy

(1/ 2,1/ 2) -regular continuous and -continuous. Since 2B is fuzzy (1/ 2,1/ 2) - open in W but 1

2( )f B is not fuzzy

(1/ 2,1/ 2) -regular open and -open in T .

Example3.4 Let { , }X a b Y , and let 1 ( )A I X , 1 2 1 2 1 2, , , ( )B B B B B B I Y defined as

1 1 1 1( ) ( ) (0.3,0.1), ( ) ( ) (0.3,0.1)A x B x A y B y , 2 2( ) (0.4,0.3), ( ) (0.4,0.3)B x B y

1 2 1 2 1 2 1 2( ) ( ) (0.4,0.1), ( ) ( ) (0.4,0.1), ( ) ( ) (0.3,0.3), ( ) ( ) (0.3,0.3)B x B x B y B y B x B x B y B y Define

: ( )T I X I I and : ( )W I Y I I by

~ ~ ~ ~

1 1 1 11 2 1 1 2 1 2 1 2 1 22 2 2 2

(1,0) if 0 ,1 , (1,0) if 0 ,1 ,

( ) ( ( ), ( )) ( , ) if ( ) ( ( ), ( )) ( , ) if , , ,

(0,1) otherwise (0,1) otherwise

A B

T A T A T A A A W B W B W B B B B B B B B

For 1/ 2, 1/ 2r s . Then the identity mapping 1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) - -continuous

which is not fuzzy (1/ 2,1/ 2) - continuous. Since 2B is fuzzy 1 1

( , )2 2

-open in W but 1

2( )f B is not fuzzy (1/ 2,1/ 2) -

open in T .

Example 3.5 Let { , }X a b Y , and let 1 ( )A I X , 1 2 1 2 1 2, , , ( )B B B B B B I Y defined as

1 1 1 1( ) ( ) (0.3,0.6), ( ) ( ) (0.3,0.5)A x B x A y B y , 2 2( ) (0.1,0.2), ( ) (0.1,0.2)B x B y



~ ~

1 11 2 12 2

(1,0) if 0 ,1 ,

( ) ( ( ), ( )) ( , ) if

(0,1) otherwise

A

T A T A T A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 11 2 1 2 1 22 2

(1,0) if 0 ,1 ,

( , ) if , , ,

(0,1) otherwise

B

B B B B B B B

For 1/ 2, 1/ 2r s . Then the identity

mapping 1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) - continuous which is not fuzzy (1/ 2,1/ 2) -semi

continuous. Since 2B is fuzzy (1/ 2,1/ 2) - open in W but 1

2( )f B is not fuzzy (1/ 2,1/ 2) -semi open in T .

Example3.6 Let { , }X a b Y , and let 1 ( )A I X , 1 2 1 2 1 2, , , ( )B B B B B B I Y defined as

1 1 1 1( ) ( ) (0.3,0.6), ( ) ( ) (0.3,0.5)A x B x A y B y , 2 2( ) (0.1,0.3), ( ) (0.1,0.3)B x B y ,



~ ~

1 11 2 12 2

(1,0) if 0 ,1 ,

( ) ( ( ), ( )) ( , ) if

(0,1) otherwise

A

T A T A T A A A




1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 11 2 1 2 1 22 2

(1,0) if 0 ,1 ,

( , ) if , , ,

(0,1) otherwise

B

B B B B B B B


mapping 1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) - continuous which is not fuzzy (1/ 2,1/ 2) -pre continuous.

Since 2B is fuzzy (1/ 2,1/ 2) - open in W but 1

2( )f B is not fuzzy (1/ 2,1/ 2) -pre open in T .

Example 3.7 Let { , }X a b Y , and let 1 ( )A I X , 1 2 1 2 1 2, , , ( )B B B B B B I Y defined as

1 1 1 1 1 2 1 2( ) ( ) (0.1,0.3), ( ) ( ) (0.1,0.3), ( ) ( ) (0.1,0.6), ( ) ( ) (0.1,0.5)A x B x A y B y B x B x B y B y

2 2( ) (0.3,0.6), ( ) (0.3,0.5)B x B y 1 2 1 2( ) ( ) (0.3,0.3), ( ) ( ) (0.3,0.3)B x B x B y B y


~ ~

1 11 2 12 2

(1,0) if 0 ,1 ,

( ) ( ( ), ( )) ( , ) if

(0,1) otherwise

A

T A T A T A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 11 2 1 2 1 22 2

(1,0) if 0 ,1 ,

( , ) if , , ,

(0,1) otherwise

B

B B B B B B B


mapping 1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) -pre continuous which is not fuzzy (1/ 2,1/ 2) - continuous.

Since 2B is fuzzy (1/ 2,1/ 2) -open in W but 1

2( )f B is not fuzzy (1/ 2,1/ 2) - open in T .

It need not be true that f and g are fuzzy ( , )r s -regular semi (resp. regular, , pre and ) continuous mappings then so

is g f . But we have the following theorem.

Theorem 3.1 Let 1 2 1 2 1 2( , , ),( , , ) and ( , , )X T T Y W W Z V V be SoIFTSs and let :f X Y and :g Y Z be mappings

and ( , )r s I I . If f is a fuzzy ( , )r s -regular semi (resp. regular, , pre and ) continuous mapping and g is a fuzzy

( , )r s -continuous mapping, then g f is a fuzzy ( , )r s -regular semi (resp. regular, , pre and ) continuous mapping.

Proof. Straighforward

Theorem 3.2 Let 1 2( , , )X T T and 1 2( , , )Y W W be SoIFTSs. Let :f X Y be a mapping. The following statements are

equivalent:

[(i)] f is ( , )r s -fuzzy regular semi continuous.[(ii)] 1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf rscl A r s rcl f A r s , for each

XA I .[(iii)]

1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wrscl f B r s f rcl B r s , for each YB I .[(iv)]

1 2 1 2

1 1

, ,( ( , , )) ( ( ), , )W W T Tf rint B r s rsint f B r s ,

for each YB I .

Proof. (i) (ii) Let (i) holds and let XA I . Suppose that there exists

XA I and ( , )r s I I such that

1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf rscl A r s rcl f A r sÚ . So there exists , (0,1]y Y t such that

1 2 1 2, ,( ( , , ))( ) ( ( ), , )( )T T W Wf rscl A r s y t rcl f A r s y . If 1( )f y , then,

1 2, ~( ( , , ))( ) 0T Tf rscl A r s y . So, there exists

1( )x f y such that: 1 2 1 2 1 2, , ,( ( , , ))( ) ( , , )( ) ( ( ), , )( ) (3.1)T T T T W Wf rscl A r s y rscl A r s x t RC f A r s y

Since 1 2, ( ( ), , )( )W Wrcl f A r s y t , there exists ( , )r s -frc set B such that ( )f A B . Then,

1 2, ( ( ), , )( )W Wrcl f A r s y B and this implies 1 2, ( ( ), , )( ( )) ( ( ))W Wrcl f A r s f x B f x t . Moreover, ( )f A B implies

1( )A f B . So, 1

~ ~1 1 ( )A f B . Since 1

~(1 )f B is ( , )r s -frso, then from (i),

1 2

1

~ , ~(1 ) (1 , , )T Tf B rsint A r s or 1 2

1

~ , ~ ~ ~1 (1 , , ) 1 (1 )T Trsint A r s f B .

Then, 1 2

1

, ( , , ) ( )T Trscl A r s f B . This gives 1 2

1

, ( , , )( ) ( )( ) ( ( ))T Trscl A r s x f B x B f x t .

But the last relation contradicts the relation (3.1) above. Hence, the result in (ii) is true.

(ii) (iii) Take 1( )( )YA f B B I and apply (ii) we have the required result.

(iii) (iv) Taking the complement of (iii), we have the result.(iv) (i) Let (iv) holds and let YB I be a ( , )r s -fro set,

since every ( , )r s -fro set is ( , )r s -fuzzy open. So 1 2( ) , and ( )W B r W B s . By (iv)




1 2 1 2

1 1

, ,( ( , , )) ( ( ), , )W W T Tf rint B r s rsint f B r s or 1 2

1 1

,( ) ( ( ), , )T Tf B rsint f B r s (since 1( )W B r , 2( )W B s ).

But 1 2

1 1

, ( ( ), , ) ( )T Trsint f B r s f B . Then, 1 2

1 1

,( ) ( ( ), , )T Tf B rsint f B r s and this means that 1( )f B

is ( , )r s -

frso. Hence, f is ( , )r s fuzzy regular semi continuous.


equivalent: [(i)] f is fuzzy ( , )r s - continuous.[(ii)] 1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf cl A r s cl f A r s , for each

XA I .[(iii)]

1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wcl f B r s f cl B r s , for each YB I .[(iv)]

1 2 1 2

1 1

, ,( ( , , )) ( ( ), , )W W T Tf int B r s int f B r s ,

for each YB I .


equivalent: [(i)] f is fuzzy ( , )r s -pre continuous.[(ii)] 1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf pcl A r s cl f A r s , for each

XA I .[(iii)]

1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wpcl f B r s f cl B r s , for each YB I .[(iv)]

1 2 1 2

1 1

, ,( ( , , )) ( ( ), , )W W T Tf int B r s pint f B r s ,

for each YB I .


equivalent: [(i)] f is fuzzy ( , )r s - continuous.

[(ii)] 1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf cl A r s cl f A r s , for each

XA I .[(iii)] 1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wcl f B r s f cl B r s

, for each YB I .[(iv)]

1 2 1 2

1 1

, ,( ( , , )) ( ( ), , )W W T Tf int B r s int f B r s , for each YB I .

Proof.Proof of Theorem 3.3,3.4,3.5 it follows from Theorem 3.2

4. Fuzzy ( , )r s -regular semi(resp. , pre, ) open mappings

Definition 4.1 Let 1 2 1 2: ( , , ) ( , , )f X T T Y W W be a mapping from a SoIFTS X to another SoIFTS Y and

( , )r s I I . Then f is said to be

[(i)] fuzzy ( , )r s -regular semi (resp. regular, , pre and ) open if ( )f A is a fuzzy ( , )r s -regular semi (resp. regular,

, pre and ) open set of Y for each fuzzy ( , )r s -open set A of X

[(ii)] fuzzy ( , )r s -regular semi (resp. regular, , pre and ) closed if ( )f A is a fuzzy ( , )r s -regular semi (resp. regular,

, pre and ) closed set of Y for each fuzzy ( , )r s -closed set A of X .

Fuzzy (r,s)-regular open map

Fuzzy (r,s)-open map Fuzzy (r,s)- regular semi open map

Fuzzy (r,s)- open map Fuzzy (r,s)- semi open map

Fuzzy (r,s)- pre open map Fuzzy (r,s)- open map

From the above definitions it is clear that the above implications are true for ( , )r s I I

The converses of the above implications are not true as the following examples show:

Example 4.1 Let { , }X a b Y . Define 1 ( )A I X 1 2, ( )B B I Y as follows

1 1 1 1 2 2( ) ( ) (0.6,0.3), ( ) ( ) (0.5,0.3), ( ) (0.8,0.1), ( ) (0.5,0.1)A x B x A y B y B x B y

Define : ( )T I X I I and : ( )W I Y I I by 1 2( ) ( ( ), ( ))T A T A T A

~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

A

A A

1 2( ) ( ( ), ( ))W B W A W A

~ ~

1 11 22 2

(1,0) if 0 ,1 ,

( , ) if ,

(0,1) otherwise

B

B B B


1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) -open mapping which is not fuzzy (1/ 2,1/ 2) -regular open mapping.

Since 1A is fuzzy (1/ 2,1/ 2) - open in T but 1( )f A is not fuzzy (1/ 2,1/ 2) -regular open in W .

Example 4.2 Let { , }X a b Y . Define 1 ( )A I X , 1 2, ( )B B I X as follows




1 1 2 1 2 1( ) (0.6,0.3), ( ) (0.5,0.3), ( ) ( ) (0.8,0.1), ( ) ( ) (0.5,0.1)B x B y B x A x B y A y


~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

A

A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 11 22 2

(1,0) if 0 ,1 ,

( , ) if ,

(0,1) otherwise

B

B B B


1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) - semi open mapping which is not fuzzy (1/ 2,1/ 2) -regular semi open

mapping. Since 1A is fuzzy (1/ 2,1/ 2) - open in T but 1( )f A is not fuzzy (1/ 2,1/ 2) -regular semi open in W .

Example 4.3 Let { , }X a b Y . Define 1 2, ( )A A I X , 1 ( )B I X , as follows

1 1 1 1 2 2( ) ( ) (0.3,0.6), ( ) ( ) (0.3,0.5), ( ) (0.4,0.3), ( ) (0.4,0.3)A x B x A y B y A x A y


~ ~

1 11 22 2

(1,0) if 0 ,1 ,

( , ) if ,

(0,1) otherwise

A

A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

B

B B

For 1/ 2, 1/ 2r s . Then the identity mapping 1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) - semi open

mapping, regular semi open mapping which is not fuzzy (1/ 2,1/ 2) -regular open mapping. Since 2A is fuzzy (1/ 2,1/ 2) -

open in T but 2( )f A is not fuzzy (1/ 2,1/ 2) -regular open and open in W .

Example 4.4 Let { , }X a b Y . Define 1 2 1 2 1 2, , , ( )A A A A A A I X , 1 ( )B I Y as follows

1 1 1 1 1 2 1 2( ) ( ) (0.3,0.1), ( ) ( ) (0.3,0.1), ( ) ( ) (0.3,0.3), ( ) ( ) (0.3,0.3)A x B x A y B y A x A x A y A y

2 2( ) (0.4,0.3), ( ) (0.4,0.3)A x A y 1 2 1 2( ) ( ) (0.4,0.1), ( ) ( ) (0.4,0.1)A x A x A y A y


~ ~

1 11 2 1 2 1 22 2

(1,0) if 0 ,1 ,

( , ) if , , ,

(0,1) otherwise

A

A A A A A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

B

B B

For 1/ 2, 1/ 2r s . Then the identity mapping 1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) - open mapping

which is not fuzzy (1/ 2,1/ 2) -open mapping. Since 2A is fuzzy (1/ 2,1/ 2) - open in T but 2( )f A is not fuzzy

(1/ 2,1/ 2) -open in W .



2 2( ) (0.1,0.2), ( ) (0.1,0.2)A x A y 1 2 1 2( ) ( ) (0.3,0.2), ( ) ( ) (0.3,0.2)A x A x A y A y


~ ~

1 11 2 1 2 1 22 2

(1,0) if 0 ,1 ,

( , ) if , , ,

(0,1) otherwise

A

A A A A A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

B

B B


which is not fuzzy (1/ 2,1/ 2) -semi open mapping. Since 2A is fuzzy (1/ 2,1/ 2) - open in T but 2( )f A is not fuzzy

(1/ 2,1/ 2) -semi open in W .






2 2( ) (0.1,0.3), ( ) (0.1,0.3)A x A y 1 2 1 2( ) ( ) (0.3,0.3), ( ) ( ) (0.3,0.3)A x A x A y A y

Define : ( )T I X I I and : ( )W I Y I I

by 1 2( ) ( ( ), ( ))T A T A T A

~ ~

1 11 2 1 2 1 22 2

(1,0) if 0 ,1 ,

( , ) if , , ,

(0,1) otherwise

A

A A A A A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

B

B B


which is not fuzzy (1/ 2,1/ 2) -pre open mapping. Since 2A is fuzzy (1/ 2,1/ 2) - open in T but 2( )f A is not fuzzy

(1/ 2,1/ 2) -pre open in W .



2 2( ) (0.3,0.6), ( ) (0.3,0.5)A x A y 1 2 1 2( ) ( ) (0.3,0.3), ( ) ( ) (0.3,0.3)A x A x A y A y

Define : ( )T I X I I and : ( )W I Y I I

by 1 2( ) ( ( ), ( ))T A T A T A

~ ~

1 11 2 1 2 1 22 2

(1,0) if 0 ,1 ,

( , ) if , , ,

(0,1) otherwise

A

A A A A A A A

1 2( ) ( ( ), ( ))W B W B W B

~ ~

1 112 2

(1,0) if 0 ,1 ,

( , ) if

(0,1) otherwise

B

B B

For 1/ 2, 1/ 2r s . Then the identity mapping 1 2 1 2: ( , , ) ( , , )f X T T Y W W is fuzzy (1/ 2,1/ 2) - pre open

mapping which is not fuzzy (1/ 2,1/ 2) - open mapping. Since 2A is fuzzy (1/ 2,1/ 2) - open in T but 2( )f A is not

fuzzy (1/ 2,1/ 2) - open in W .

It need not be true that f and g are fuzzy ( , )r s -regular semi (resp. regular, , pre and ) open (resp. ( , )r s -regular

semi (resp. regular, , pre and ) closed) mappings then so is g f . But we have the following theorem.

Theorem 4.1 Let 1 2 1 2 1 2( , , ),( , , ) and ( , , )X T T Y W W Z V V be SoIFTSs and let :f X Y and :g Y Z be mappings

and ( , )r s I I . Then the following statements are true:

[(i)] If f is a fuzzy ( , )r s -open mapping and g is a fuzzy ( , )r s -regular semi (resp. regular, , pre and ) open

mapping, then g f is a fuzzy ( , )r s -regular semi (resp. regular, , pre and ) open mapping.[(ii)] If f is a fuzzy ( , )r s -

closed mapping and g is a fuzzy ( , )r s -regular semi (resp. regular, , pre and ) closed mapping, then g f is a fuzzy

( , )r s -regular semi (resp. regular, , pre and ) closed mapping.

Proof:. Straighforward

Theorem 4.2 Let 1 2( , , )X T T and 1 2( , , )Y W W be smooth topological space's. Let :f X Y be a mapping. The

following statements are equivalent: [(i)] f is fuzzy ( , )r s -regular semi open. [(ii)]

1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf int A r s rsint f A r s , for each XA I .[(iii)]

1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wint f B r s f rsint B r s , for

each YB I .[(iv)] For any

YA I and any XB I such that 1 ~(1 )T B r ,

1

2 ~(1 ) , ( )T B s B f A , there exists

( , )r s -frsc set YC I with A C and

1( )B f C .




Proof. (i) (ii) Since 1 2, ( , , ) , ( )X

T Tint A r s A A I , then 1 2,( ( , , )) ( )T Tf int A r s f A . But

1 2,( ( , , ))T Tf int A r s is

( , )r s -frso (by (i)), then 1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf I A r s rsintI f A r s .

(ii) (iii) Let (ii) holds. Put 1( ), YA f B B I and apply (ii), we have

1 2 1 2 1 2

1 1

, , ,( ( ( ), , )) ( ( ), , ) ( , , )T T W W W Wf int f B r s rsint ff B r s rsint B r s . Hence,

1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wint f B r s f rsint B r s (iii) (iv) Let (iii) holds and let YA I and

XB I such that

1(1 )T B r and 2(1 )T B s and

1( )B f A . Since 1 1

~ ~ ~1 1 ( ) (1 )B f A f A , it implies

1 2 1 2

1

, ~ ~ , ~(1 ) 1 ( (1 ), , )T T T TI B B I f A r s or 1 2

1

~ , ~1 ( (1 , , ))W WB f rsint A r s . Then,

1 2 1 2

1 1

~ , ~ ,1 ( (1 , , )) ( ( , , ))W W W WB f rsint A r s f rscl A r s . Take 1 2, ( , , )W WC rscl A r s . Hence C is ( , )r s -frsc and

C A such that 1( )B f C .(iv) (i) Let D be a fuzzy set such that

XD I and 1( )T D r and 2( )T D s . Put

~1B D and ~1 ( )A f D . Then 1( )B f A . So there exists ( , )r s -frsc set C such that C A and

1( )B f C . Then 1

~1 ( )D f C implies 1

~(1 )D f C . Thus ~( ) 1f w C . Also, A C ,

~ ~( ) 1 1f D A C . Hence ~( ) 1f D C and therefore ( )f D is ( , )r s -frso. So f is fuzzy ( , )r s -regular semi

open.


following statements are equivalent:[(i)] f is fuzzy ( , )r s - open.

[(ii)] 1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf int A r s int f A r s , for each

XA I .[(iii)]

1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wint f B r s f int B r s , for each YB I .[(iv)] For any

YA I and any XB I such that

1 ~(1 )T B r , 1

2 ~(1 ) , ( )T B s B f A , there exists fuzzy ( , )r s - open set YC I with A C and

1( )B f C .


following statements are equivalent: [(i)] f is fuzzy ( , )r s -pre open.

[(ii)] 1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf int A r s pint f A r s , for each

XA I .[(iii)]

1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wint f B r s f pint B r s , for each YB I .[(iv)] For any

YA I and any XB I such that

1 ~(1 )T B r , 1

2 ~(1 ) , ( )T B s B f A , there exists fuzzy ( , )r s -pre open set YC I with A C and

1( )B f C .


following statements are equivalent:

[(i)] f is fuzzy ( , )r s - open. [(ii)] 1 2 1 2, ,( ( , , )) ( ( ), , )T T W Wf int A r s int f A r s , for each

XA I .

[(iii)] 1 2 1 2

1 1

, ,( ( ), , ) ( ( , , ))T T W Wint f B r s f int B r s , for each YB I .[(iv)] For any

YA I and any XB I such

that 1 ~(1 )T B r , 1

2 ~(1 ) , ( )T B s B f A , there exists fuzzy ( , )r s - open set YC I with A C and

1( )B f C .

Proof. Proof of theorem 4.3,4.4,4.5 it follows from Theorem 4.2

Theorem 4.6 Let 1 2( , , )X T T and 1 2( , , )Y W W be SoIFTS`s. Let :f X Y be a mapping. The following statements are

equivalent: [(i)] f is fuzzy ( , )r s -regular semi closed.

[(ii)] 1 2 1 2, ,( ( ), , )) ( ( , , ))W W T Trscl f A r s f rcl A r s , for each

XA I .

Proof. (i) (ii) Let (i) holds and let XA I . Since

1 2, ( , , )T TA rcl A r s , then 1 2,( ) ( ( , , ))T Tf A f rcl A r s and

therefore, 1 2 1 2, ,( ( ), , ) ( ( , , ))W W T Trscl f A r s f RC A r s .

(ii) (i) Let (ii) holds and let XA I be a ( , )r s -frc set. Then

1 2 1 2, ,( ( ), , ) ( ( , , ))W W T Trscl f A r s f rcl A r s . It implies

1 2, ( ( ), , ) ( )W Wrscl f A r s f A . But 1 2, ( ( ), , ) ( )W Wrscl f A r s f A . Then,

1 2,( ) ( ( ), , )W Wf A rscl f A r s and it follows

that ( )f A is ( , )r s -frsc. Hence, f is fuzzy ( , )r s -regular semi closed.





equivalent: [(i)] f is fuzzy ( , )r s - closed. [(ii)] 1 2 1 2, ,( ( ), , )) ( ( , , ))W W T Tcl f A r s f cl A r s , for each

XA I .


equivalent: [(i)] f is fuzzy ( , )r s -pre closed. [(ii)] 1 2 1 2, ,( ( ), , )) ( ( , , ))W W T Tpcl f A r s f cl A r s , for each

XA I .


equivalent: [(i)] f is fuzzy ( , )r s - closed. [(ii)] 1 2 1 2, ,( ( ), , )) ( ( , , ))W W T Tcl f A r s f cl A r s , for each

XA I .

Proof.Proof of theorem 4.7,4.8,4.9 it follows from Theorem 4.6

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