An extension of the concept of $\boldsymbol\gamma$-continuity for multifunctions

M. Przemski

A function $f:(X,\tau) \rightarrow (Y,\tau^{\ast})$ between topological spaces is called $\gamma$-continuous if $f^{-1}(W) \subset Cl(Int(f^{-1}(W))) \cup Int(Cl(f^{-1}(W)))$ for each open $W \subset Y$, 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: $F^{+}(W) \subset Cl(Int(F^{+}(W))) \cup Int(Cl(F^{+}(W)))$ called $u.\gamma .c.$ or, $F^{-}(W) \subset Cl(Int(F^{-}(W))) \cup Int(Cl(F^{-}(W)))$ called $l.\gamma .c.$, where F$^{+}$(W) = $\left\{x \in X: F(x) \subset W \right\}$ and F$^{-}$(W) = $\left\{x \in X: F(x) \cap W \neq\emptyset\right\}$. 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.