AN EXTENSION OF THE CONCEPT OF -γ-CONTINUITY FOR MULTIFUNCTIONS

A function f : (X, tau) -> (Y, tau*) between topological spaces is called -gamma-continuous if f(-1)(W) subset of Cl(Int(f(-)(1)(W))) boolean OR Int(Cl(f(-1)(W))) for each open W subset of Y, where Cl (resp. Int ) denotes the closure (resp. interior) operator on X. When we use the other possible operators obtained by multiple composing Cl and Int, then this condition boils down to the definitions of known types of generalized continuity. The case of multifunctions is quite different. The appropriate condition have two forms: F+ (W) subset of Cl(Int(F+(W))) boolean OR Int (Cl(F+(W))) called u.gamma.c. or, F-(W) subset of Cl(Int(F-(W))) boolean OR Int (Cl(F-(W))) called l.gamma.c., where F+(W) = {x is an element of X : F(x) subset of W} and F-(W) = {x is an element of X : F(x) boolean AND w not equal empty set}. So, one can consider the simultaneous use of the two different inverse images namely, F+(W) and F-(W). We will show that in this case the usage of all possible multiple compositions of Cl and Int leads to the new different types of continuity for multifunctions, which together with the previous defined types of continuity forms a collection which is complete in a certain topological sense.