Distances from selectors to spaces of Baire one functions

Given a metric space X and a Banach space ( E , ‖ ⋅ ‖ ) we study distances from the set of selectors Sel ( F ) of a set-valued map F : X → P ( E ) to the space B 1 ( X , E ) of Baire one functions from X into E. For this we introduce the d-τ-semioscillation of a set-valued map with values in a topological space ( Y , τ ) also endowed with a metric d. Being more precise we obtain that d ( Sel ( F ) , B 1 ( X , E ) ) ⩽ 2 osc w ∗ ( F ) , where osc w ∗ ( F ) is the ‖ ⋅ ‖ -w-semioscillation of F. In particular, when F takes closed values and osc w ∗ ( F ) = 0 we get that then F has a Baire one selector: we point out that if F is weakly upper semicontinuous then osc w ∗ ( F ) = 0 and therefore our results strengthen a Srivatsa selection theorem when F takes closed set. We also obtain similar results when τ is the topology of convergence on some boundary B or τ is the w ∗ topology of a bidual Banach space.