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Chapter5Generalized Binary mContinuous and bContinuous Paper

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Chapter-5

Generalized Binary m-Continuous and b-Continuous Maps

5.2. Generalized Binary m-Continuous Maps

Definition 5.2.1: Let (A, B) be the subset of a g-binary topological space (X,Y,?), then it is called ?-binary regular open (A,B)=I_? (Cl_? (A,B) ). The ?-binary delta (theta) interior of subset (A, B) of a g-binary topological space (X,Y,?) is the union of all ?-binary regular open sets of (X,Y,?) contained in (A, B) and is denoted by ??I?_? (A,B). A subset (A, B) of a (X,Y,?) is called ?-binary ?-open if (A,B)=??I?_? (A,B), i.e. a set is ?-binary ?-open if it is the union of ?-binary regular open sets. The complement of ?-binary ?-open set is called ?-binary ?-closed (A,B)=??Cl?_? (A,B), where ??Cl?_? (A,B)={(x,y)??(X)??(Y): I_? (Cl_? (U,V) )?(A,B)??,(U,V)?? and (x,y)?(U,V)}.

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Definition 5.2.2: Let (X,Y,?) be a g-binary topological space and (A,B) be a subset of ?(X)??(Y), then ??Cl?_? (A,B)={(x,y)??(X)??(Y):Cl_? (U,V)?(A,B)??, (U,V)?? and (x,y)?(U,V)}. A subset (A, B) of a (X,Y,?) is called ?-binary ?-closed if (A,B)=??Cl?_? (A,B).

Definition 5.2.3: Let (X,Y,?) be a g-binary topological space and (A,B) be a subset of ?(X)??(Y), then (A,B) is called

?-binary ?-semi-closed if I_? (??Cl?_? (A,B))?(A,B).

?-binary ?-pre-closed if Cl_? (??I?_? (A,B))?(A,B).

?-binary ?-?-closed if Cl_? (I_? (??Cl?_? (A,B) ))?(A,B).

?-binary ?-?-closed if I_? (Cl_? (??I?_? (A,B) ))?(A,B).

Definition 5.2.4: Let (X,Y,?) be a g-binary topological space and (A,B) be a subset of ?(X)??(Y), then (A,B) is called

?-binary m-open set if (A,B)?Cl_? (??I?_? (A,B) ) ? I_? (??Cl?_? (A,B))

?-binary m-closed set if (A,B)?I_? (??Cl?_? (A,B) ) ? Cl_? (??I?_? (A,B))

Proposition 5.2.1: In a g-binary topological space (X,Y,?)

Every ?-binary ?-semi-open set is ?-binary m-open.

Every ?-binary ?-pre-open set is ?-binary m-open.

Proof: Obvious

Remark 5.2.1: Converse of Proposition 5.2.1 is not true in general as shown in Example 5.2.1

Example 5.2.1: Let X={1,2,3} and Y={a,b,c}. Then ? ={(?,?),({1},{a,b} ),({2,3},{c} ),({1,3},{Y} ),(X,Y)}. Clearly ? is g-binary topology from X to Y. Therefore the set ({1,3},{a,b} ) is ?-binary m-open but not ?-binary ?-semi-open or ?-binary ?-pre-open set.

Definition 5.2.5: Let (Z,?_g ) be a g-topological space and (X,Y,?) be g-binary topological space. Then the map f:Z?X?Y is said to be ?-binary m-continuous if f^(-1) (A,B) is ?_g-m-open in (Z,?_g) for every ?-binary open set (A, B) in (X,Y,?).

Example 5.2.2: Let Z={1,2,3,4}, X={a_1,a_2,a_3 } and Y={b_1,b_2,b_3 }.Then ?_g={?,{1},{3,4},{1,2,4},{1,3,4} Z} and ?={(?,?),({a_1 },{b_1 } ),({a_2 },{b_2 } ),({a_2 },{Y} ), (X,Y)}. Clearly ?_g is a g-topology on Z and ? is g-binary topology from X to Y. Define f:Z?X?Y by f(1)=(a_1,b_1 )= f(3) and f(2)=f(4)=(a_2,?). Now f^(-1) (?,?)=?, f^(-1) ({a_1 },{b_1 })={1,3}, f^(-1) ({a_2 },{b_2 })={?}, f^(-1) ({a_2 },{Y})={?} and f^(-1) (X,Y)=Z. This shows that the inverse image of every ?-binary open set in (X,Y,?) is ?_g-m-open in (Z,?_g ). Hence f is ?-binary m-continuous map.

Proposition 5.2.2: Every ?-binary ?-semi-continuous map is ?-binary m-continuous map.

Proof: Obvious from the definition

Remark 5.2.2: Converse of Proposition 5.2.2 is not true in general as shown in Example 5.2.3.

Example 5.2.3: In Example 5.2.2 f is ?-binary m-continuous map but not ?-binary ?-semi-continuous because the set {1,3} is ?_g-m-open in (Z,?_g ) but not ?_g-?-semi-open.

Proposition 5.2.3: Every ?-binary ?-pre-continuous map is ?-binary m-continuous map.

Proof: Obvious from the definition

Remark 5.2.3: Converse of Proposition 5.2.3 is not true in general as shown in Example 5.2.6

Example 5.2.4: In Example 5.2.2 f is ?-binary m-continuous map but not ?-binary ?-pre-continuous because the set {1,3} is ?_g-m-open in (Z,?_g ) but not ?_g-?-pre-open.

Remark 5.2.4: Every ?-binary continuous map is ?-binary m-continuous but not converse as shown in Example 5.2.5.

Example 5.2.5: In Example 5.2.2 f is ?-binary m-continuous map but not ?-binary continuous map because the set {1,3} is ?_g-m-open in (Z,?_g ) but not ?_g-open.

Remark 5.2.5: Every ?-binary pre-continuous map is ?-binary m-continuous but not converse as shown in Example 5.2.6.

Example 5.2.6: Let Z={1,2,3,4}, X={a_1,a_2,a_3 } and Y={b_1,b_2,b_3 }.Then ?_g={?,{4},{1,2},{2,3},{1,2,3},{2,3,4},{1,2,4}, Z} and ?={(?,?),({a_1 },{b_1 } ), ({a_2 },{b_2 } ),({a_2 },{Y} ),(X,Y)}. Clearly ?_g is a g-topology on Z and ? is g-binary topology from X to Y. Define f:Z?X?Y by f(1)=(a_1,b_1 )= f(4) and f(2)=f(3)=(a_2,?). Now f^(-1) (?,?)=?, f^(-1) ({a_1 },{b_1 })={1,4}, f^(-1) ({a_2 },{b_2 })={?}, f^(-1) ({a_2 },{Y})={?} and f^(-1) (X,Y)=Z. This shows that the inverse image of every ?-binary open set in (X,Y,?) is ?_g-m-open in (Z,?_g). Hence f is ?-binary m-continuous but not ?-binary pre-continuous because the set {1,4} is ?_g-m-open but not ?_g-pre-open in (Z,?_g).

Remark 5.2.6: Every ?-binary semi-continuous map is ?-binary m-continuous but not converse as shown in Example 5.2.7.

Example 5.2.7: In Example 5.2.6 f is ?-binary m-continuous but not ?-binary semi-continuous because the set {1,4} is ?_g-m-open but not ?_g-semi-open in (Z,?_g).

Remark 5.2.7: Every ?-binary ?-continuous map is ?-binary m-continuous but not converse as shown in Example 5.2.8.

Example 5.2.8: In Example 5.3.6 f is ?-binary m-continuous but not ?-binary ?-continuous because the set {1,4} is ?_g-m-open but not ?_g-?-open in (Z,?_g).

Remark 5.2.8: Every ?-binary ?-continuous map is ?-binary m-continuous but not converse as shown in Example 5.2.9.

Example 5.2.9: In Example 5.2.6 f is ?-binary m-continuous but not ?-binary ?-continuous because the set {1,4} is ?_g-m-open but not ?_g-?-open in (Z,?_g).

5.3. m-Totally Generalized Binary Continuous Maps

Definition 5.3.1: Let (Z,?_g ) be a g-topological space and (X,Y,?) be g-binary topological space. Then the mapping f:Z?X?Y is said to be m-totally ?-binary continuous if f^(-1) (A,B) is ?_g-clopen in (Z,?_g ) for every ?-binary m-open set (A, B) in (X,Y,?).

Example 5.3.1: Let Z={1,2,3}, X={a_1,a_2 } and Y={b_1,b_2 }. Then ?_g={?,{1},{2},{1,2},{1,3},{2,3},Z}and?={(?,?),({a_1 },{b_1 } ),({a_2 },{Y} ),(X,Y) }. Clearly ?_g is a g-topology on Z and ? is g-binary topology from X to Y. Define f:Z?X?Y by f(1)=(a_1,b_1 ),f(2)=(a_2,b_2 )=f(3). Now f^(-1) (?,?)=?, f^(-1) ({a_1 },{b_1 })={1}, f^(-1) ({a_2 },{b_2 })={2,3}, f^(-1) ({a_2 },{Y})={2,3} and f^(-1) (X,Y)=Z. This shows that the inverse image of every ?-binary m-open sets in (X,Y,?) is ?_g-clopen in (Z,?). Hence f is m-totally ?-binary continuous.

Proposition 5.3.1: Let (Z,?_g ) be a g-topological space and (X,Y,?) be g-binary topological space. Then the map f:Z?X?Y is said to be m-totally ?-binary continuous if and only if f^(-1) (A,B) is ?_g-clopen in (Z,?_g ) for every ?-binary m-closed set (A, B) in (X,Y,?).

Definition 5.3.2: Let (Z,?_g ) be a g-topological space and (X,Y,?) be g-binary topological space. Then the mapping f:Z?X?Y is said to be

Totally ?-binary m-continuous if f^(-1) (A,B) is ?_g-m-clopen in (Z,?_g ) for every ?-binary open set (A, B) in (X,Y,?).

Strongly ?-binary m-continuous if f^(-1) (A,B) is ?_g-m-clopen in (Z,?_g ) for every ?-binary set (A, B) in (X,Y,?).

Proposition 5.3.2: Every m-totally ?-binary continuous map is totally ?-binary continuous.

Proof: Let (Z,?_g ) be a g-topological space and (X,Y,?) be g-binary topological space and the map f:Z?X?Y be m-totally ?-binary continuous. Since every ?-binary open set in g-binary topology is ?-binary m-open. Let (A, B) be any ?-binary m-open set in (X,Y,?). This implies f^(-1) (A,B) is ?_g-clopen in (Z,?_g ). Thus, inverse image of every ?-binary open set in (X,Y,?) is ?_g-clopen in (Z,?_g ) . Therefore f is totally ?-binary continuous.

Remark 5.3.1: Converse of Proposition 5.3.2 is not true shown in Example 5.3.2.

Example 5.3.2: Let Z={1,2,3}, X={a_1,a_2 } and Y={b_1,b_2 }. Then ?_g={?,{1},{2},{1,2},{1,3},{2,3},Z} and ?={(?,?),({a_1 },{b_1 } ),({a_1 },{b_2 } ),(X,Y) }. Clearly ?_g is a g-topology on Z and ? is g-binary topology from X to Y. Define f:Z?X?Y by f(1)=(a_1,b_1 ),f(2)=(a_1,b_2 ),f(3)=(a_2,b_2). Now f^(-1) (?,?)=?, f^(-1) ({a_1 },{b_1 })={1}, f^(-1) ({a_1 },{b_2 })={2}, and f^(-1) (X,Y)=Z. This shows that the inverse image of every ?-binary open sets in (X,Y,?) is ?_g-clopen in (Z,?_g ). Hence f is totally ?-binary continuous but not m-totally ?-binary continuous because f^(-1) ({a_2 },{b_2 } )={3} i.e. inverse image of ?-binary m-open set ({a_2 },{b_2 } ) is not ?_g-clopen in (Z,?_g ).

Proposition 5.3.3: Every strongly ?-binary continuous map is m-totally ?-binary continuous map.

Proof: Let (Z,?_g ) be a g-topological space, (X,Y,?) be g-binary topological space and the map f:Z?X?Y be strongly ?-binary continuous. Let (A, B) be any ?-binary m-open set in (X,Y,?). Then by definition f^(-1) (A,B) is ?_g-clopen in (Z,?_g ). Thus the inverse image of every ?-binary m-open set in (X,Y,?) is ?_g-clopen in (Z,?_g ). Therefore f is m-totally ?-binary continuous.

Remark 5.3.2: Converse of Proposition 5.3.3 is not true shown in Example 5.3.3.

Example 5.3.3: Let Z={1,2,3}, X={a_1,a_2 } and Y={b_1,b_2 }. Then ?_g={?,{1},{2},{1,2},{1,3},{2,3},Z} and ?={(?,?),({a_1 },{b_1 } ),({a_2 },{Y} ),(X,Y) }. Clearly ?_g is a g-topology on Z and ? is g-binary topology from X to Y. Define f:Z?X?Y by f(1)=(a_1,b_1 ),f(2)=(a_2,b_2 ),f(3)=(X,b_1). Now f^(-1) (?,?)=?, f^(-1) ({a_1 },{b_1 })={1}, f^(-1) ({a_2 },{b_2 })={2}, f^(-1) ({a_2 },{Y})={2} and f^(-1) (X,Y)=Z. This shows that the inverse image of every ?-binary m-open sets in (X,Y,?) is ?_g-clopen in (Z,?_g ). Hence f is m-totally ?-binary continuous but not strongly ?-binary continuous because f^(-1) ({a_2 },{b_1 })={3} i.e. inverse image of ?-binary set ({a_2 },{b_1 }) is not ?_g-clopen.

Proposition 5.3.4: Every m-totally ?-binary continuous map is totally ?-binary m-continuous map.

Proof: Let (Z,?_g ) be a g-topological space and (X,Y,?) be g-binary topological space and the map f:Z?X?Y be m-totally ?-binary continuous. Suppose (A, B) be any ?-binary open set in (X,Y,?). Since every ?-binary open set in g-binary topological space is ?-binary m-open. Therefore f^(-1) (A,B) is ?_g-clopen and hence ?_g-m-clopen in (Z,?_g ). Thus the inverse image of every ?-binary open set in (X,Y,?) is ?_g-m-clopen in (Z,?_g ). Therefore f is totally ?-binary m-continuous.

Remark 5.3.3: Converse of Proposition 5.3.4 is not true shown in Example 5.3.4.

Example 5.3.4: Let Z={1,2,3}, X={a_1,a_2 } and Y={b_1,b_2 }. Then ?_g={?,{1},{2},{1,2},{1,3},{2,3},Z} and ?={(?,?),({a_1 },{b_1 } ),({a_1 },{b_2 } ),(X,Y) }.. Clearly ?_g is a g-topology on Z and ? is g-binary topology from X to Y. Define f:Z?X?Y by f(1)=(a_1,b_1 ),f(2)=(a_2,b_2 ),f(3)=(a_2,b_2). Now f^(-1) (?,?)=?, f^(-1) ({a_1 },{b_1 })={1}, f^(-1) ({a_1 },{b_2 })={2} and f^(-1) (X,Y)=Z. This shows that the inverse image of every ?-binary open set in (X,Y,?) is ?_g-m-clopen in (Z,?_g ). Hence f is totally ?-binary m-continuous but not m-totally ?-binary continuous because f^(-1) ({a_2 },{b_2 } )={3} i.e. inverse image of ?-binary m-open set ({a_2 },{b_2 } ) is not ?_g-clopen.

Proposition 5.3.5: Every m-totally ?-binary continuous map is ?-binary m-continuous map.

Proof: Let (Z,?_g ) be a g-topological space and (X,Y,?) be g-binary topological space and the map f:Z?X?Y be m-totally ?-binary continuous. Suppose (A, B) be any g-binary open set in (X,Y,?). Therefore f^(-1) (A,B) is ?_g-clopen and hence ?-m-clopen in

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