Extension of Functions with Small Oscillation

Abstract

A classical theorem of Kuratowski says that every Baire one function on a G_\delta subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index \beta(f). We prove a refinement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that \beta_{Y}(f)<\omega^{\alpha}, \alpha < \omega_1, then f has an extension F onto X so that \beta_X(F)is not more than \omega^{\alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that \beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.