32 Totally generalized binary continuous Maps TGB CMDefinition 321 Essay

3.2. Totally generalized binary continuous Maps (TG_B CM)

Definition 3.2.1: Let (Z,T_G ) be G_T S and (X,Y,?) be G_B TS. Then the mapping F:Z?X?Y is called totally generalized binary continuous map (TG_B CM) if F^(-1) (A,B) is T_G-clopen in (Z,T_G ) for every G_B OS (A, B) in (X,Y,?).

Example 3.2.1: Let Z={1,2,3}, X={a_1,a_2 } and Y={b_1,b_2 }. Then T_G={?,{1},{1,2},{2,3},Z}and?={(?,?),({a_1 },{b_1 } ),({a_1 },{Y} ),({a_2 },{Y} ),(X,Y) }.

Clearly T_G is G_T on Z and ? is G_B T from X to Y. Define F:Z?X?Y by F(1)=(a_1,b_1 ) and F(2)=(a_2,b_2 )=F(3). The T_G-clopen sets in (Z,?_g ) are ?,{1},{2,3},Z. Now F^(-1) (?,?)=?, F^(-1) ({a_1 },{b_1 })={1}, F^(-1) ({a_1 },{Y})={1}, F^(-1) ({a_2 },{Y})={2,3} and F^(-1) (X,Y)=Z. This shows that the inverse image of every G_B OS in (X,Y,?) is T_G-clopen in (Z,T_G ).

Hence F is TG_B CM.

Definition 3.2.2: Let (Z,T_G ) be G_T S and (X,Y,?) be G_B TS. Then the mapping F:Z?X?Y is called totally generalized binary semi-continuous map (TG_B SCM) if F^(-1) (A,B) is T_G-semi-clopen in (Z,T_G ) for every G_B OS (A, B) in (X,Y,?).

Example 3.2.2: In Example 3.2.1, ?,{1},{2,3},Z are T_G-semi-clopen sets in (Z,T_G ) and the inverse image of every G_B OS in (X,Y,C) is T_G-semi-clopen in (Z,T_G ). Hence F is TG_B SCM.

Proposition 3.2.1: TG_B CM ?? TG_B SCM

Proof: Let (A,B) be G_B OS in (X,Y,?). Since F is TG_B CM, we have F^(-1) (A,B) is T_G-clopen in (Z,T_G ). We know that every T_G-clopen set in G_T S is T_G-semi-clopen. Hence F^(-1) (A,B) is T_G-semi-open in (Z,T_G ). Thus F is TG_B SCM

Remark 3.2.1: The converse of Proposition 3.2.1 is illustrated in Example 3.2.3.

Example 3.2.3: Let Z={1,2,3,4}, X={a_1,? a?_2 } and Y={b_1,b_2 }. Then T_G={?,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3},{1,3,4},{1,2,4},Z} and ?={(?,?), ({a_1 },{b_1 } ),({a_1 },{Y} ),({a_2 },{Y} ),(X,Y)}. Clearly T_G is G_T on Z and ? is G_B T from X to Y. Define F:Z?X?Y by F(1)=(a_1,b_1 ) and F(2)=F(3)=F(4)=(a_2,b_2 ). The T_G-semi-clopen sets in (Z,T_G ) are ?,{1},{2},{3},{1,2,4}, {1,3,4},{2,3,4},Z. Now F^(-1) (?,?)=?, F^(-1) ({a_1 },{b_1 })={1},F^(-1) ({a_1 },{Y})={1}, F^(-1) ({a_2 },{Y})={2,3,4} and F^(-1) (X,Y)=Z. This shows that the inverse image of every G_B OS in (X,Y,?) is T_G-semi-clopen in (Z,T_G ). Hence F is TG_B SCM but not TG_B CM because {2,3,4} is T_G-semi-clopen but not T_G-clopen in (Z,T_G ).