On weakly Gibson -measurable mappings
A function f: X → Y between topological spaces is said to be a weakly Gibson function if for any open connected set U ⊆ X. We prove that if X is a locally connected hereditarily Baire space and Y is a T₁-space then an -measurable mapping f: X → Y is weakly Gibson if and only if for any connected set C ⊆ X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson -measurable mapping f: ℝⁿ → Y, where Y is a T₁-space, has a connected graph.