On dual topologies

For two fixed topological spaces Y and Z we consider the set C(Y,Z) of all continuous maps from Y to Z and the set OZ(Y) consisting of all open subsets V of Y such that V=f−1(U), where f∈C(Y,Z) and U is an open subset of Z. We define and study some relations between the topologies on C(Y,Z) and the topologies on OZ(Y) concerning the notions of A-splitting topologies and A-jointly continuous topologies, where A is a fixed family of topological spaces, and a new notion of mutually dual topologies.