REGULAR SEMI CONTINUOUS AND OPEN MAPPINGS ON INTUITIONISTIC FUZZY … · 2018-09-30 ·...
Transcript of REGULAR SEMI CONTINUOUS AND OPEN MAPPINGS ON INTUITIONISTIC FUZZY … · 2018-09-30 ·...
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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
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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
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) -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
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Define : ( )T I X I I and : ( )W I Y I I by
~ ~
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
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 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 2 1 2 1 2 1 2( ) ( ) (0.3,0.2), ( ) ( ) (0.3,0.2), ( ) ( ) (0.1,0.6), ( ) ( ) (0.1,0.5)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 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 2 1 2 1 2 1 2( ) ( ) (0.3,0.3), ( ) ( ) (0.3,0.3), ( ) ( ) (0.1,0.6), ( ) ( ) (0.1,0.5)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 11 2 12 2
(1,0) if 0 ,1 ,
( ) ( ( ), ( )) ( , ) if
(0,1) otherwise
A
T A T A T A A A
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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) -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
Define : ( )T I X I I and : ( )W I Y I I by
~ ~
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) -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)
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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.
Theorem 3.3 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 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 .
Theorem 3.4 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 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 .
Theorem 3.5 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 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
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) -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
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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
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 B W B
~ ~
1 11 22 2
(1,0) if 0 ,1 ,
( , ) if ,
(0,1) otherwise
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) - 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
Define : ( )T I X I I and : ( )W I Y I I by 1 2( ) ( ( ), ( ))T A T A T A
~ ~
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
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) - 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 .
Example 4.5 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( ) ( ) (0.3,0.6), ( ) ( ) (0.3,0.5), ( ) ( ) (0.1,0.6), ( ) ( ) (0.1,0.5)A x B x A y B y A x A x A y A y
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
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) - open mapping
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 .
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Example 4.6 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( ) ( ) (0.3,0.6), ( ) ( ) (0.3,0.5), ( ) ( ) (0.1,0.6), ( ) ( ) (0.1,0.5)A x B x A y B y A x A x A y A y
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
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) -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 .
Example 4.7 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( ) ( ) (0.1,0.3), ( ) ( ) (0.1,0.3), ( ) ( ) (0.1,0.6), ( ) ( ) (0.1,0.5)A x B x A y B y A x A x A y A y
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 .
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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.
Theorem 4.3 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 - 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 .
Theorem 4.4 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 -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 .
Theorem 4.5 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 - 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.
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Theorem 4.7 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 - closed. [(ii)] 1 2 1 2, ,( ( ), , )) ( ( , , ))W W T Tcl f A r s f cl A r s , for each
XA I .
Theorem 4.8 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 -pre closed. [(ii)] 1 2 1 2, ,( ( ), , )) ( ( , , ))W W T Tpcl f A r s f cl A r s , for each
XA I .
Theorem 4.9 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 - 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
References
[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.
[2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190.
[3] K. C. Chattopadhyay, R. N. Harza and S. K. Samanta, Gradation of openness: fuzzy topology, Fuzzy Sets and Systems 49
(1992), 237-242.
[4] D. Coker and A. Haydar Es, On fuzzy compactness in intuitionistic fuzzy topological spaces, J. Fuzzy Math 3 (1995), 899-
909.
[5] D. Coker, M. Demirci An introduction to intuitionistic fuzzy topologica spaces in Sostak's sense,BUSEFAL, 67 (1996), 67-76.
[6] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems 88 (1997), 81-89.
[7] H. Gurcay, D. Coker and A. Haydar Es, On fuzzy continuity in intuitionistic fuzzy topological spaces, J. Fuzzy Math 5 (1997),
365-378.
[8] E.P. Lee and Y.B. Im, Mated fuzzy topological spaces, J. of Fuzzy Logic and Intelligent Systems 11 (2001), 161-165.
[9] E.P. Lee, Semiopen sets on intuitionistic fuzzy topological spaces in S ostak's sense, Journal of Korean institute of Intelligent
Systems 11 (2) (2004), 234-238.
[10] A. A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems, 48 (1992), 371-375.
[11]G. Saravanakumar, S. Tamilselvan and A. Vadivel, Regular Semi open sets on intuitionistic fuzzy topological spaces in Sostak`s sense. (submitted)
[12]A. S ostak, On a fuzzy topological structure, Supp. Rend. Circ. Math. palermo (Ser.II), 11 (1985), 89-103.
[13]L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.