Admissible topologies on C(Y,Z) and OZ(Y)

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Let Y and Z be two given topological spaces, O(Y) (respectively, O(Z)) the set of all open subsets of Y (respectively, Z), and C(Y,Z) the set of all continuous maps from Y to Z. We study Scott type topologies on O(Y) and we construct admissible topologies on C(Y,Z) and OZ(Y)={f⁻¹(U)∈ O(Y): f∈ C(Y,Z) andU∈ O(Z)}, introducing new problems in the field.