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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

This paper example is written by Benjamin, a student from St. Ambrose University with a major in Management. All the content of this paper consists of his personal thoughts on Chapter5Generalized Binary mContinuous and bContinuous and his way of presenting arguments and should be used only as a possible source of ideas and arguments.

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