Distances to spaces of Baire one functions

Given a metric space X and a Banach space (E, ||·||) we use an index of σ-fragmentability for maps to estimate the distance of f to the space B ₁(X, E) of Baire one functions from X into (E, ||·||). When X is Polish we use our estimations for these distances to give a quantitative version of the well known Rosenthal’s result stating that in the pointwise relatively countably compact sets are pointwise relatively compact. We also obtain a quantitative version of a Srivatsa’s result that states that whenever X is metric any weakly continuous function belongs to B ₁(X, E): our result here says that for an arbitrary we havewhere osc stands for the supremum of the oscillations of at all points . As a consequence of the above we prove that for functions in two variables , X complete metric and K compact, there exists a G _δ-dense set such that the oscillation of f at each is bounded by the oscillations of the partial functions f _x and f ^k . A representative result in this direction, that we prove using games, is the following: if X is a σ–β-unfavorable space and K is a compact space, then there exists a dense G _δ-subset D of X such that, for each ,When the right hand side of the above inequality is zero we are dealing with separately continuous functions and we obtain as a particular case some well-known results obtained by the third named author in the mid 1970s. C. Angosto, B. Cascales and I. Namioka are supported by the Spanish grants MTM2005-08379 (MEC & FEDER) and 00690/PI/04 (Fund. Séneca). C. Angosto is also supported by the FPU grant AP2003-4443 (MEC & FEDER).