Completely B# Continuous Mappings in Intuitionistic Fuzzy Topological Spaces

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Completely B# Continuous Mappings in Intuitionistic Fuzzy Topological Spaces

S. Dhivya1

Master of Philosophy (Mathematics) Avinashilingam (Deemed to be) University Coimbatore, India

Dr. D. Jayanthi2

Assistant Professor of Mathematics Avinashilingam (Deemed to be) University Coimbatore, India

Abstract In this chapter we have introduced two types of b# continuous mappings namely intuitionistic fuzzy completely b# continuous mappings and intuitionistic fuzzy perfectly b# continuous mappings. Also we have provided some interesting results based on these continuous mappings.

Keywords Intuitionistic fuzzy sets, intuitionistic fuzzy topology, intuitionistic fuzzy completely b# continuous mapping.

  1. INTRODUCTIONIntuitionistic fuzzy set is introduced by Atanassov in 1986. Using the notion of intuitionistic fuzzy sets, Coker [1997] has constructed the basic concepts of intuitionistic fuzzy topological spaces. The concept of b# closed sets and b# continuous mappings in intuitionistic fuzzy topological spaces are introduced by Gomathi and Jayanthi (2018). In this paper we have introduced intuitionistic fuzzy completely b# continuous mappings and intuitionistic fuzzy perfectly b# continuous mappings. Also we have provided some interesting results based on these continuous mappings.
  2. PRELIMINARIESDefinition 2.1: [Atanassov 1986] An intuitionistic fuzzy set(IFS) A is an object having the form A= {x, µA(x), A(x): x X},where the functions µA: X [0, 1] and A: X [0,1] denote the degree of membership and the degree of non-membership of each element x X to the set A respectively , and 0 µA(x) + A(x) 1 for each x X. Denote by IFS(X) , the set of all intuitionistic fuzzy sets in

    1. An IFS A in X is simply denoted by A = x, µA, A instead of denoting A = {x, µA(x), A(x): x X}.Definition 2.2: [Atanassov 1986] Let A and B be two IFSs of the form A = {x, µA(x), A(x): x X} and B = {x,

      µA(x), A(x): x X}. Then the following properties hold:

      1. AB if and only if µA(x) µB(x) and A(x) B(x) for all x X,
      2. A=B if and only if A B and A B,
      3. Ac = {x, µA(x), A(x) : x X},

    iv. A B = {x, µA(x) µB(x), A(x) B(x) : x X},

    v. A B = { x, µA(x) µB(x) , A(x) B(x) : x

    X}.

    The IFSs 0~= x, 0, 1 and 1~= x, 1, 0 are respectively the empty set and whole set of X.

    Definition 2.3: [Coker, 1997] An intuitionistic fuzzy topology (IFT) on X is a family of IFSs in X satisfying the following axioms:

    i. 0~,1~

    1. G1G2 for any G1, G2
    2. Gi for any {Gi : i J} .

    In this case the pair (X, ) is called the intuitionistic fuzzy topological space (IFTS) and any IFS in is known as an intuitionistic fuzzy open set (IFOS) in X. Then the complement Ac of an IFOS A in an IFTS (X, ) is called an intuitionistic fuzzy closed set (IFCS) in X.

    Definition 2.4: [Coker, 1997] Let (X,) be an IFTS and A =

    x, µA, A be an IFS in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure are defined by

    int(A) = {G/G is an IFOS in X and G A}, cl(A) = {K/K is an IFCS in X and A K}.

    Definition 2.5: [Gurcay, Coker and Hayder, 1997] An IFS A = x, µA, A in an IFTS (X, ) is said to be an

    1. intuitionistic fuzzy semi closed set if int(cl(A))A
    2. intuitionistic fuzzy pre closed set if cl(int(A)) A
    3. intuitionistic fuzzy regular closed set if cl(int(A)) = A
    4. intuitionistic fuzzy closed set if cl(int(cl(A)))A
    5. intuitionistic fuzzy closed set if int(cl(int(A)))

    A

    Definition 2.6: [Hanafy, 2009] An IFS A=x, µA, A in an IFTS (X, ) is said to be an intuitionistic fuzzy closed set if int(cl(A)) cl(int(A)) A.

    Definition 2.7: [Gomathi and Jayanthi, 2018] An IFS A =

    x, µA, A in an IFTS (X, ) is said to be an intuitionistic fuzzy b# closed set (IFb#CS) if int(cl(A)) cl(int(A)) = A.

    Definition 2.8: [Coker, 1997] Let X and Y be two non empty sets and f: XY be a mapping. If B = {y, B(y), B(y) / y Y} is an IFS in Y, then the preimage of B under f is denoted and defined by f-1(B)= { x, f-1(B)(x), f-1(B)(x)

    / x X }, where f-1 (B)(x) = B(f(x)) for every x X.

    Definition 2.9: [Gurcay, Coker and Hayder, 1997] Let f be a mapping from an IFTS (X,) into an IFTS (Y,). Then f said to be an intuitionistic fuzzy continuous mapping if f- 1(V) is an IFCS in (X,) for every IFCS V of (Y,).

    Definition 2.10: [Joung Kon Jeon, 2005] Let f be a mapping from an IFTS (X, ) into an IFTS (Y, ). Then f said to be an

    1. intuitionistic fuzzy semi continuous mapping if f-1(V) is an IFSCS in (X, ) for every IFCS V of (Y,).
    2. intuitionistic fuzzy continuous mapping if f-1(V) is an IFCS in (X, ) for every IFCS V of (Y, ).
    3. intuitionistic fuzzy pre continuous mapping if f-1(V) is an IFPCS in (X, ) for every IFCS V of (Y, ).
    4. intuitionistic fuzzy continuous mapping if f-1(V) is an IFCS in (X, ) for every IFCS V of (Y, ).

    Definition 2.11: [Gomathi and Jayanthi, 2018] Let f be a mapping from an IFTS (X, ) into an IFTS (Y, ). Then f is said to be an

    1. intuitionistic fuzzy b# continuous mapping if f-1 (V) is an IFb#CS in (X, ) for every IFCS V of (Y, ).
    2. intuitionistic fuzzy contra b# continuous mapping if f-1 (V) is an IFb#CS in (X, ) for every IFOS V of (Y, ).
    3. intuitionistic fuzzy b# irresolute mapping if f-1 (V)
  3. COMPLETELY b# CONTINUOUS MAPPINGS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES

In this chapter we have introduced and investigated intuitionistic fuzzy completely b# continuous mappings and intuitionistic fuzzy perfectly b# continuous mappings. We have provided many interesting results using these continuous mappings.

Definition 3.1: A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy completely b# continuous mapping if f- 1(V) is an IFRCS in (X, ) for every IFb#CS V of (Y, ).

Example 3.2: Let X = {a, b}, Y = {u, v}. Then = {0~, G1, G2 1~} and = {0~, G3, G4 1~} are IFS on X and Y respectively, where, G1 = x, (0.2a, 0.3b), (0.4a, 0.5b), G2 =

x, (0.4a, 0.5b), (0.2a, 0.3b), G3 = y, (0.2u, 0.3v), (0.4u, 0.5v)

and G4 = y, (0.4u, 0.5v), (0.2u, 0.3v). Define a mapping f: (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an intuitionistic fuzzy completely b# continuous mapping.

Proposition 3.3: A mapping f: (X, ) (Y, ) is an intuitionistic fuzzy completely b# continuous mapping if and only if the inverse image of each IFb#OS in Y is an IFROS in X.

Proof: Obviously.

Proposition 3.4: If f: (X, ) (Y, ) is an intuitionistic fuzzy completely b# continuous mapping where Y is an IFT

is an IFb#CS in (X, ) for every IFb#CS V of (Y, ).

b# space[4], then for each IFP (, )

X and for every

Definition 2.12: [Hanafy and El-Arish, 2003] Let f be a mapping from an IFTS (X, ) into an IFTS (Y, ). Then f is said to be an intuitionistic fuzzy completely continuous mapping if f-1(V) is an IFROS in (X, ) for every IFOS V of (Y, ).

Definition 2.13: [Coker and Demirci, 1995] Intuitionistic

intuitionistic fuzzy neighbourhood A of f((, )), there exists an IFROS B of X such that (, ) B and f(B) A.

Proof: Let (, ) be an IFP of X and let A be an intuitionistic fuzzy neighbourhood of f((, )) sch that f((, )) C A, where C is an IFOS in X. Since every

fuzzy point (IFP), written as p(, ), is defined to be an IFS of

IFOS is an IFb#OS in an IFT

#

#

 

b

space, C is an IFb#OS in Y

(, )

X given by p(, )(x) = {

=

. An IFP p(, ) is said

as Y is an IFT

space. Hence by hypothesis, f -1(C) is an

(0,1)

to belong to a set A if µA and A.

Definition 2.14: [Thakur and Rekha Chaturvedi, 2008] Two IFSs A and B are said to be q-coincident (A q B) if and only if there exist an element x X such that µA(x) B(x) or A(x) < µB(x).

Definition 2.15: [Seok Jong Lee and Eun Pyo Lee, 2000] Let p(, ) be an IFP in (X, ). An IFS A of X is called an intuitionistic fuzzy neighbourhood of p(, ) if there exist an

b#

IFROS in X and (, ) f -1(C). Put B = f -1(C). Therefore

(, ) B = f-1(C) f-1(A).Thus f(B) f(f-1(A)) A. That is f(B) A.

Proposition 3.5: A mapping f: (X, ) (Y, ) is an intuitionistic fuzzy completely b# continuous mapping then cl(int(f-1(cl(B)))) f-1(B) for every IFS B in Y where Y is

an IFT b# space.

IFOS B in X such that p(, ) B A. Proof: Let B Y be an IFS. Then cl(B) is an IFCS in Y and

Definition 2.16: [Dhivya and Jayanthi, 2019] Let f be a

hence an IFb#CS in Y as Y is an IFT #

space. By

b

b

 

mapping from an IFTS (X, ) into an IFTS (Y, ). Then f is said to be an intuitionistic fuzzy almost b# continuous mapping if f-1(V) is an IFb#CS in (X, ) for every IFRCS V of (Y, ).

hypothesis, f-1(cl(B)) is an IFRCS in X. Hence cl(int(f- 1(cl(B)))) = f-1(cl(B)) f-1(B).

Proposition 3.6: Let f: (X, ) (Y, ) be an mapping. Then the following are equivalent:

  1. f is an intuitionistic fuzzy completely b# continuous mappingProof: Let × be an IFb#CS of × . Then (f1, f2)- 1( × )(x)=( × )(f1(x),f2(x))=
  2. f -1(V) is an IFROS in X for every IFb#OS V in Yx, min(

    ( f (x)),

    ( f (x))), max(

    ( f (x)),

    ( f (x))) =

  3. for every IFP

and for every IFb#OS B

A 1 B 2

A 1 B 2

(, )

x, min( f 1 ( )(x), f 1 ( )(x)),max( f 1 (

)(x), f 1 (

)(x) =

in Y such that f(

) B there exists an IFROS

1 A 2 B

1 A 2 B

(, )

f 1(A) f 1(B)(x). Since f and f are an intuitionistic

in X such that (, ) and f(A) B

Proof: (i) (ii): Let V be an IFb#OS in Y. Then Vc is an

1 2 1 2

fuzzy completely b# continuous mapping, f-1(A) and f-1(B) are IFROSs in X. Since the intersection of two IFROSs is an

IFb#CS in Y. Since f is an intuitionistic fuzzy completely b#

IFROS,

f 1 (A) f 1 (B)

is an IFROS in X. Hence

continuous mapping, f-1(Vc) is an IFRCS in X. Since f-1(Vc) 1 2

-1 c -1

(f1,f2) is an intuitionistic fuzzy completely b# continuous

= (f (V)) , f

(V) is an IFROS in X.

mapping.

(ii) (iii): Let (, ) and B Y such that f((, ))

B. This implies (, ) f-1(B). Since B is an IFb#OS in Y, by hypothesis f-1(B) is an IFROS in X. Let A = f-1(B). Then

(, ) f-1 (f((, ))) f-1(B)=A. Therefore (, )

and f(A) = f(f-1(B)) B.This implies f(A) B.

(iii) (ii): Let B Y be an IFb#OS. Let (, ) and f((, )) B. By hypothesis, there exists an IFROS C in X such that (, ) and f(C) B. This implies C f-1(f(C)) f-1(B). Therefore (, ) f-1(B). That is

Proposition 3.9: Let f : X Y and g : Y Z be any two mappings. If f and g are intuitionistic fuzzy completely b# continuous mapping, then g f is also an intuitionistic fuzzy

#

#

 

completely b# continuous mapping, where Y is an IFT

b

space.

Proof: Let B be an IFb#CS in Z. Since g is an intuitionistic fuzzy completely b# continuous mapping, g-1(B) is an IFRCS in Y. Since every IFRCS is an IFCS, g-1(B) is an

IFCS in Y. As Y is an IFT space, g-1(B) is an IFb#CS in

f-1(B) =

p( , ) C

f-1(B). This implies

b#

  1. Now as f is an intuitionistic fuzzy completely b#p( , )f -1 ( B)

     

    f-1(B) = C

    p( , )f -1 ( B)

    p( , )f -1 ( B)

    . Since the union IFROSs is an IFROS,

    continuous mapping, f-1(g-1(B)) = (g f )-1(B) is an IFRCS in X. Hence g f is an intuitionistic fuzzy completely b# continuous mapping.

    f-1(B) is an IFROS in X. Hence f is intuitionistic fuzzy completely b# continuous mapping.

    Proposition 3.7: A mapping f : X Y is an intuitionistic fuzzy completely b# continuous mapping then the following are equivalent:

    1. For any IFb#OS A in Y and for any IFP (, ), if f((, ))q A, then (, )q int(f-1(A)).
    2. For any IFb#OS A in Y and for any (, ) , if f((, ))q A , then there exists an IFOS B such that

(, )q B and f(B) A.

Proof: (i) (ii): Let A Y be an IFb#OS and let (, )

. Let f((, ))q A. Then (, )qf-1(A) (i) implies that

Proposition 3.10: Let f : X Y and g : Y Z be any two mappings. If f is an intuitionistic fuzzy completely b# continuous mapping and g is an intuitionistic fuzzy b# irresolute mapping then g f is also an intuitionistic fuzzy completely b# continuous mapping.

Proof: Let B be an IFb#CS in Z. Since g is an intuitionistic fuzzy b# irresolute mapping, g-1(B) is an IFb#CS in Y. Also, since f is an intuitionistic fuzzy completely b# continuous mapping, f-1(g-1(B)) is an IFRCS in X. Since (g f)-1(B) = f-1(g-1(B)), g f is an intuitionistic fuzzy completely b# continuous mapping.

Proposition 3.11: Let f : X Y and g : Y Z be any two mappings. If f is an intuitionistic fuzzy completely b#

(, )

q int(f-1( A)) where int(f-1( A)) is an IFOS in X. Let B

continuous mapping and g is an intuitionistic fuzzy b#

= int(f-1( A)). Since int(f-1( A)) f-1( A), B f-1( A). Then f(B) f(f-1( A)) A.

(ii) (i): Let A Y be an IFb#OS and let (, ) .

continuous mapping then g f is also an intuitionistic fuzzy

completely continuous mapping.

Proof: Let B be an IFCS in Z. Since g is an intuitionistic

Suppose f (

(, ))q

A, then by (ii) there exists an IFOS B in

fuzzy b# continuous mapping, g-1(B) is an IFb#CS in Y. Also, since f is an intuitionistic fuzzy completely b#

X such that (, )q B and f(B) A. Now B f-1(f( B))

f-1(A). That is B = int(B) int(f-1( A)). Therefore (, )qB implies (, )q int(f-1( A)).

Proposition 3.8: Let f1: (X, ) (Y, ) and f2 : (X, ) (Y, ) be any two intuitionistic fuzzy completely b# continuous mappings. Then the mapping (f1,f2) : (X, ) ( × , × ) is also an intuitionistic fuzzy completely b# continuous mapping.

continuous mapping, f-1(g-1(B)) is an IFRCS in X. Since (g f)-1(B) = f-1(g-1(B)), (g f) is an intuitionistic fuzzy completely continuous mapping.

Proposition 3.12: Let f : X Y and g : Y Z be any two mappings. If f is an intuitionistic fuzzy completely b# continuous mapping and g is an intuitionistic fuzzy b# continuous mapping then g f is also an intuitionistic fuzzy completely continuous mapping.

Proof: Let B be an IFCS in Z. Since g is an intuitionistic Proof: Let B be an IFRCS in Y. Since every IFRCS is an

fuzzy b# continuous mapping, g-1(B) is an IFb#CS in Y. Also, since f is an intuitionistic fuzzy completely b# continuous mapping, f-1(g-1(B)) is an IFRCS in X. Since (g f)-1(B) = f-1(g-1(B)), g f is an intuitionistic fuzzy completely continuous mapping.

IFCS, B is an IFCS in Y. Since Y is an IFT b# space, B is an IFb#CS in Y. Since f is an intuitionistic fuzzy perfectly b# continuous mapping, f-1(B) is an intuitionistic fuzzy clopen set in X. Thus f-1(B) is an IFCS in X. Since every IFCS is an

p>Proposition 3.13: Let f : X Y and g : Y Z be any two

IFb#CS in an IFT

#

#

 

b

space, f-1(B) is an IFb#CS in X, as X is

mappings. If f is an intuitionistic fuzzy almost b# continuous mapping and g is an intuitionistic fuzzy completely b# continuous mapping then g f is also an intuitionistic fuzzy b# irresolute mapping.

Proof: Let B be an IFb#CS in Z. Since g is an intuitionistic fuzzy completely b# continuous mapping, g-1(B) is an IFRCS in Y. Also, since f is an intuitionistic fuzzy almost b# continuous mapping, f-1(g-1(B)) is an IFb#CS in X. Since (g

o f)-1(B) = f-1(g-1(B)), g f is an intuitionistic fuzzy b# irresolute mapping.

Definition 3.14: A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy perfectly b# continuous mapping if f-1(V) is an intuitionistic fuzzy clopen set in (X, ) for every

an IFT b# space. Hence f is an intuitionistic fuzzy almost b continuous mapping.

#

#

 

Proposition 3.19: A mapping f:(X, ) (Y, ) is an intuitionistic fuzzy perfectly b# continuous mapping and then f is an intuitionistic fuzzy b# continuous mapping

where X and Y are IFT b# spaces.

#

#

 

Proof: Let B be an IFCS in Y. Since every IFCS is an IFb#CS in an IFT space, B is an IFb#CS in Y, as Y is an

b

#

#

 

IFT space. Since f is an intuitionistic fuzzy perfectly b#

b

continuous mapping, f-1(B) is an intuitionistic fuzzy clopen set in X. Thus f-1(B) is an IFCS in X. Since every IFCS is an

IFb#CS V of (Y, ).

IFb#CS in an IFT

#

#

 

b

space, f-1(B) is an IFb#CS in X, as X is

Example 3.15: Let X = {a, b}, Y = {u, v}. Then = {0~, G1, G2, 1~} and = {0~ , G3, G4, 1~} are IFS on X and Y respectively, where, G1 = x, (0.2a, 0.3b), (0.4a, 0.5b), G2 =

x, (0.4a, 0.5b), (0.2a, 0.3b), G3 = y, (0.2u, 0.3v), (0.4u, 0.5v)

and G4 = y, (0.4u, 0.5v), (0.2u, 0.3v).Define a mapping f: (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an intuitionistic fuzzy perfectly b# continuous mapping.

Proposition 3.16: A mapping f : (X,) (Y,) is an intuitionistic fuzzy perfectly b# continuous mapping if and

an IFT b# space. Hence f is an intuitionistic fuzzy b continuous mapping.

#

#

 

Proposition 3.20: A mapping f : (X,) (Y,) is an intuitionistic fuzzy perfectly b# continuous mapping, then f is an intuitionistic fuzzy semi continuous mapping, where Y

is an IFT b# space.

Proof: Let B be an IFCS in Y. Since every IFCS is an

only if the inverse image of each IFb#OS in Y is an intuitionistic fuzzy clopen in X.

IFb#CS in an IFT

#

#

 

b

space, B is an IFb#CS in Y as Y is an

Proof: Straight forward.

Proposition 3.17: A mapping f : (X,) (Y,) is an intuitionistic fuzzy perfectly b# continuous mapping then f is an intuitionistic fuzzy continuous mapping where Y is an

IFT b# space.

Proof: Let B be an IFCS in Y. Since every IFCS is an

IFT space. Since f is an intuitionistic fuzzy perfectly b#

#

#

 

b

continuous mapping, f-1(B) is an intuitionistic fuzzy clopen set in X. Thus f-1(B) is an IFCS in X. Since every IFCS is an IFSCS, f-1(B) is an IFSCS in X. Hence f is an intuitionistic fuzzy semi continuous mapping.

Proposition 3.21: A mapping f : (X, ) (Y,) is an intuitionistic fuzzy perfectly b# continuous mapping, then f is an intuitionistic fuzzy continuous mapping, where Y is

#

#

 

IFb#CS in an IFT

b

space, B is an IFb#CS in Y, as Y is an

an IFT b# space.

#

#

 

IFT b# space. Since f is an intuitionistic fuzzy perfectly b continuous mapping, f-1(B) is an intuitionistic fuzzy clopen set in X. Thus f-1(B) is an IFCS in X. Hence f is an intuitionistic fuzzy continuous mapping.

Proposition 3.18: A mapping f:(X,) (Y,) is an intuitionistic fuzzy perfectly b# continuous mapping, then f is an intuitionistic fuzzy almost b# continuous mapping,

Proof: Let B be an IFCS in Y. Since every IFCS is an IFb#CS in an IFT # space, B is an IFb#CS in Y, as Y is an

b

b

 

#

#

 

IFT b# space. Since f is an intuitionistic fuzzy perfectly b continuous mapping, f-1(B) is an intuitionistic fuzzy clopen set in X. Thus f-1(B) is an IFCS in X. Since every IFCS is an IFCS, f-1(B) is an IFCS in X. Hence f is an intuitionistic

where X and Y are IFT b#

spaces.

fuzzy continuous mapping.

Proposition 3.22: A mapping f : (X,) (Y,) is an intuitionistic fuzzy perfectly b# continuous mapping then f is

an intuitionistic fuzzy pre continuous mapping, where Y is

an IFT b#

space.

#

#

 

Proof: Let B be an IFCS in Y. Since every IFCS is an IFb#CS in an IFT space. B is an IFb#CS in Y as Y is an

b

#

#

 

IFT b# space. Since f is an intuitionistic fuzzy perfectly b continuous mapping, f-1(B) is an intuitionistic fuzzy clopen set in X. Thus f-1(B) is an IFCS in X. Since every IFCS is an IFPCS, f-1(B) is an IFPCS in X. Hence f is an intuitionistic fuzzy pre continuous mapping.

Proposition 3.23: Let f : X Y and g : Y Z be any two intuitionistic fuzzy perfectly b# continuous mappings where

Y is an IFT b# space. Then their composition g f : X Z is an intuitionistic fuzzy perfectly b# continuous mapping.

Proof: Let A be an IFb#CS in Z. Then by hypothesis, g-1(A) is an intuitionistic fuzzy clopen set in Y. Since Y is an IFT

-1 #

-1 #

 

b# space, g (A) is an IFb CS in Y. Again by hypothesis, f-1(g-1(A)) is an intuitionistic fuzzy clopen set in X. Since f-1(g-1(A)) = (g f )-1(A), (g f)-1(A) is an intuitionistic fuzzy clopen set in X. Hence g f is an intuitionistic fuzzy perfectly b# continuous mapping.

REFERENCES

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  3. Coker, D. and Demirci, M., On intuitionistic fuzzy points, Notes on Intuitionistic Fuzzy Sets, 1, 1995, 79-84.
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