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On weakly Gibson F σ -measurable mappings

Olena Karlova, Volodymyr Mykhaylyuk (2013)

Colloquium Mathematicae

A function f: X → Y between topological spaces is said to be a weakly Gibson function if f ( Ū ) f ( U ) ¯ 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 F σ -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 F σ -measurable mapping f: ℝⁿ → Y, where Y is a T₁-space, has a connected graph.

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