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Sci.Int.(Lahore),27(5),3899-3904,2015 ISSN 1013-5316; CODEN: SINTE 8 3899
Sept-Oct.
INTUITIONISTIC FUZZY TOPOLOGY: FUZZY -STRONG SEMI CONTINUITY
AND FUZZY -STRONG SEMI RETRACTS
Mohammed M. Khalaf
Department of Mathematics, Faculty of Science, Al-Azahar University, Assuit, Egypt.
Current Address : Al-zulfi College of Science, Majmaah University, KSA
Khalfmohammed2003@yahoo.com
ABSTRACT The concept of a fuzzy retract was introduced by Rodabaugh in 1981 and The concept of a intuitionistic fuzzy
topology (IFT) was introduced by Coker 1997. The aim of this paper is to introduce a new concepts of fuzzy of Intuitionistic
fuzzy -strongly semi open set of a nonempty set and define an Intuitionistic fuzzy -strong semi continuity and Intuitionistic
fuzzy -strongly semi retract. Also we prove that the product and the graph of two Intuitionistic fuzzy -strong semi continuity
are Intuitionistic fuzzy -strong semi continuity . The concept of Intuitionistic fuzzy -strongly semi retract are introduced , the
relations between these new concepts are discussed.
Keywords -strongly semiopen, - strongly semi continuous, -retract and - neighbourhood retract, – strongly
semiretract
1-INTRODUCTION
The notions of Intuitionistic fuzzy retracts are introduced by
Hanafy and khalaf [6]. In [4,5] weaker forms of
Intuitionistic fuzzy continuity between of Intuitionistic fuzzy
topological space are introduced. In this work we introduced
and explain in section 2 a new notions of Intuitionistic fuzzy
open sets -open sets ) are studied. in section 3 many
results of - strongly semi continuous are obtained, finely
in section 4 we define -retract and - neighborhood
retract Finley in section 5 - strongly semi retract as
applications of - strongly semi continuous. The relations
between all these concepts are discussed.
Definition 1.1 [1] Let X be a nonempty set. An IF-set A is an
object of the form . where the
functions and denote
respectively, the degree of membership function (namely
and the degree of non-membership function
(namely of , + , for each
). An IF-set A = can be
written in the form A=
Definition 1.2 [1] Let
be IF -set on X and a
function Then,
for each
[7]
(v) [7]
Definition 1.3 [6] Let A be an IF-set of an IF-ts Then
A is called :
(i) An IF-regular open ( IF-ro, for short ) set if
(ii) An IF-semi open ( IF-so, for short ) set if
(iii) An IF-preopen ( IF-po, for short ) set if
(iv) An IF-strongly semi open (IF-so, for short )set if
(v) An IF-semi-preopen (IF-spo, for short ) set if
Their complements are called IF-semi closed, IF-pre
closed, IF-strongly semi closed and IF-semi-pre closed sets
Definition 1.3 [1] Let X and Y be two nonempty sets
and be a function
is an IFS in Y, then
the pre image of B under (B)) is defined
by
If A= is an IFS in X , then
the Image of A under f( denoted f(A)) is defined by
Definition 1.4 [6] Let be a IF –ts, and Then,
the F-subspace is called a IF – retract ( for short,
IFR) of if there exists a IF-continuous mapping
such that for all . In
this case is called an IF-retraction
Definition 1.5 [6] Let be a IF –ts .Then is
said to be IF-Neighborhood retract (IF – nbd R) of
if is a IFR of , Such that
.
Definition 1.6 [6] Let be a IF –ts , and ,
then the IF - subspace is called a IF - semi retract
(for short, IFSR ) ( resp. IF – pre retract , IF – strongly
semi retract and IF –semi pre retract ) ( IFPR , IFSSR,
IFSPR) of if there exists a IF-semicontinuous
(resp. IF-pre continuous, IF –strongly Semi continuous, IF-
semi pre continuous ) mapping such
that In this case, is called an IF-
semi retraction (resp., -IF pre retraction, IF-strongly semi
retraction, IF- semi pre retraction)
Definition 1.7 [6] Let be an IF –ts .Then
is said to be an IF- neighborhood semi retract , ( for short,
IF-nbd SR ) ( resp. IF-nbd pre retract , IF-nbd strongly
semi retract, IF-nbd semi pre retract.) ( for short, IF-nbd
PR, IF-nbd SSR, IF-nbd SPR) of . is
IFSR ( resp. IFPR,IFSSR,IFSPR. ) of
, such that
2. -semiopen, -preopen , -strongly semiopen
and - semi preopen sets
Definition 2.1 Let be a IF-ts ,
. Then is called
(i) a -semi open ( briefly , ) set if there
exist . such that
.
(ii) a -preopen ( briefly , ) set if
.
(iii) a -regular open ( briefly , ) set if
.
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(iv ) a - strongly semi open ( briefly , ) set
if there exists such that . ( or
,
(v) a -semi preopen ( briefly , ) set if there
exists a - preopen set such that
Their complements are called -semi closed (briefly,
) ,-pre closed (briefly , ) , -regular closed
(briefly, , -strongly semi closed (briefly ,
) , -semipre closed ( briefly, set .
, , , and (resp.
will always
denote the family of -semi open, -preopen, -
regular open , - strongly semi open, - semi preopen
(resp. - semi closed, - preclosed, - regular closed
, - strongly semiclosed, - semi preclosed) sets
Remark 2.1 The implications between these different
notions of - sets are given by the following diagram.
ro ( rc )
so (sc )
o ( c ) sso ( ssc )
spo (spc)
po (pc )
But the converse need not to be true, in general as
shown by the following examples
Example 2.1 Let IF-ts where ,
.Then is an - semi preopen set but not
-preopen set, is an - semi open set ut not
-strongly semi open set, is an - semi
preopen set but not - semi open set,
Example 2.2 Let and ,
are defined by,
is an - preopen set but not - strongly semi open
set .
Example 2.3 Let and,
are defined by ,
is an - strongly semi open set but not - open set . ,
- open set but not -regular open set.
Theorem 2.1
(a) The -closure of a - preopen set is a -regular
closed set ,
(b) The -interior of a preclosed set is a -
regular open set
Proof . It is obvious
Theorem 2.2
(i) The intersection of two - regular open sets is -
regular open ,
(ii) The union of two - regular closed sets is a -
regular closed
Proof . It is obvious.
Proposition 2.1
(i) The intersection of any -semi closed sets is also
-semi closed .
(ii) Any union of any -semi open sets is also -semi
open
Proof . It is obvious
Theorem 2.3
(i) Arbitrary union of -strongly semi open sets is -
strongly semi open
(ii) Arbitrary intersection of -strongly semi closed sets
is -strongly semi closed (iii) Arbitrary union
(intersection) of - semi preopen ( - semi preclosed )
sets is - semi preopen ( - preclosed )
Proof . It is obvious
Remark 2.2 Let be a -closed set and
be a - closed set . Then need not be
a - closed set.
Example 2.4 Let and be IF – sets on ,
defined by
and .Then is a - closed set . Let
and be IF – sets on , defined by ,
and . Then is a - closed set. But -
is not a – closed set .
Remark 2.2 An - semiopen set and a - preopen set
are independent concepts.
Example 2.5 Let and be IF- sets on
, defined by
and . Then is a - preopen set, but
not - semiopen set .
Sci.Int.(Lahore),27(5),3899-3904,2015 ISSN 1013-5316; CODEN: SINTE 8 3901
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Example 2.6 Let and be IF- sets on ,
defined by
and . Then is a - semiopen set, but
not - preopen set .
Theorem 2.4. Let be a F-ts, .
Then the following are equivalent .
(i) is a - semi closed set ,
(ii) is a - semiopen
(iii)
(iv)
Proof . It is obvious
3-semi continuous, - precontinuous , -
strongly semi continuous and - semi
precontinuous mappings
Definition 3.1 Let be a mapping
from a IF-ts to another IF-ts .
Then , is called :
(i) a -semicontinuous ( briefly , sc ) mapping if for
each , we have .
(ii) a - precontinuous ( briefly , pc ) mapping if
for each , we have .
(iii) a - strongly semi continuous ( briefly , ssc )
mapping if for each
, we have .
(iv) a - semi precontinuous ( briefly , pc )
mapping if for each , we have
Remark 3.1 The implications between these different
concepts are given by the following diagram
pc
c ssc
sp
sc
Example 3.1 Let and
be two IF-ts's where,
and
. Then is an
–continuous but not IF- continuous mapping .
Example 3.2 Let , and
be two IF-ts's where,
and
,
. Then is an –
semi precontinuous but not - precontinuous mapping ,
also , – semi continuous but not –strongly semi
continuous mapping .
Example 3.3 Let , and
be two IF-ts's where,
and
. Then is an
–strongly semi continuous but not –continuous
mapping, also, –semi precontinuous but not –
semicontinuous mapping .
Example 3.4 Let , and
be two IF-ts's where,
and
Then,
is an
- precontinuous but not - strongly semi continuous
mapping .
Theorem 3.1 Let be an -
strongly semi continuous mapping . The following
statements are equivalent:
(i) The inverse image of each -closed set is -
strongly semi closed
(ii) . for each
-open set of
(iii) . for each
-open set of
Proof Since is - closed set of ,
is - strongly semi closed and hence
)
Let –open set of . and .
By (ii)
) . Thus
Let be an – closed set of , and
. By (iii)
. Hence, , i .e. ,
and -
strongly semiclosed.
Definition 3.2 .Let and be IF-ts's,
Then we define as
follows
Lemma 2.1 Let and . be IF-ts's,
Then,
Proof Follows directly from definition (3.2 )
Theorem 3.2 ( Coker 1996) Let and
be IF-ts's,
and be mappings . If is -
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continuous and is
- continuous , then is - continuous
Theorem 3.3 Let and
be IF-ts's,
and . Then , is -
continuous and is -continuous iff the
product . is
continuous .
Proof Let , i .e . ,
where
' s and 's are IF- open sets of and
, respectively , we want to show that
. Since
is – continuous ,
, then
. Also , since
, is IF- continuous , ,
then
. By using lemma (2.1 ). We get
)
, hence,
Conversely , Let , i. e.
where ,
and
, since
is a - continuous , we
have
i . e ,
.
Hence , is a - continuous . The proof with respect to
in the same fashion .
Theorem 3.4 Let be IF-ts's . and
be a mapping . Then , the graph
of is - continuous iff
is - continuous , where is the F – product topology
generated by and
Proof suppose the graph is
- continuous . Let
where ,
we want to show that ,
. since
,
then
. so is
- continuous
Conversely , suppose is - continuous , let
, , where
' and ' are F- open set of and respectively.
Now
. So is - continuous
Proposition 3.1 Let
be an injective and - continuous mapping. Then for
each
we have is -
semi open ( res . , -preopen , -strongly semi open ,
-semi preopen set
Proof . It is obvious
Proposition 3.2 Let , be F-ts's ,
and be a bijective map. Then ,
if is a -homeomorphism , then is a -
semi continuous ( resp. -precontinuous , -strongly
semi continuous , - semi pre continuous) mappings
Proof . It is obvious
Theorem 3.5 Let , be IF-ts's ,
, If . is - semi
continuous and - precontinuous , then is -
strongly semi- continuous
Proof The proof is simple and hence omitted
Remark 3.2 a - semicontinuity and -
precontinuity are independent concepts .
Example 3.5 Let
and
and are
defined by ,
Then , is - pre continuous but
not - semicontinuous mapping
Example 3.6 Let , Let and
be two IF-ts's where ,
and
. Then is an -
semicontinuous but not -
Pre continuous mapping .
4. -retract and - neighbourhood retract
Definition 4.1 Let be IF-ts , and , Then
, the IF- subspace is called a -retract
of if there exists a -continuous
mapping such that
. In this case is called a -retraction
Remark 4.1 Every IF – retract is a - R , but the
converse is not true
Example 4.1 Let and be IF-sets on
Sci.Int.(Lahore),27(5),3899-3904,2015 ISSN 1013-5316; CODEN: SINTE 8 3903
Sept-Oct.
, defined by
, and Then , is a
-R of , but not a
IF retract
Remark 4.2 Let be a IF-ts. Since the identity
map is -continuous , then is a - R
or itself .
Proposition 3.1 . Let
be -retraction ,
be retraction . Then
is a -retraction
Proof It follows from theorem 3.2
Proposition 4.2 Let be a IF-ts , and
. Then the function
is -retraction iff for any IF-ts
, every - continuous function
has a -continuous function
such that
Proof Let be -retraction
be continuous function,
By Theorem 2.2.2. is -
continuous and ,
Conversely , let , then .
Since is - continuous , then g has a fuzzy -
continuous and .
Theorem 4.1 Let be a IF- ts , and
be a mapping such that
. Then the graph
of is -continuous iff is a - retraction, where
is the product topology generated by and
Proof. It follows directly from Theorem 3.3.
Definition 4.2 Let be a IF-ts , .
Then is said to be a -neighborhood retract
- nbd R of if is a – R of
, such that .
Remark 4.3 Every -R is a -nbd R , but the
converse is not true .
Example 4.2 Let
and be IF-sets on , defined by
Consider . Then
is a -nbd R of but not a -R of
.
Proposition 4.3 Let and be IF-ts's,
and . If is a -nbd R of
and is a
-nbd R of , then is a
-nbd R of .
Proof Since is a -nbd R of , then
is a - R of such that
, this implies that , there exists a -
continuous mapping such that
. Also since is a -nbd R of
, then is a –R of such
that , this implies that , there
exists a - continuous mapping
, such that , by using
Theorem 3.3 and Lemma 3.1
we have
is a
- continuous mapping , , and
. Hence , is a -nbd
R of .
Proposition 3.4. Let and be IF-ts's
. If is a - R of and
is a – R of , then
is a -R of
Proof The proof is much simpler than that of
Proposition 4.3. and hence omitted
5 - – semiretract , – preretract , – strongly
semiretract and – semi preretract
Definition 5.1 Let be a IF-ts , and
. Then the F – subspace is called a –
semi retract (resp . - pre retract,- strongly
semiretract and - semi preretract ) ( -PR , - SSR
, - SPR ) of if there exists a - semi
continuous (resp . - precontinuous , -strongly semi
continuous , - semi pre continuous) mapping
such that . In
this case , is called a -semi retraction ( resp . -
- perpetration , - strongly semiretraction , - semi
preretraction )
-PR
-R -R -SSR
- SPR
-SR
But , the converse is not true in general , as we indicate the
following examples .
Remark 5.1 Example 3.1 shows that a - R need not to
be a IF- retract.
Example 5.1 Let and
and be IF- sets on defined by
Consider . Then is a - SPR ,
but not a -PR, also , - SR , but not a - SSR .
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Example 5.2. Let and
be IF- sets on
defined by
Consider . Then is a - PR ,
but not a -SSR .
Example 5.3. Let and
and be IF- sets on defined by
Consider . Then is a -
SSR , but not a -R
Example 5.4 Let and
and be IF- sets on defined by
Consider . Then is a - SPR , but
not a -SR
Proposition 5.1 Let , be a IF-ts,
and
be a mapping such that
. If is a - precontinuous and -
semicontinuous , then is a -SSR of
Proof . It follows directly from Theorem 3.5
Remark 5.2 A - semiretract and - pre retract are
independent concepts
Example 5.6 Let and and
be IF- sets on defined by
onsider .Then is a - SR of
, but not a -PR
Example 5.7 Let and and
be IF- sets on defined by
Consider .Then is a - PR of
, but not a -SR
CONCLUSION
The purpose of this paper is to define-open sets )
and many results of - strongly semi continuous Also, we
define -retract and - neighborhood retract and -
strongly semi retract as applications of - strongly semi
continuous.
REFERENCES
1. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and
Systems 20(1986),87-96
2. D. Coker, An introduction to intuitionistic fuzzy
Topological Spaces, Fuzzy Sets and Systems
88(1997)81-89
3. D. Coker, An introduction to fuzzy subspace in
intuitionistic fuzzy topological spaces, J. Fuzzy Math., 4
(4)(1996) 749-764
4. H. Gurcay, D. Coker and A. Hayder. On fuzzy continuity
in intuitionistic fuzzy topological spaces, J. Fuzzy Math.
5(2) (1997), 365-378
5. I. M. Hanafy, Completely continuous functions in
intuitionistic fuzzy topological spaces, Czechoslovak
Mathematical Journal, 53(4) (2003), 793-803
6. I. M. Hanafy, F.S. Mahmoud and Mohammed .M. Khalaf
intuitionistic fuzzy retracts, International journal of
logic and intelligent system, 5(1) (2005), 40-45
7.S.K.Samanta and T.K. Mondal, Topology on interval
valued intuitionistic fuzzy sets, Fuzzy Sets and Systems
119 (2001) 483-494
8.L.A Zadeh, Fuzzy sets, Inform. and Control 8(1965)338-
353
