AN EXTENSION OF THE CONCEPT OF γ-CONTINUITY FOR MULTIFUNCTIONS

Abstract

A function $f:(X,\tau )\to (Y,{\tau }^{\ast })$ 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) = $\{x\in X:F(x)\subset W\}$ and F${}^{-}$(W) = $\{x\in X:F(x)\cap W\ne \mathrm{\varnothing }\}$.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 completein a certain topological sense.