2.4. ?^*-binary Closed Sets

Definition 2.4.1: Let (X,Y,?) be a g-binary topological space. Let (A,B)? ?(X)??(Y). Then (A, B) is ?^*-binary closed if Cl_? (A,B)?(V,U) whenver (A,B)?(V,U) and (V,U) is ?-binary open in (X,Y,?).

Proposition 2.4.1: ?-binary closed sets in (X,Y,?) are ?^*-binary closed.

Remark 2.4.1: The converse of Proposition 2.4.1 is need not true for example the ?-binary set (A, B) and (C, D) in Example 2.4.1 are ?^*-binary closed but not ?-binary closed.

Example 2.4.1: Let X={1,2,3} and Y={a,b}. Then ?={(?,?),({1,2},{?} ), ({1},{a} ),({1,3},{Y} ),({1,2},{a} ),(X,Y)} is g-binary topology. Clearly (X,Y), ({3},{Y} ),({2,3},{b,c} ),({2},{?} ),({3},{b,c} ) and (?,?)are ?-binary closed sets in (X,Y,?). Suppose (A,B)=({1,2},{Y}) and (A,B)?({X},{Y} ). Also Cl_? (A,B)=({X},{Y} )?({X},{Y} ). Therefore (A,B)=({1,2},{Y}) is ?^*-binary closed but not ?-binary closed.

Proposition 2.4.2: Let (A, B) be a ?^*-binary closed set in (X,Y,?). Suppose (C,D)?(??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B)) where (C,D) is ?-binary closed in (X,Y,?).

Then (C,D)=(?,?).

Proof: Let (C,D)?(??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B)) where (C,D) is ? -binary closed in (X,Y,?). Therefore (C,D)?(?(A,B)^(1^* )?_? ,?(A,B)^(2^* )?_?)). Now (A,B)?(XC ,YD) and since (A, B) is ?^*-binary closed, we have Cl_? (A,B)?(XC ,YD) i.e., (?(A,B)^(1^* )?_? ,?(A,B)^(2^* )?_?))?(XC ,YD) or (C,D)?(X??(A,B)?^(1^* )?_? ,?Y?(A,B)?^(2^* )?_?)). This gives (C,D)?(?(A,B)^(1^* )?_??(X?(A,B)^(1^* )?_? ) ),(?(A,B)^(2^* )?_??(Y?(A,B)^(2^* )?_? ) )=(?,?). This implies (C,D)=(?,?).

Proposition 2.4.3: Let (X,Y,?) be a g-binary topological space. Let (A,B)??(X)??(Y). Suppose (??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B)) contains the only ?-binary closed set(?,?). Then (A, B) is ?^*-binary closed.

Proof: (A,B)?(U,V), where (U,V) is ?-binary open in (X,Y,?). Suppose ?(A,B)^(1^* )?_? is not subset of U and ?(A,B)^(2^* )?_?) is not subset of V. Hence (?(A,B)^(1^* )?_? ,?(A,B)^(2^* )?_?)) is not a subset of (U, V). This implies (?(A,B)^(1^* )?_? ,?(A,B)^(2^* )?_?))?(XU,YV). Therefore (?(A,B)^(1^* )?_??(XU) ),(?(A,B)^(2^* )?_??(YV) ) is a ?-binary closed set and (?(A,B)^(1^* )?_??(XU) )??, (?(A,B)^(2^* )?_??(YV) )=?. Also (?(A,B)^(1^* )?_??(XU) ),(?(A,B)^(2^* )?_??(YV) )?(??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B)). This is contradiction. Hence Cl_? (A,B)?(U,V).

Proposition 2.4.4: Let (X,Y,?) be a g-binary topological space. Then a ?^*-binary closed set (A,B) is ?-binary iff (??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B)) is ?-binary closed.

Proof: Let (A, B) be a ?^*-binary closed set. Assume that (A, B) is a ?-binary closed. We shall show that (??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B)) is ?-binary closed. Since (A, B) is ?-binary, we have Cl_? (A,B)=(A,B). Therefore (??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B))=(A,B). This implies (??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B))=(?,?) which is ?-binary closed. Conversely assume (??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B)) is ?-binary closed. But (A, B) is ?^*-binary closed and (??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B)) is ?-binary closed subset of itself. By Proposition 2.3.3. (??(A,B)?^(1^* )?_?A ,??(A,B)?^(2^* )?_?B))=(?,?). Therefore (??(A,B)?^(1^* )?_? ,??(A,B)?^(2^* )?_?))=(A,B) that is Cl_? (A,B)=(A,B). By Proposition 1.5.3 we have (A, B) is ?-binary closed.

Definition 2.4.2: Let (X,Y,?) be a g-binary topological space. Let (A,B)? ?(X)??(Y). Then (A, B) is ?^*-binary open if (XA,YB) is ?^*-binary closed in (X,Y,?).

Proposition 2.4.5: Every ?-binary open set in a g-binary topological space is ?^*-binary open.

Proof: Let (X,Y,?) be a g-binary topological space. Let (A,B) be ?-binary open in (X,Y,?). Then (XA,YB) is ?-binary closed in (X,Y,?). By proposition 1.4.1 (XA,YB) is ?^*-binary closed in (X,Y,?). Therefore (A, B) is ?^*-binary open in (X,Y,?).

Remark 2.4.2: The converse of Proposition 2.4.5 is need not true which can be shown by example 2.4.2.

Example 2.4.2: Let X={1,2,3} and Y={a,b}. Then ?={(?,?),({?},{b} ), ({1,3},{ Y} ),({X},{a} ),(X,Y)} is g-binary topology. Clearly (?,?),({?},{b} ),({X},{a} ) and (X,Y) are ?-binary closed sets in (X,Y,?). Suppose (A,B)=({?},{Y}) and (XA,YB)=({X},{?} ). Clearly (XA,YB)=({X},{?} )?({X},{a} ) and gbcl(XA,YB)=gbcl({X},{?} )=({X},{a} )?({X},{a} ). Therefore (XA,YB)=({X},{?} ) is ?^*-binary closed, implies (A,B)=({?},{Y}) is ?^*-binary open but not ?-binary open.

2.5. Conclusion

### From the above discussion we conclude

?-binary closed ?(?) ?^*-binary closed

?-binary open ?(?) ?^*-binary open