Collectionwise weak continuity duals

A function f : ( X, τ ) → ( Y, σ ) is weakly collectionwise continuous if for some C ⊆ 2 X with τ ⊆ C we have f −1 ( V ) ∈ C for each V ∈ σ . In this case, f is said to be C -continuous. If also τ ⊆ C * ⊆ 2 X , C *-continuity is a dual to C -continuity if C ⋂ C * = τ and then the pair ( C -continuity, C *-continuity) is a decomposition of continuity. In this paper, two natural topological methods are found for construction of a dual to any collectionwise weak continuity. Some known decompositions are improved.